purewhite42/putnambench_solving · Datasets at Hugging Face

informal_problem stringlengths 47 738	informal_answer stringlengths 1 204	informal_solution stringlengths 16 303	header stringlengths 8 63 ⌀	intros listlengths 0 2	formal_answer stringlengths 20 24.1k	formal_answer_type stringlengths 1 88	outros listlengths 1 13	metainfo dict
Find every real-valued function $f$ whose domain is an interval $I$ (finite or infinite) having 0 as a left-hand endpoint, such that for every positive member $x$ of $I$ the average of $f$ over the closed interval $[0, x]$ is equal to the geometric mean of the numbers $f(0)$ and $f(x)$.	the set of functions $f(x) = \frac{a}{(1 - c x)^2}$ where $a \geq 0$	Show that \[ f(x) = \frac{a}{(1 - cx)^2} \begin{cases} \text{for } 0 \le x < \frac{1}{c}, & \text{if } c > 0\\ \text{for } 0 \le x < \infty, & \text{if } c \le 0, \end{cases} \] where $a > 0$.	open MeasureTheory Set	[]	@Eq (Set (Real → Real)) answer (@setOf (Real → Real) fun (f : Real → Real) => @Exists Real fun (a : Real) => @Exists Real fun (c : Real) => And (@GE.ge Real Real.instLE a (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero))) (@Eq (Real → Real) f fun (x : Real) => @HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) a (@HPow.hPow Real Nat Real (@instHPow Real Nat (@Monoid.toNatPow Real Real.instMonoid)) (@HSub.hSub Real Real Real (@instHSub Real Real.instSub) (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)) (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) c x)) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))))	Set (ℝ → ℝ)	[ { "t": "Set ℝ → (ℝ → ℝ) → Prop", "v": null, "name": "P" }, { "t": "∀ s f, P s f ↔ 0 ≤ f ∧ ∀ x ∈ s, ⨍ t in Ico 0 x, f t = √(f 0 * f x)", "v": null, "name": "P_def" }, { "t": "answer = {f : ℝ → ℝ \| P (Ioi 0) f ∨ (∃ e > 0, P (Ioo 0 e) f)}", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1962_a2", "tags": [ "analysis" ] }
Evaluate in closed form \[ \sum_{k=1}^n {n \choose k} k^2. \]	$n(n+1)2^{n-2}$	Show that the expression equals $n(n+1)2^{n-2}$.	null	[]	@Eq (Nat → Nat) answer fun (n_1 : Nat) => @HMul.hMul Nat Nat Nat (@instHMul Nat instMulNat) (@HMul.hMul Nat Nat Nat (@instHMul Nat instMulNat) n_1 (@HAdd.hAdd Nat Nat Nat (@instHAdd Nat instAddNat) n_1 (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))) (@HPow.hPow Nat Nat Nat (@instHPow Nat Nat (@Monoid.toNatPow Nat Nat.instMonoid)) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) (@HSub.hSub Nat Nat Nat (@instHSub Nat instSubNat) n_1 (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))	ℕ → ℕ	[ { "t": "ℕ", "v": null, "name": "n" }, { "t": "n ≥ 2", "v": null, "name": "hn" }, { "t": "answer n = ∑ k in Finset.Icc 1 n, Nat.choose n k * k^2", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1962_a5", "tags": [ "algebra", "combinatorics" ] }
Find an integral formula (i.e., a function $z$ such that $y(x) = \int_{1}^{x} z(t) dt$) for the solution of the differential equation $$\delta (\delta - 1) (\delta - 2) \cdots (\delta - n + 1) y = f(x)$$ with the initial conditions $y(1) = y'(1) = \cdots = y^{(n-1)}(1) = 0$, where $n \in \mathbb{N}$, $f$ is continuous for all $x \ge 1$, and $\delta$ denotes $x\frac{d}{dx}$.	$(x - t)^{n-1} \cdot f(t) / ((n-1)! \cdot t^n)$	Show that the solution is $$y(x) = \int_{1}^{x} \frac{(x - t)^{n - 1} f(t)}{(n - 1)!t^n} dt$$.	open Nat Set Topology Filter	[]	@Eq ((Real → Real) → Nat → Real → Real → Real) answer fun (f_1 : Real → Real) (n_1 : Nat) (x t : Real) => @HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) (@HPow.hPow Real Nat Real (@instHPow Real Nat (@Monoid.toNatPow Real Real.instMonoid)) (@HSub.hSub Real Real Real (@instHSub Real Real.instSub) x t) (@HSub.hSub Nat Nat Nat (@instHSub Nat instSubNat) n_1 (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))) (f_1 t)) (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) (@Nat.cast Real Real.instNatCast (Nat.factorial (@HSub.hSub Nat Nat Nat (@instHSub Nat instSubNat) n_1 (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))) (@HPow.hPow Real Nat Real (@instHPow Real Nat (@Monoid.toNatPow Real Real.instMonoid)) t n_1))	(ℝ → ℝ) → ℕ → ℝ → ℝ → ℝ	[ { "t": "ℕ → (ℝ → ℝ) → (ℝ → ℝ)", "v": null, "name": "P" }, { "t": "P 0 = id ∧ ∀ i y, P (i + 1) y = P i (fun x ↦ x * deriv y x - i * y x)", "v": null, "name": "hP" }, { "t": "ℕ", "v": null, "name": "n" }, { "t": "0 < n", "v": null, "name": "hn" }, { "t": "ℝ → ℝ", "v": null, "name": "f" }, { "t": "ℝ → ℝ", "v": null, "name": "y" }, { "t": "ContinuousOn f (Ici 1)", "v": null, "name": "hf" }, { "t": "ContDiffOn ℝ n y (Ici 1)", "v": null, "name": "hy" }, { "t": "∀ x ≥ 1, y x = ∫ t in (1 : ℝ)..x, answer f n x t", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1963_a3", "tags": [ "analysis" ] }
For what integer $a$ does $x^2-x+a$ divide $x^{13}+x+90$?	2	Show that $a=2$.	open Topology Filter Polynomial	[]	@Eq Int answer (@OfNat.ofNat Int (nat_lit 2) (@instOfNat (nat_lit 2)))	ℤ	[ { "t": "ℤ", "v": null, "name": "a" }, { "t": "Polynomial.X^2 - Polynomial.X + (Polynomial.C a) ∣ (Polynomial.X ^ 13 + Polynomial.X + (Polynomial.C 90))", "v": null, "name": "h_div" }, { "t": "answer = a", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1963_b1", "tags": [ "algebra" ] }
Let $S$ be the set of all numbers of the form $2^m3^n$, where $m$ and $n$ are integers, and let $P$ be the set of all positive real numbers. Is $S$ dense in $P$?	True	Show that $S$ is dense in $P$.	open Topology Filter Polynomial	[]	@Eq Prop answer True	Prop	[ { "t": "Set ℝ", "v": null, "name": "S" }, { "t": "S = {2 ^ m * 3 ^ n \| (m : ℤ) (n : ℤ)}", "v": null, "name": "hS" }, { "t": "answer = (closure S ⊇ Set.Ioi (0 : ℝ))", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1963_b2", "tags": [ "analysis" ] }
Find every twice-differentiable real-valued function $f$ with domain the set of all real numbers and satisfying the functional equation $(f(x))^2-(f(y))^2=f(x+y)f(x-y)$ for all real numbers $x$ and $y$.	the set of functions of the form $A \sinh(k u)$, $A u$, or $A \sin(k u)$	Show that the solution is the sets of functions $f(u)=A\sinh ku$, $f(u)=Au$, and $f(u)=A\sin ku$ with $A,k \in \mathbb{R}$.	open Topology Filter Polynomial	[]	@Eq (Set (Real → Real)) answer (@Union.union (Set (Real → Real)) (@Set.instUnion (Real → Real)) (@Union.union (Set (Real → Real)) (@Set.instUnion (Real → Real)) (@setOf (Real → Real) fun (x : Real → Real) => @Exists Real fun (A : Real) => @Exists Real fun (k : Real) => @Eq (Real → Real) (fun (u : Real) => @HMul.hMul Real Real Real (@instHMul Real Real.instMul) A (Real.sinh (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) k u))) x) (@setOf (Real → Real) fun (x : Real → Real) => @Exists Real fun (A : Real) => @Eq (Real → Real) (fun (u : Real) => @HMul.hMul Real Real Real (@instHMul Real Real.instMul) A u) x)) (@setOf (Real → Real) fun (x : Real → Real) => @Exists Real fun (A : Real) => @Exists Real fun (k : Real) => @Eq (Real → Real) (fun (u : Real) => @HMul.hMul Real Real Real (@instHMul Real Real.instMul) A (Real.sin (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) k u))) x))	Set (ℝ → ℝ)	[ { "t": "ℝ → ℝ", "v": null, "name": "f" }, { "t": "f ∈ answer", "v": null, "name": "h_answer" }, { "t": "ContDiff ℝ 1 f", "v": null, "name": "h_cont_diff" }, { "t": "Differentiable ℝ (deriv f)", "v": null, "name": "h_diff" }, { "t": "∀ x y : ℝ, (f x) ^ 2 - (f y) ^ 2 = f (x + y) * f (x - y)", "v": null, "name": "h_func_eq" } ]	{ "problem_name": "putnam_1963_b3", "tags": [ "analysis" ] }
Let $\alpha$ be a real number. Find all continuous real-valued functions $f : [0, 1] \to (0, \infty)$ such that \begin{align} \int_0^1 f(x) dx &= 1, \\ \int_0^1 x f(x) dx &= \alpha, \\ \int_0^1 x^2 f(x) dx &= \alpha^2. \\ \end{align}	the empty set	Prove that there are no such functions.	open Set	[]	@Eq (Real → Set (Real → Real)) answer fun (x : Real) => @EmptyCollection.emptyCollection (Set (Real → Real)) (@Set.instEmptyCollection (Real → Real))	ℝ → Set (ℝ → ℝ)	[ { "t": "ℝ", "v": null, "name": "α" }, { "t": "answer α = {f : ℝ → ℝ \| (∀ x ∈ Icc 0 1, f x > 0) ∧ ContinuousOn f (Icc 0 1) ∧ ∫ x in (0)..1, f x = 1 ∧ ∫ x in (0)..1, x * f x = α ∧ ∫ x in (0)..1, x^2 * f x = α^2}", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1964_a2", "tags": [ "analysis", "algebra" ] }
Let $\triangle ABC$ satisfy $\angle CAB < \angle BCA < \frac{\pi}{2} < \angle ABC$. If the bisector of the external angle at $A$ meets line $BC$ at $P$, the bisector of the external angle at $B$ meets line $CA$ at $Q$, and $AP = BQ = AB$, find $\angle CAB$.	π / 15	Show that the solution is $\angle CAB = \frac{\pi}{15}$.	open EuclideanGeometry Real	[]	@Eq Real answer (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) Real.pi (@OfNat.ofNat Real (nat_lit 15) (@instOfNatAtLeastTwo Real (nat_lit 15) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 13) (instOfNatNat (nat_lit 13)))))))	ℝ	[ { "t": "EuclideanSpace ℝ (Fin 2)", "v": null, "name": "A" }, { "t": "EuclideanSpace ℝ (Fin 2)", "v": null, "name": "B" }, { "t": "EuclideanSpace ℝ (Fin 2)", "v": null, "name": "C" }, { "t": "EuclideanSpace ℝ (Fin 2)", "v": null, "name": "X" }, { "t": "EuclideanSpace ℝ (Fin 2)", "v": null, "name": "Y" }, { "t": "¬Collinear ℝ {A, B, C}", "v": null, "name": "hABC" }, { "t": "∠ C A B < ∠ B C A ∧ ∠ B C A < π/2 ∧ π/2 < ∠ A B C", "v": null, "name": "hangles" }, { "t": "Collinear ℝ {X, B, C} ∧ ∠ X A B = (π - ∠ C A B)/2 ∧ dist A X = dist A B", "v": null, "name": "hX" }, { "t": "Collinear ℝ {Y, C, A} ∧ ∠ Y B C = (π - ∠ A B C)/2 ∧ dist B Y = dist A B", "v": null, "name": "hY" }, { "t": "∠ C A B = answer", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1965_a1", "tags": [ "geometry" ] }
How many orderings of the integers from $1$ to $n$ satisfy the condition that, for every integer $i$ except the first, there exists some earlier integer in the ordering which differs from $i$ by $1$?	$2^{n-1}$	There are $2^{n-1}$ such orderings.	open EuclideanGeometry Topology Filter Complex	[]	@Eq (Nat → Nat) answer fun (n_1 : Nat) => @HPow.hPow Nat Nat Nat (@instHPow Nat Nat (@Monoid.toNatPow Nat Nat.instMonoid)) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) (@HSub.hSub Nat Nat Nat (@instHSub Nat instSubNat) n_1 (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))	ℕ → ℕ	[ { "t": "ℕ", "v": null, "name": "n" }, { "t": "n > 0", "v": null, "name": "npos" }, { "t": "{p ∈ permsOfFinset (Finset.Icc 1 n) \| ∀ m ∈ Finset.Icc 2 n, ∃ k ∈ Finset.Ico 1 m, p m = p k + 1 ∨ p m = p k - 1}.card = answer n", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1965_a5", "tags": [ "combinatorics" ] }
Find $$\lim_{n \to \infty} \int_{0}^{1} \int_{0}^{1} \cdots \int_{0}^{1} \cos^2\left(\frac{\pi}{2n}(x_1 + x_2 + \cdots + x_n)\right) dx_1 dx_2 \cdots dx_n.$$	1 / 2	Show that the limit is $\frac{1}{2}$.	open EuclideanGeometry Topology Filter Complex	[]	@Eq Real answer (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)) (@OfNat.ofNat Real (nat_lit 2) (@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0)))))))	ℝ	[ { "t": "Tendsto (fun n : ℕ ↦ ∫ x in {x : Fin (n+1) → ℝ \| ∀ k : Fin (n+1), x k ∈ Set.Icc 0 1}, (Real.cos (Real.pi/(2 * (n+1)) * ∑ k : Fin (n+1), x k))^2) atTop (𝓝 answer)", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1965_b1", "tags": [ "analysis" ] }
Consider polynomial forms $ax^2-bx+c$ with integer coefficients which have two distinct zeros in the open interval $0<x<1$. Exhibit with a proof the least positive integer value of $a$ for which such a polynomial exists.	5	Show that the minimum possible value for $a$ is $5$.	open Polynomial	[]	@Eq Nat answer (@OfNat.ofNat Nat (nat_lit 5) (instOfNatNat (nat_lit 5)))	ℕ	[ { "t": "Set ℤ", "v": null, "name": "S" }, { "t": "S = {a \| ∃ P : Polynomial ℤ, P.degree = 2 ∧ (∃ z1 z2 : Set.Ioo (0 : ℝ) 1, z1 ≠ z2 ∧ aeval (z1 : ℝ) P = 0 ∧ aeval (z2 : ℝ) P = 0) ∧P.coeff 2 = a ∧ a > 0}", "v": null, "name": "hS" }, { "t": "IsLeast S answer", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1967_a3", "tags": [ "algebra" ] }
Given real numbers $\{a_i\}$ and $\{b_i\}$, ($i=1,2,3,4$), such that $a_1b_2-a_2b_1 \neq 0$. Consider the set of all solutions $(x_1,x_2,x_3,x_4)$ of the simultaneous equations $a_1x_1+a_2x_2+a_3x_3+a_4x_4=0$ and $b_1x_1+b_2x_2+b_3x_3+b_4x_4=0$, for which no $x_i$ ($i=1,2,3,4$) is zero. Each such solution generates a $4$-tuple of plus and minus signs $(\text{signum }x_1,\text{signum }x_2,\text{signum }x_3,\text{signum }x_4)$. Determine, with a proof, the maximum number of distinct $4$-tuples possible.	8	Show that the maximum number of distinct $4$-tuples is eight.	open Nat Topology Filter	[]	@Eq Nat answer (@OfNat.ofNat Nat (nat_lit 8) (instOfNatNat (nat_lit 8)))	ℕ	[ { "t": "Fin 4 → ℝ", "v": null, "name": "a" }, { "t": "Fin 4 → ℝ", "v": null, "name": "b" }, { "t": "a 0 * b 1 - a 1 * b 0 ≠ 0", "v": null, "name": "abneq0" }, { "t": "ℕ", "v": null, "name": "numtuples" }, { "t": "numtuples = {s : Fin 4 → ℝ \| ∃ x : Fin 4 → ℝ, (∀ i : Fin 4, x i ≠ 0) ∧ (∑ i : Fin 4, a i * x i) = 0 ∧ (∑ i : Fin 4, b i * x i) = 0 ∧ (∀ i : Fin 4, s i = Real.sign (x i))}.encard", "v": null, "name": "hnumtuples" }, { "t": "numtuples = answer", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1967_a6", "tags": [ "algebra", "geometry" ] }
Let $V$ be the set of all quadratic polynomials with real coefficients such that $\|P(x)\| \le 1$ for all $x \in [0, 1]$. Find the supremum of $\|P'(0)\|$ across all $P \in V$.	8	The supremum is $8$.	open Finset Polynomial	[]	@Eq Real answer (@OfNat.ofNat Real (nat_lit 8) (@instOfNatAtLeastTwo Real (nat_lit 8) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))))))	ℝ	[ { "t": "Set ℝ[X]", "v": null, "name": "V" }, { "t": "V = {P : ℝ[X] \| P.degree = 2 ∧ ∀ x ∈ Set.Icc 0 1, \|P.eval x\| ≤ 1}", "v": null, "name": "V_def" }, { "t": "sSup {\|(derivative P).eval 0\| \| P ∈ V} = answer", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1968_a5", "tags": [ "algebra" ] }
Find all polynomials of the form $\sum_{0}^{n} a_{i} x^{n-i}$ with $n \ge 1$ and $a_i = \pm 1$ for all $0 \le i \le n$ whose roots are all real.	{X - 1, -(X - 1), X + 1, -(X + 1), X^2 + X - 1, -(X^2 + X - 1), X^2 - X - 1, -(X^2 - X - 1), X^3 + X^2 - X - 1, -(X^3 + X^2 - X - 1), X^3 - X^2 - X + 1, -(X^3 - X^2 - X + 1)}	The set of such polynomials is $$\{\pm (x - 1), \pm (x + 1), \pm (x^2 + x - 1), \pm (x^2 - x - 1), \pm (x^3 + x^2 - x - 1), \pm (x^3 - x^2 - x + 1)\}.$$	open Finset Polynomial	[]	@Eq (Set (@Polynomial Complex Complex.instSemiring)) answer (@Insert.insert (@Polynomial Complex Complex.instSemiring) (Set (@Polynomial Complex Complex.instSemiring)) (@Set.instInsert (@Polynomial Complex Complex.instSemiring)) (@HSub.hSub (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@instHSub (@Polynomial Complex Complex.instSemiring) (@Polynomial.sub Complex Complex.instRing)) (@Polynomial.X Complex Complex.instSemiring) (@OfNat.ofNat (@Polynomial Complex Complex.instSemiring) (nat_lit 1) (@One.toOfNat1 (@Polynomial Complex Complex.instSemiring) (@Polynomial.one Complex Complex.instSemiring)))) (@Insert.insert (@Polynomial Complex Complex.instSemiring) (Set (@Polynomial Complex Complex.instSemiring)) (@Set.instInsert (@Polynomial Complex Complex.instSemiring)) (@Neg.neg (@Polynomial Complex Complex.instSemiring) (@Polynomial.neg' Complex Complex.instRing) (@HSub.hSub (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@instHSub (@Polynomial Complex Complex.instSemiring) (@Polynomial.sub Complex Complex.instRing)) (@Polynomial.X Complex Complex.instSemiring) (@OfNat.ofNat (@Polynomial Complex Complex.instSemiring) (nat_lit 1) (@One.toOfNat1 (@Polynomial Complex Complex.instSemiring) (@Polynomial.one Complex Complex.instSemiring))))) (@Insert.insert (@Polynomial Complex Complex.instSemiring) (Set (@Polynomial Complex Complex.instSemiring)) (@Set.instInsert (@Polynomial Complex Complex.instSemiring)) (@HAdd.hAdd (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@instHAdd (@Polynomial Complex Complex.instSemiring) (@Polynomial.add' Complex Complex.instSemiring)) (@Polynomial.X Complex Complex.instSemiring) (@OfNat.ofNat (@Polynomial Complex Complex.instSemiring) (nat_lit 1) (@One.toOfNat1 (@Polynomial Complex Complex.instSemiring) (@Polynomial.one Complex Complex.instSemiring)))) (@Insert.insert (@Polynomial Complex Complex.instSemiring) (Set (@Polynomial Complex Complex.instSemiring)) (@Set.instInsert (@Polynomial Complex Complex.instSemiring)) (@Neg.neg (@Polynomial Complex Complex.instSemiring) (@Polynomial.neg' Complex Complex.instRing) (@HAdd.hAdd (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@instHAdd (@Polynomial Complex Complex.instSemiring) (@Polynomial.add' Complex Complex.instSemiring)) (@Polynomial.X Complex Complex.instSemiring) (@OfNat.ofNat (@Polynomial Complex Complex.instSemiring) (nat_lit 1) (@One.toOfNat1 (@Polynomial Complex Complex.instSemiring) (@Polynomial.one Complex Complex.instSemiring))))) (@Insert.insert (@Polynomial Complex Complex.instSemiring) (Set (@Polynomial Complex Complex.instSemiring)) (@Set.instInsert (@Polynomial Complex Complex.instSemiring)) (@HSub.hSub (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@instHSub (@Polynomial Complex Complex.instSemiring) (@Polynomial.sub Complex Complex.instRing)) (@HAdd.hAdd (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@instHAdd (@Polynomial Complex Complex.instSemiring) (@Polynomial.add' Complex Complex.instSemiring)) (@HPow.hPow (@Polynomial Complex Complex.instSemiring) Nat (@Polynomial Complex Complex.instSemiring) (@instHPow (@Polynomial Complex Complex.instSemiring) Nat (@Monoid.toNatPow (@Polynomial Complex Complex.instSemiring) (@MonoidWithZero.toMonoid (@Polynomial Complex Complex.instSemiring) (@Semiring.toMonoidWithZero (@Polynomial Complex Complex.instSemiring) (@Polynomial.semiring Complex Complex.instSemiring))))) (@Polynomial.X Complex Complex.instSemiring) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) (@Polynomial.X Complex Complex.instSemiring)) (@OfNat.ofNat (@Polynomial Complex Complex.instSemiring) (nat_lit 1) (@One.toOfNat1 (@Polynomial Complex Complex.instSemiring) (@Polynomial.one Complex Complex.instSemiring)))) (@Insert.insert (@Polynomial Complex Complex.instSemiring) (Set (@Polynomial Complex Complex.instSemiring)) (@Set.instInsert (@Polynomial Complex Complex.instSemiring)) (@Neg.neg (@Polynomial Complex Complex.instSemiring) (@Polynomial.neg' Complex Complex.instRing) (@HSub.hSub (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@instHSub (@Polynomial Complex Complex.instSemiring) (@Polynomial.sub Complex Complex.instRing)) (@HAdd.hAdd (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@instHAdd (@Polynomial Complex Complex.instSemiring) (@Polynomial.add' Complex Complex.instSemiring)) (@HPow.hPow (@Polynomial Complex Complex.instSemiring) Nat (@Polynomial Complex Complex.instSemiring) (@instHPow (@Polynomial Complex Complex.instSemiring) Nat (@Monoid.toNatPow (@Polynomial Complex Complex.instSemiring) (@MonoidWithZero.toMonoid (@Polynomial Complex Complex.instSemiring) (@Semiring.toMonoidWithZero (@Polynomial Complex Complex.instSemiring) (@Polynomial.semiring Complex Complex.instSemiring))))) (@Polynomial.X Complex Complex.instSemiring) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) (@Polynomial.X Complex Complex.instSemiring)) (@OfNat.ofNat (@Polynomial Complex Complex.instSemiring) (nat_lit 1) (@One.toOfNat1 (@Polynomial Complex Complex.instSemiring) (@Polynomial.one Complex Complex.instSemiring))))) (@Insert.insert (@Polynomial Complex Complex.instSemiring) (Set (@Polynomial Complex Complex.instSemiring)) (@Set.instInsert (@Polynomial Complex Complex.instSemiring)) (@HSub.hSub (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@instHSub (@Polynomial Complex Complex.instSemiring) (@Polynomial.sub Complex Complex.instRing)) (@HSub.hSub (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@instHSub (@Polynomial Complex Complex.instSemiring) (@Polynomial.sub Complex Complex.instRing)) (@HPow.hPow (@Polynomial Complex Complex.instSemiring) Nat (@Polynomial Complex Complex.instSemiring) (@instHPow (@Polynomial Complex Complex.instSemiring) Nat (@Monoid.toNatPow (@Polynomial Complex Complex.instSemiring) (@MonoidWithZero.toMonoid (@Polynomial Complex Complex.instSemiring) (@Semiring.toMonoidWithZero (@Polynomial Complex Complex.instSemiring) (@Polynomial.semiring Complex Complex.instSemiring))))) (@Polynomial.X Complex Complex.instSemiring) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) (@Polynomial.X Complex Complex.instSemiring)) (@OfNat.ofNat (@Polynomial Complex Complex.instSemiring) (nat_lit 1) (@One.toOfNat1 (@Polynomial Complex Complex.instSemiring) (@Polynomial.one Complex Complex.instSemiring)))) (@Insert.insert (@Polynomial Complex Complex.instSemiring) (Set (@Polynomial Complex Complex.instSemiring)) (@Set.instInsert (@Polynomial Complex Complex.instSemiring)) (@Neg.neg (@Polynomial Complex Complex.instSemiring) (@Polynomial.neg' Complex Complex.instRing) (@HSub.hSub (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@instHSub (@Polynomial Complex Complex.instSemiring) (@Polynomial.sub Complex Complex.instRing)) (@HSub.hSub (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@instHSub (@Polynomial Complex Complex.instSemiring) (@Polynomial.sub Complex Complex.instRing)) (@HPow.hPow (@Polynomial Complex Complex.instSemiring) Nat (@Polynomial Complex Complex.instSemiring) (@instHPow (@Polynomial Complex Complex.instSemiring) Nat (@Monoid.toNatPow (@Polynomial Complex Complex.instSemiring) (@MonoidWithZero.toMonoid (@Polynomial Complex Complex.instSemiring) (@Semiring.toMonoidWithZero (@Polynomial Complex Complex.instSemiring) (@Polynomial.semiring Complex Complex.instSemiring))))) (@Polynomial.X Complex Complex.instSemiring) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) (@Polynomial.X Complex Complex.instSemiring)) (@OfNat.ofNat (@Polynomial Complex Complex.instSemiring) (nat_lit 1) (@One.toOfNat1 (@Polynomial Complex Complex.instSemiring) (@Polynomial.one Complex Complex.instSemiring))))) (@Insert.insert (@Polynomial Complex Complex.instSemiring) (Set (@Polynomial Complex Complex.instSemiring)) (@Set.instInsert (@Polynomial Complex Complex.instSemiring)) (@HSub.hSub (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@instHSub (@Polynomial Complex Complex.instSemiring) (@Polynomial.sub Complex Complex.instRing)) (@HSub.hSub (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@instHSub (@Polynomial Complex Complex.instSemiring) (@Polynomial.sub Complex Complex.instRing)) (@HAdd.hAdd (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@instHAdd (@Polynomial Complex Complex.instSemiring) (@Polynomial.add' Complex Complex.instSemiring)) (@HPow.hPow (@Polynomial Complex Complex.instSemiring) Nat (@Polynomial Complex Complex.instSemiring) (@instHPow (@Polynomial Complex Complex.instSemiring) Nat (@Monoid.toNatPow (@Polynomial Complex Complex.instSemiring) (@MonoidWithZero.toMonoid (@Polynomial Complex Complex.instSemiring) (@Semiring.toMonoidWithZero (@Polynomial Complex Complex.instSemiring) (@Polynomial.semiring Complex Complex.instSemiring))))) (@Polynomial.X Complex Complex.instSemiring) (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) (@HPow.hPow (@Polynomial Complex Complex.instSemiring) Nat (@Polynomial Complex Complex.instSemiring) (@instHPow (@Polynomial Complex Complex.instSemiring) Nat (@Monoid.toNatPow (@Polynomial Complex Complex.instSemiring) (@MonoidWithZero.toMonoid (@Polynomial Complex Complex.instSemiring) (@Semiring.toMonoidWithZero (@Polynomial Complex Complex.instSemiring) (@Polynomial.semiring Complex Complex.instSemiring))))) (@Polynomial.X Complex Complex.instSemiring) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))) (@Polynomial.X Complex Complex.instSemiring)) (@OfNat.ofNat (@Polynomial Complex Complex.instSemiring) (nat_lit 1) (@One.toOfNat1 (@Polynomial Complex Complex.instSemiring) (@Polynomial.one Complex Complex.instSemiring)))) (@Insert.insert (@Polynomial Complex Complex.instSemiring) (Set (@Polynomial Complex Complex.instSemiring)) (@Set.instInsert (@Polynomial Complex Complex.instSemiring)) (@Neg.neg (@Polynomial Complex Complex.instSemiring) (@Polynomial.neg' Complex Complex.instRing) (@HSub.hSub (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@instHSub (@Polynomial Complex Complex.instSemiring) (@Polynomial.sub Complex Complex.instRing)) (@HSub.hSub (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@instHSub (@Polynomial Complex Complex.instSemiring) (@Polynomial.sub Complex Complex.instRing)) (@HAdd.hAdd (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@instHAdd (@Polynomial Complex Complex.instSemiring) (@Polynomial.add' Complex Complex.instSemiring)) (@HPow.hPow (@Polynomial Complex Complex.instSemiring) Nat (@Polynomial Complex Complex.instSemiring) (@instHPow (@Polynomial Complex Complex.instSemiring) Nat (@Monoid.toNatPow (@Polynomial Complex Complex.instSemiring) (@MonoidWithZero.toMonoid (@Polynomial Complex Complex.instSemiring) (@Semiring.toMonoidWithZero (@Polynomial Complex Complex.instSemiring) (@Polynomial.semiring Complex Complex.instSemiring))))) (@Polynomial.X Complex Complex.instSemiring) (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) (@HPow.hPow (@Polynomial Complex Complex.instSemiring) Nat (@Polynomial Complex Complex.instSemiring) (@instHPow (@Polynomial Complex Complex.instSemiring) Nat (@Monoid.toNatPow (@Polynomial Complex Complex.instSemiring) (@MonoidWithZero.toMonoid (@Polynomial Complex Complex.instSemiring) (@Semiring.toMonoidWithZero (@Polynomial Complex Complex.instSemiring) (@Polynomial.semiring Complex Complex.instSemiring))))) (@Polynomial.X Complex Complex.instSemiring) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))) (@Polynomial.X Complex Complex.instSemiring)) (@OfNat.ofNat (@Polynomial Complex Complex.instSemiring) (nat_lit 1) (@One.toOfNat1 (@Polynomial Complex Complex.instSemiring) (@Polynomial.one Complex Complex.instSemiring))))) (@Insert.insert (@Polynomial Complex Complex.instSemiring) (Set (@Polynomial Complex Complex.instSemiring)) (@Set.instInsert (@Polynomial Complex Complex.instSemiring)) (@HAdd.hAdd (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@instHAdd (@Polynomial Complex Complex.instSemiring) (@Polynomial.add' Complex Complex.instSemiring)) (@HSub.hSub (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@instHSub (@Polynomial Complex Complex.instSemiring) (@Polynomial.sub Complex Complex.instRing)) (@HSub.hSub (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@instHSub (@Polynomial Complex Complex.instSemiring) (@Polynomial.sub Complex Complex.instRing)) (@HPow.hPow (@Polynomial Complex Complex.instSemiring) Nat (@Polynomial Complex Complex.instSemiring) (@instHPow (@Polynomial Complex Complex.instSemiring) Nat (@Monoid.toNatPow (@Polynomial Complex Complex.instSemiring) (@MonoidWithZero.toMonoid (@Polynomial Complex Complex.instSemiring) (@Semiring.toMonoidWithZero (@Polynomial Complex Complex.instSemiring) (@Polynomial.semiring Complex Complex.instSemiring))))) (@Polynomial.X Complex Complex.instSemiring) (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) (@HPow.hPow (@Polynomial Complex Complex.instSemiring) Nat (@Polynomial Complex Complex.instSemiring) (@instHPow (@Polynomial Complex Complex.instSemiring) Nat (@Monoid.toNatPow (@Polynomial Complex Complex.instSemiring) (@MonoidWithZero.toMonoid (@Polynomial Complex Complex.instSemiring) (@Semiring.toMonoidWithZero (@Polynomial Complex Complex.instSemiring) (@Polynomial.semiring Complex Complex.instSemiring))))) (@Polynomial.X Complex Complex.instSemiring) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))) (@Polynomial.X Complex Complex.instSemiring)) (@OfNat.ofNat (@Polynomial Complex Complex.instSemiring) (nat_lit 1) (@One.toOfNat1 (@Polynomial Complex Complex.instSemiring) (@Polynomial.one Complex Complex.instSemiring)))) (@Singleton.singleton (@Polynomial Complex Complex.instSemiring) (Set (@Polynomial Complex Complex.instSemiring)) (@Set.instSingletonSet (@Polynomial Complex Complex.instSemiring)) (@Neg.neg (@Polynomial Complex Complex.instSemiring) (@Polynomial.neg' Complex Complex.instRing) (@HAdd.hAdd (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@instHAdd (@Polynomial Complex Complex.instSemiring) (@Polynomial.add' Complex Complex.instSemiring)) (@HSub.hSub (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@instHSub (@Polynomial Complex Complex.instSemiring) (@Polynomial.sub Complex Complex.instRing)) (@HSub.hSub (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@instHSub (@Polynomial Complex Complex.instSemiring) (@Polynomial.sub Complex Complex.instRing)) (@HPow.hPow (@Polynomial Complex Complex.instSemiring) Nat (@Polynomial Complex Complex.instSemiring) (@instHPow (@Polynomial Complex Complex.instSemiring) Nat (@Monoid.toNatPow (@Polynomial Complex Complex.instSemiring) (@MonoidWithZero.toMonoid (@Polynomial Complex Complex.instSemiring) (@Semiring.toMonoidWithZero (@Polynomial Complex Complex.instSemiring) (@Polynomial.semiring Complex Complex.instSemiring))))) (@Polynomial.X Complex Complex.instSemiring) (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) (@HPow.hPow (@Polynomial Complex Complex.instSemiring) Nat (@Polynomial Complex Complex.instSemiring) (@instHPow (@Polynomial Complex Complex.instSemiring) Nat (@Monoid.toNatPow (@Polynomial Complex Complex.instSemiring) (@MonoidWithZero.toMonoid (@Polynomial Complex Complex.instSemiring) (@Semiring.toMonoidWithZero (@Polynomial Complex Complex.instSemiring) (@Polynomial.semiring Complex Complex.instSemiring))))) (@Polynomial.X Complex Complex.instSemiring) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))) (@Polynomial.X Complex Complex.instSemiring)) (@OfNat.ofNat (@Polynomial Complex Complex.instSemiring) (nat_lit 1) (@One.toOfNat1 (@Polynomial Complex Complex.instSemiring) (@Polynomial.one Complex Complex.instSemiring)))))))))))))))))	Set ℂ[X]	[ { "t": "{P : ℂ[X] \| P.natDegree ≥ 1 ∧ (∀ k ∈ Set.Icc 0 P.natDegree, P.coeff k = 1 ∨ P.coeff k = -1) ∧\n ∀ z : ℂ, P.eval z = 0 → ∃ r : ℝ, r = z} = answer", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1968_a6", "tags": [ "algebra" ] }
Let $p$ be a prime number. Find the number of distinct $2 \times 2$ matrices $$\begin{pmatrix} a & b \\ c & d \end{pmatrix}$$ such that $a, b, c, d \in \{0, 1, ..., p - 1\}$, $a + d \equiv 1 \pmod p$, and $ad - bc \equiv 0 \pmod p$.	$p^2 + p$	There are $p^2 + p$ such matrices.	open Finset Polynomial Topology Filter Metric	[]	@Eq (Nat → Nat) answer fun (p_1 : Nat) => @HAdd.hAdd Nat Nat Nat (@instHAdd Nat instAddNat) (@HPow.hPow Nat Nat Nat (@instHPow Nat Nat (@Monoid.toNatPow Nat Nat.instMonoid)) p_1 (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) p_1	ℕ → ℕ	[ { "t": "ℕ", "v": null, "name": "p" }, { "t": "Nat.Prime p", "v": null, "name": "hp" }, { "t": "{M : Matrix (Fin 2) (Fin 2) (ZMod p) \| M 0 0 + M 1 1 = 1 ∧ M 0 0 * M 1 1 - M 0 1 * M 1 0 = 0}.ncard = answer p", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1968_b5", "tags": [ "linear_algebra", "number_theory", "combinatorics" ] }
What are the possible ranges (across all real inputs $x$ and $y$) of a polynomial $f(x, y)$ with real coefficients?	{{x} \| x : ℝ} ∪ {Set.Ici x \| x : ℝ} ∪ {Set.Iic x \| x : ℝ} ∪ {Set.Iio x \| x : ℝ} ∪ {Set.Ioi x \| x : ℝ} ∪ {Set.univ}	Show that the possibles ranges are a single point, any half-open or half-closed semi-infinite interval, or all real numbers.	open Matrix Filter Topology Set Nat	[]	@Eq (Set (Set Real)) answer (@Union.union (Set (Set Real)) (@Set.instUnion (Set Real)) (@Union.union (Set (Set Real)) (@Set.instUnion (Set Real)) (@Union.union (Set (Set Real)) (@Set.instUnion (Set Real)) (@Union.union (Set (Set Real)) (@Set.instUnion (Set Real)) (@Union.union (Set (Set Real)) (@Set.instUnion (Set Real)) (@setOf (Set Real) fun (x : Set Real) => @Exists Real fun (x_1 : Real) => @Eq (Set Real) (@Singleton.singleton Real (Set Real) (@Set.instSingletonSet Real) x_1) x) (@setOf (Set Real) fun (x : Set Real) => @Exists Real fun (x_1 : Real) => @Eq (Set Real) (@Set.Ici Real Real.instPreorder x_1) x)) (@setOf (Set Real) fun (x : Set Real) => @Exists Real fun (x_1 : Real) => @Eq (Set Real) (@Set.Iic Real Real.instPreorder x_1) x)) (@setOf (Set Real) fun (x : Set Real) => @Exists Real fun (x_1 : Real) => @Eq (Set Real) (@Set.Iio Real Real.instPreorder x_1) x)) (@setOf (Set Real) fun (x : Set Real) => @Exists Real fun (x_1 : Real) => @Eq (Set Real) (@Set.Ioi Real Real.instPreorder x_1) x)) (@Singleton.singleton (Set Real) (Set (Set Real)) (@Set.instSingletonSet (Set Real)) (@Set.univ Real)))	Set (Set ℝ)	[ { "t": "answer = {{z : ℝ \| ∃ x : Fin 2 → ℝ, MvPolynomial.eval x f = z} \| f : MvPolynomial (Fin 2) ℝ}", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1969_a1", "tags": [ "algebra", "set_theory" ] }
Show that a finite group can not be the union of two of its proper subgroups. Does the statement remain true if 'two' is replaced by 'three'?	False	Show that the statement is no longer true if 'two' is replaced by 'three'.	open Matrix Filter Topology Set Nat	[]	@Eq Prop answer False	Prop	[ { "t": "ℕ → Prop", "v": null, "name": "P" }, { "t": "∀ n, P n ↔ ∀ (G : Type) [Group G] [Finite G],\n ∀ H : Fin n → Subgroup G, (∀ i, H i < ⊤) → ⋃ i, (H i : Set G) < ⊤", "v": null, "name": "P_def" }, { "t": "answer ↔ (P 3)", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1969_b2", "tags": [ "abstract_algebra" ] }
Find the length of the longest possible sequence of equal nonzero digits (in base 10) in which a perfect square can terminate. Also, find the smallest square that attains this length.	(3, 1444)	The maximum attainable length is $3$; the smallest such square is $38^2 = 1444$.	open Metric Set EuclideanGeometry	[]	@Eq (Prod Nat Nat) answer (@Prod.mk Nat Nat (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3))) (@OfNat.ofNat Nat (nat_lit 1444) (instOfNatNat (nat_lit 1444))))	ℕ × ℕ	[ { "t": "ℕ → ℕ", "v": null, "name": "L" }, { "t": "∀ n : ℕ, L n ≤ (Nat.digits 10 n).length ∧\n(∀ k : ℕ, k < L n → (Nat.digits 10 n)[k]! = (Nat.digits 10 n)[0]!) ∧\n(L n ≠ (Nat.digits 10 n).length → (Nat.digits 10 n)[L n]! ≠ (Nat.digits 10 n)[0]!)", "v": null, "name": "hL" }, { "t": "(∃ n : ℕ, (Nat.digits 10 (n^2))[0]! ≠ 0 ∧ L (n^2) = answer.1) ∧\n(∀ n : ℕ, (Nat.digits 10 (n^2))[0]! ≠ 0 → L (n^2) ≤ answer.1) ∧\n(∃ m : ℕ, m^2 = answer.2) ∧\nL (answer.2) = answer.1 ∧\n(Nat.digits 10 answer.2)[0]! ≠ 0 ∧\n∀ n : ℕ, (Nat.digits 10 (n^2))[0]! ≠ 0 ∧ L (n^2) = answer.1 → n^2 ≥ answer.2", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1970_a3", "tags": [ "number_theory" ] }
Evaluate the infinite product $\lim_{n \to \infty} \frac{1}{n^4} \prod_{i = 1}^{2n} (n^2 + i^2)^{1/n}$.	$e^{2 \ln 5 - 4 + 2 \arctan 2}$	Show that the solution is $e^{2 \log(5) - 4 + 2 arctan(2)}$.	open Metric Set EuclideanGeometry Filter Topology	[]	@Eq Real answer (Real.exp (@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd) (@HSub.hSub Real Real Real (@instHSub Real Real.instSub) (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) (@OfNat.ofNat Real (nat_lit 2) (@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0)))))) (Real.log (@OfNat.ofNat Real (nat_lit 5) (@instOfNatAtLeastTwo Real (nat_lit 5) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))))))) (@OfNat.ofNat Real (nat_lit 4) (@instOfNatAtLeastTwo Real (nat_lit 4) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))))) (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) (@OfNat.ofNat Real (nat_lit 2) (@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0)))))) (Real.arctan (@OfNat.ofNat Real (nat_lit 2) (@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))))))))))	ℝ	[ { "t": "Tendsto (fun n => 1/(n^4) * ∏ i in Finset.Icc (1 : ℤ) (2*n), ((n^2 + i^2) : ℝ)^((1 : ℝ)/n)) atTop (𝓝 answer)", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1970_b1", "tags": [ "analysis" ] }
Determine all polynomials $P(x)$ such that $P(x^2 + 1) = (P(x))^2 + 1$ and $P(0) = 0$.	{Polynomial.X}	Show that the only such polynomial is the identity function.	open Set	[]	@Eq (Set (@Polynomial Real Real.semiring)) answer (@Singleton.singleton (@Polynomial Real Real.semiring) (Set (@Polynomial Real Real.semiring)) (@Set.instSingletonSet (@Polynomial Real Real.semiring)) (@Polynomial.X Real Real.semiring))	Set (Polynomial ℝ)	[ { "t": "Polynomial ℝ", "v": null, "name": "P" }, { "t": "∀ P : Polynomial ℝ, P ∈ answer ↔ (P.eval 0 = 0 ∧ (∀ x : ℝ, P.eval (x^2 + 1) = (P.eval x)^2 + 1))", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1971_a2", "tags": [ "algebra" ] }
After each play of a certain game of solitaire, the player receives either $a$ or $b$ points, where $a$ and $b$ are positive integers with $a > b$; scores accumulate from play to play. If there are $35$ unattainable scores, one of which is $58$, find $a$ and $b$.	(11, 8)	Show that the solution is $a = 11$ and $b = 8$.	open Set MvPolynomial	[]	@Eq (Prod Int Int) answer (@Prod.mk Int Int (@OfNat.ofNat Int (nat_lit 11) (@instOfNat (nat_lit 11))) (@OfNat.ofNat Int (nat_lit 8) (@instOfNat (nat_lit 8))))	ℤ × ℤ	[ { "t": "ℤ", "v": null, "name": "a" }, { "t": "ℤ", "v": null, "name": "b" }, { "t": "a > 0 ∧ b > 0 ∧ a > b", "v": null, "name": "hab" }, { "t": "ℤ → ℤ → Prop", "v": null, "name": "pab" }, { "t": "∀ x y, pab x y ↔\n {s : ℕ \| ¬∃ m n : ℕ, mx + ny = s}.ncard = 35 ∧ \n ¬∃ m n : ℕ, mx + ny = 58", "v": null, "name": "hpab" }, { "t": "pab a b ↔ a = answer.1 ∧ b = answer.2", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1971_a5", "tags": [ "number_theory" ] }
Find all functions $F : \mathbb{R} \setminus \{0, 1\} \to \mathbb{R}$ that satisfy $F(x) + F\left(\frac{x - 1}{x}\right) = 1 + x$ for all $x \in \mathbb{R} \setminus \{0, 1\}$.	$\left\{x \mapsto \frac{x^3 - x^2 - 1}{2x(x - 1)}\right\}$	The only such function is $F(x) = \frac{x^3 - x^2 - 1}{2x(x - 1)}$.	open Set MvPolynomial	[]	@Eq (Set (Real → Real)) answer (@Singleton.singleton (Real → Real) (Set (Real → Real)) (@Set.instSingletonSet (Real → Real)) fun (x : Real) => @HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) (@HSub.hSub Real Real Real (@instHSub Real Real.instSub) (@HSub.hSub Real Real Real (@instHSub Real Real.instSub) (@HPow.hPow Real Nat Real (@instHPow Real Nat (@Monoid.toNatPow Real Real.instMonoid)) x (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) (@HPow.hPow Real Nat Real (@instHPow Real Nat (@Monoid.toNatPow Real Real.instMonoid)) x (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))) (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))) (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) (@OfNat.ofNat Real (nat_lit 2) (@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0)))))) x) (@HSub.hSub Real Real Real (@instHSub Real Real.instSub) x (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)))))	Set (ℝ → ℝ)	[ { "t": "Set ℝ", "v": null, "name": "S" }, { "t": "S = univ \\ {0, 1}", "v": null, "name": "hS" }, { "t": "(ℝ → ℝ) → Prop", "v": null, "name": "P" }, { "t": "P = fun (F : ℝ → ℝ) => ∀ x ∈ S, F x + F ((x - 1)/x) = 1 + x", "v": null, "name": "hP" }, { "t": "∀ F ∈ answer, P F ∧ ∀ f : ℝ → ℝ, P f → ∃ F ∈ answer, (∀ x ∈ S, f x = F x)", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1971_b2", "tags": [ "algebra" ] }
We call a function $f$ from $[0,1]$ to the reals to be supercontinuous on $[0,1]$ if the Cesaro-limit exists for the sequence $f(x_1), f(x_2), f(x_3), \dots$ whenever it does for the sequence $x_1, x_2, x_3 \dots$. Find all supercontinuous functions on $[0,1]$.	the set of all linear functions on [0,1]	Show that the solution is the set of affine functions.	open EuclideanGeometry Filter Topology Set	[]	@Eq (Set (Real → Real)) answer (@setOf (Real → Real) fun (f : Real → Real) => @Exists Real fun (A : Real) => @Exists Real fun (B : Real) => ∀ (x : Real), @Membership.mem Real (Set Real) (@Set.instMembership Real) (@Set.Icc Real Real.instPreorder (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))) x → @Eq Real (f x) (@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd) (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) A x) B))	Set (ℝ → ℝ)	[ { "t": "(ℕ → ℝ) → Prop", "v": null, "name": "climit_exists" }, { "t": "(ℝ → ℝ) → Prop", "v": null, "name": "supercontinuous" }, { "t": "∀ x, climit_exists x ↔ ∃ C : ℝ, Tendsto (fun n => (∑ i in Finset.range n, (x i))/(n : ℝ)) atTop (𝓝 C)", "v": null, "name": "hclimit_exists" }, { "t": "∀ f, supercontinuous f ↔ ∀ (x : ℕ → ℝ), (∀ i : ℕ, x i ∈ Icc 0 1) → climit_exists x → climit_exists (fun i => f (x i))", "v": null, "name": "hsupercontinuous" }, { "t": "{f \| supercontinuous f} = answer", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1972_a3", "tags": [ "analysis" ] }
Let $x : \mathbb{R} \to \mathbb{R}$ be a twice differentiable function whose second derivative is nonstrictly decreasing. If $x(t) - x(0) = s$, $x'(0) = 0$, and $x'(t) = v$ for some $t > 0$, find the maximum possible value of $t$ in terms of $s$ and $v$.	$2s / v$	Show that the maximum possible time is $t = \frac{2s}{v}$.	open EuclideanGeometry Filter Topology Set MeasureTheory Metric	[]	@Eq (Real → Real → Real) answer fun (s_1 v_1 : Real) => @HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) (@OfNat.ofNat Real (nat_lit 2) (@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0)))))) s_1) v_1	ℝ → ℝ → ℝ	[ { "t": "ℝ", "v": null, "name": "s" }, { "t": "ℝ", "v": null, "name": "v" }, { "t": "s > 0", "v": null, "name": "hs" }, { "t": "v > 0", "v": null, "name": "hv" }, { "t": "ℝ → (ℝ → ℝ) → Prop", "v": null, "name": "valid" }, { "t": "∀ t x, valid t x ↔\n DifferentiableOn ℝ x (Set.Icc 0 t) ∧ DifferentiableOn ℝ (deriv x) (Set.Icc 0 t) ∧\n AntitoneOn (deriv (deriv x)) (Set.Icc 0 t) ∧\n deriv x 0 = 0 ∧ deriv x t = v ∧ x t - x 0 = s", "v": null, "name": "hvalid" }, { "t": "IsGreatest {t \| ∃ x : ℝ → ℝ, valid t x} (answer s v)", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1972_b2", "tags": [ "analysis" ] }
Consider an infinite series whose $n$th term is given by $\pm \frac{1}{n}$, where the actual values of the $\pm$ signs repeat in blocks of $8$ (so the $\frac{1}{9}$ term has the same sign as the $\frac{1}{1}$ term, and so on). Call such a sequence balanced if each block contains four $+$ and four $-$ signs. Prove that being balanced is a sufficient condition for the sequence to converge. Is being balanced also necessary for the sequence to converge?	True	Show that the condition is necessary.	open Nat Set MeasureTheory Topology Filter	[]	@Eq Prop answer True	Prop	[ { "t": "List ℝ", "v": null, "name": "L" }, { "t": "L.length = 8 ∧ ∀ i : Fin L.length, L[i] = 1 ∨ L[i] = -1", "v": null, "name": "hL" }, { "t": "ℕ", "v": null, "name": "pluses" }, { "t": "pluses = {i : Fin L.length \| L[i] = 1}.ncard", "v": null, "name": "hpluses" }, { "t": "ℕ → ℝ", "v": null, "name": "S" }, { "t": "S = fun n : ℕ ↦ ∑ i in Finset.Icc 1 n, L[i % 8]/i", "v": null, "name": "hS" }, { "t": "answer ↔ ((∃ l : ℝ, Tendsto S atTop (𝓝 l)) → pluses = 4)", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1973_a2", "tags": [ "analysis" ] }
How many zeros does the function $f(x) = 2^x - 1 - x^2$ have on the real line?	3	Show that the solution is 3.	open Nat Set MeasureTheory Topology Filter	[]	@Eq Nat answer (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))	ℕ	[ { "t": "ℝ → ℝ", "v": null, "name": "f" }, { "t": "f = fun x => 2^x - 1 - x^2", "v": null, "name": "hf" }, { "t": "answer = {x : ℝ \| f x = 0}.ncard", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1973_a4", "tags": [ "analysis" ] }
Suppose $f$ is a function on $[0,1]$ with continuous derivative satisfying $0 < f'(x) \leq 1$ and $f 0 = 0$. Prove that $\left[\int_0^1 f(x) dx\right]]^2 \geq \int_0^1 (f(x))^3 dx$, and find an example where equality holds.	the identity function $f(x) = x$	Show that one such example where equality holds is the identity function.	open Nat Set MeasureTheory Topology Filter	[]	@Eq (Real → Real) answer fun (x : Real) => x	ℝ → ℝ	[ { "t": "ℝ → ℝ", "v": null, "name": "f" }, { "t": "(ℝ → ℝ) → Prop", "v": null, "name": "hprop" }, { "t": "hprop = fun g => ContDiff ℝ 1 g ∧ (∀ x : ℝ, 0 < deriv g x ∧ deriv g x ≤ 1) ∧ g 0 = 0", "v": null, "name": "hprop_def" }, { "t": "hprop f", "v": null, "name": "hf" }, { "t": "hprop answer ∧ (∫ x in Icc 0 1, answer x)^2 = ∫ x in Icc 0 1, (answer x)^3", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1973_b4", "tags": [ "analysis" ] }
Call a set of positive integers 'conspiratorial' if no three of them are pairwise relatively prime. What is the largest number of elements in any conspiratorial subset of the integers 1 through 16?	11	Show that the answer is 11.	open Set	[]	@Eq Nat answer (@OfNat.ofNat Nat (nat_lit 11) (instOfNatNat (nat_lit 11)))	ℕ	[ { "t": "Set ℤ → Prop", "v": null, "name": "conspiratorial" }, { "t": "∀ S, conspiratorial S ↔ ∀ a ∈ S, ∀ b ∈ S, ∀ c ∈ S, (a > 0 ∧ b > 0 ∧ c > 0) ∧ ((a ≠ b ∧ b ≠ c ∧ a ≠ c) → (Int.gcd a b > 1 ∨ Int.gcd b c > 1 ∨ Int.gcd a c > 1))", "v": null, "name": "hconspiratorial" }, { "t": "IsGreatest {k \| ∃ S, S ⊆ Icc 1 16 ∧ conspiratorial S ∧ S.encard = k} answer", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1974_a1", "tags": [ "number_theory" ] }
A well-known theorem asserts that a prime $p > 2$ can be written as the sum of two perfect squres if and only if $p \equiv 1 \bmod 4$. Find which primes $p > 2$ can be written in each of the following forms, using (not necessarily positive) integers $x$ and $y$: (a) $x^2 + 16y^2$, (b) $4x^2 + 4xy + 5y^2$.	({p : ℕ \| p.Prime ∧ p ≡ 1 [MOD 8]}, {p : ℕ \| p.Prime ∧ p ≡ 5 [MOD 8]})	Show that that the answer to (a) is the set of primes which are $1 \bmod 8$, and the solution to (b) is the set of primes which are $5 \bmod 8$.	open Set	[]	@Eq (Prod (Set Nat) (Set Nat)) answer (@Prod.mk (Set Nat) (Set Nat) (@setOf Nat fun (p_1 : Nat) => And (Nat.Prime p_1) (Nat.ModEq (@OfNat.ofNat Nat (nat_lit 8) (instOfNatNat (nat_lit 8))) p_1 (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))) (@setOf Nat fun (p_1 : Nat) => And (Nat.Prime p_1) (Nat.ModEq (@OfNat.ofNat Nat (nat_lit 8) (instOfNatNat (nat_lit 8))) p_1 (@OfNat.ofNat Nat (nat_lit 5) (instOfNatNat (nat_lit 5))))))	(Set ℕ) × (Set ℕ)	[ { "t": "ℕ", "v": null, "name": "p" }, { "t": "∀ p : ℕ, p.Prime ∧ p > 2 → ((∃ m n : ℤ, p = m^2 + n^2) ↔ p ≡ 1 [MOD 4])", "v": null, "name": "h_assumption" }, { "t": "∀ p : ℕ,\n ((p.Prime ∧ p > 2 ∧ (∃ x y : ℤ, p = x^2 + 16y^2)) ↔ p ∈ answer.1) ∧\n ((p.Prime ∧ p > 2 ∧ (∃ x y : ℤ, p = 4x^2 + 4xy + 5*y^2)) ↔ p ∈ answer.2)", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1974_a3", "tags": [ "number_theory" ] }
Evaluate in closed form: $\frac{1}{2^{n-1}} \sum_{k < n/2} (n-2k)*{n \choose k}$.	(fun n ↦ (1 : ℚ) / ((2 : ℚ) ^ ((n :ℕ) - 1)) * (n * (n - 1).choose ⌊n / 2⌋₊))	Show that the solution is $\frac{n}{2^{n-1}} * {(n-1) \choose \left[ (n-1)/2 \right]}$.	open Set Nat	[]	@Eq (Nat → Rat) answer fun (n_1 : Nat) => @HMul.hMul Rat Rat Rat (@instHMul Rat Rat.instMul) (@HDiv.hDiv Rat Rat Rat (@instHDiv Rat Rat.instDiv) (@OfNat.ofNat Rat (nat_lit 1) (@Rat.instOfNat (nat_lit 1))) (@HPow.hPow Rat Nat Rat (@instHPow Rat Nat Rat.instPowNat) (@OfNat.ofNat Rat (nat_lit 2) (@Rat.instOfNat (nat_lit 2))) (@HSub.hSub Nat Nat Nat (@instHSub Nat instSubNat) n_1 (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))) (@HMul.hMul Rat Rat Rat (@instHMul Rat Rat.instMul) (@Nat.cast Rat Rat.instNatCast n_1) (@Nat.cast Rat Rat.instNatCast (Nat.choose (@HSub.hSub Nat Nat Nat (@instHSub Nat instSubNat) n_1 (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))) (@Nat.floor Nat Nat.instOrderedSemiring instFloorSemiringNat (@HDiv.hDiv Nat Nat Nat (@instHDiv Nat Nat.instDiv) n_1 (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))))))	ℕ → ℚ	[ { "t": "ℕ", "v": null, "name": "n" }, { "t": "0 < n", "v": null, "name": "hn" }, { "t": "(1 : ℚ) / (2 ^ (n - 1)) * ∑ k in Finset.Icc 0 ⌊n / 2⌋₊, (n - 2 * k) * (n.choose k) = answer n", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1974_a4", "tags": [ "algebra" ] }
Given $n$, let $k(n)$ be the minimal degree of any monic integral polynomial $f$ such that the value of $f(x)$ is divisible by $n$ for every integer $x$. Find the value of $k(1000000)$.	25	Show that the answer is 25.	open Set Nat Polynomial	[]	@Eq Nat answer (@OfNat.ofNat Nat (nat_lit 25) (instOfNatNat (nat_lit 25)))	ℕ	[ { "t": "Polynomial ℤ → Prop", "v": null, "name": "hdivnallx" }, { "t": "hdivnallx = fun f => Monic f ∧ (∀ x : ℤ, (10^6 : ℤ) ∣ f.eval x)", "v": null, "name": "hdivnallx_def" }, { "t": "sInf {d : ℕ \| ∃ f : Polynomial ℤ, hdivnallx f ∧ d = f.natDegree} = answer", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1974_a6", "tags": [ "algebra" ] }
For a set with $1000$ elements, how many subsets are there whose candinality is respectively $\equiv 0 \bmod 3, \equiv 1 \bmod 3, \equiv 2 \bmod 3$?	((2^1000 - 1)/3, (2^1000 - 1)/3, 1 + (2^1000 - 1)/3)	Show that there answer is that there are $(2^1000-1)/3$ subsets of cardinality $\equiv 0 \bmod 3$ and $\equiv 1 \bmod 3$, and $1 + (2^1000-1)/3$ subsets of cardinality $\equiv 2 \bmod 3$.	open Set Nat Polynomial Filter Topology	[]	@Eq (Prod Nat (Prod Nat Nat)) answer (@Prod.mk Nat (Prod Nat Nat) (@HDiv.hDiv Nat Nat Nat (@instHDiv Nat Nat.instDiv) (@HSub.hSub Nat Nat Nat (@instHSub Nat instSubNat) (@HPow.hPow Nat Nat Nat (@instHPow Nat Nat (@Monoid.toNatPow Nat Nat.instMonoid)) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) (@OfNat.ofNat Nat (nat_lit 1000) (instOfNatNat (nat_lit 1000)))) (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))) (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) (@Prod.mk Nat Nat (@HDiv.hDiv Nat Nat Nat (@instHDiv Nat Nat.instDiv) (@HSub.hSub Nat Nat Nat (@instHSub Nat instSubNat) (@HPow.hPow Nat Nat Nat (@instHPow Nat Nat (@Monoid.toNatPow Nat Nat.instMonoid)) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) (@OfNat.ofNat Nat (nat_lit 1000) (instOfNatNat (nat_lit 1000)))) (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))) (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) (@HAdd.hAdd Nat Nat Nat (@instHAdd Nat instAddNat) (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))) (@HDiv.hDiv Nat Nat Nat (@instHDiv Nat Nat.instDiv) (@HSub.hSub Nat Nat Nat (@instHSub Nat instSubNat) (@HPow.hPow Nat Nat Nat (@instHPow Nat Nat (@Monoid.toNatPow Nat Nat.instMonoid)) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) (@OfNat.ofNat Nat (nat_lit 1000) (instOfNatNat (nat_lit 1000)))) (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))) (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))))))	ℕ × ℕ × ℕ	[ { "t": "ℤ", "v": null, "name": "n" }, { "t": "n = 1000", "v": null, "name": "hn" }, { "t": "ℕ", "v": null, "name": "count0" }, { "t": "ℕ", "v": null, "name": "count1" }, { "t": "ℕ", "v": null, "name": "count2" }, { "t": "count0 = {S \| S ⊆ Finset.Icc 1 n ∧ S.card ≡ 0 [MOD 3]}.ncard", "v": null, "name": "hcount0" }, { "t": "count1 = {S \| S ⊆ Finset.Icc 1 n ∧ S.card ≡ 1 [MOD 3]}.ncard", "v": null, "name": "hcount1" }, { "t": "count2 = {S \| S ⊆ Finset.Icc 1 n ∧ S.card ≡ 2 [MOD 3]}.ncard", "v": null, "name": "hcount2" }, { "t": "(count0, count1, count2) = answer", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1974_b6", "tags": [ "set_theory" ] }
If an integer $n$ can be written as the sum of two triangular numbers (that is, $n = \frac{a^2 + a}{2} + \frac{b^2 + b}{2}$ for some integers $a$ and $b$), express $4n + 1$ as the sum of the squares of two integers $x$ and $y$, giving $x$ and $y$ in terms of $a$ and $b$. Also, show that if $4n + 1 = x^2 + y^2$ for some integers $x$ and $y$, then $n$ can be written as the sum of two triangular numbers.	(fun (a, b) => a + b + 1, fun (a, b) => a - b)	$x = a + b + 1$ and $y = a - b$ (or vice versa).	open Polynomial	[]	@Eq (Prod (Prod Int Int → Int) (Prod Int Int → Int)) answer (@Prod.mk (Prod Int Int → Int) (Prod Int Int → Int) (fun (x : Prod Int Int) => _example.match_2 (fun (x_1 : Prod Int Int) => Int) x fun (a b : Int) => @HAdd.hAdd Int Int Int (@instHAdd Int Int.instAdd) (@HAdd.hAdd Int Int Int (@instHAdd Int Int.instAdd) a b) (@OfNat.ofNat Int (nat_lit 1) (@instOfNat (nat_lit 1)))) fun (x : Prod Int Int) => _example.match_2 (fun (x_1 : Prod Int Int) => Int) x fun (a b : Int) => @HSub.hSub Int Int Int (@instHSub Int Int.instSub) a b)	((ℤ × ℤ) → ℤ) × ((ℤ × ℤ) → ℤ)	[ { "t": "(ℤ × ℤ × ℤ) → Prop", "v": null, "name": "nab" }, { "t": "(ℤ × ℤ × ℤ) → Prop", "v": null, "name": "nxy" }, { "t": "nab = fun (n, a, b) => n = (a^2 + (a : ℚ))/2 + (b^2 + (b : ℚ))/2", "v": null, "name": "hnab" }, { "t": "nxy = fun (n, x, y) => 4*n + 1 = x^2 + y^2", "v": null, "name": "hnxy" }, { "t": "(∀ n a b : ℤ, nab (n, a, b) → nxy (n, answer.1 (a, b), answer.2 (a, b))) ∧ ∀ n : ℤ, (∃ x y : ℤ, nxy (n, x, y)) → ∃ a b : ℤ, nab (n, a, b)", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1975_a1", "tags": [ "algebra", "number_theory" ] }
For which ordered pairs $(b, c)$ of real numbers do both roots of $z^2 + bz + c$ lie strictly inside the unit disk (i.e., $\{\|z\| < 1\}$) in the complex plane?	$c < 1 \land c - b > -1 \land c + b > -1$	The desired region is the strict interior of the triangle with vertices $(0, -1)$, $(2, 1)$, and $(-2, 1)$.	open Polynomial	[ { "t": "ℝ", "v": null, "name": "b" }, { "t": "ℝ", "v": null, "name": "c" } ]	@Eq Prop (answer (@Prod.mk Real Real b c)) (And (@LT.lt Real Real.instLT c (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))) (And (@GT.gt Real Real.instLT (@HSub.hSub Real Real Real (@instHSub Real Real.instSub) c b) (@Neg.neg Real Real.instNeg (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)))) (@GT.gt Real Real.instLT (@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd) c b) (@Neg.neg Real Real.instNeg (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))))))	(ℝ × ℝ) → Prop	[ { "t": "(∀ z : ℂ, (Polynomial.X^2 + (Polynomial.C (b : ℂ)) * Polynomial.X + (Polynomial.C (c : ℂ)) : Polynomial ℂ).eval z = 0 → ‖z‖ < 1) ↔ answer (b, c)", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1975_a2", "tags": [ "algebra" ] }
If $a$, $b$, and $c$ are real numbers satisfying $0 < a < b < c$, at what points in the set $$\{(x, y, z) \in \mathbb{R}^3 : x^b + y^b + z^b = 1, x \ge 0, y \ge 0, z \ge 0\}$$ does $f(x, y, z) = x^a + y^b + z^c$ attain its maximum and minimum?	(fun (a, b, c) => ((a/b)^(1/(b - a)), (1 - ((a/b)^(1/(b - a)))^b)^(1/b), 0), fun (a, b, c) => (0, (1 - ((b/c)^(1/(c - b)))^b)^(1/b), (b/c)^(1/(c - b))))	$f$ attains its maximum at $\left(x_0, (1 - x_0^b)^{\frac{1}{b}}, 0\right)$ and its minimum at $\left(0, (1 - z_0^b)^{\frac{1}{b}}, z_0\right)$, where $x_0 = \left(\frac{a}{b}\right)^{\frac{1}{b-a}}$ and $z_0 = \left(\frac{b}{c}\right)^{\frac{1}{c-b}}$.	open Polynomial	[]	@Eq (Prod (Prod Real (Prod Real Real) → Prod Real (Prod Real Real)) (Prod Real (Prod Real Real) → Prod Real (Prod Real Real))) answer (@Prod.mk (Prod Real (Prod Real Real) → Prod Real (Prod Real Real)) (Prod Real (Prod Real Real) → Prod Real (Prod Real Real)) (fun (x : Prod Real (Prod Real Real)) => _example.match_1 (fun (x_1 : Prod Real (Prod Real Real)) => Prod Real (Prod Real Real)) x fun (a_1 b_1 c_1 : Real) => @Prod.mk Real (Prod Real Real) (@HPow.hPow Real Real Real (@instHPow Real Real Real.instPow) (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) a_1 b_1) (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)) (@HSub.hSub Real Real Real (@instHSub Real Real.instSub) b_1 a_1))) (@Prod.mk Real Real (@HPow.hPow Real Real Real (@instHPow Real Real Real.instPow) (@HSub.hSub Real Real Real (@instHSub Real Real.instSub) (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)) (@HPow.hPow Real Real Real (@instHPow Real Real Real.instPow) (@HPow.hPow Real Real Real (@instHPow Real Real Real.instPow) (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) a_1 b_1) (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)) (@HSub.hSub Real Real Real (@instHSub Real Real.instSub) b_1 a_1))) b_1)) (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)) b_1)) (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)))) fun (x : Prod Real (Prod Real Real)) => _example.match_1 (fun (x_1 : Prod Real (Prod Real Real)) => Prod Real (Prod Real Real)) x fun (a_1 b_1 c_1 : Real) => @Prod.mk Real (Prod Real Real) (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) (@Prod.mk Real Real (@HPow.hPow Real Real Real (@instHPow Real Real Real.instPow) (@HSub.hSub Real Real Real (@instHSub Real Real.instSub) (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)) (@HPow.hPow Real Real Real (@instHPow Real Real Real.instPow) (@HPow.hPow Real Real Real (@instHPow Real Real Real.instPow) (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) b_1 c_1) (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)) (@HSub.hSub Real Real Real (@instHSub Real Real.instSub) c_1 b_1))) b_1)) (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)) b_1)) (@HPow.hPow Real Real Real (@instHPow Real Real Real.instPow) (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) b_1 c_1) (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)) (@HSub.hSub Real Real Real (@instHSub Real Real.instSub) c_1 b_1)))))	((ℝ × ℝ × ℝ) → (ℝ × ℝ × ℝ)) × ((ℝ × ℝ × ℝ) → (ℝ × ℝ × ℝ))	[ { "t": "ℝ", "v": null, "name": "a" }, { "t": "ℝ", "v": null, "name": "b" }, { "t": "ℝ", "v": null, "name": "c" }, { "t": "0 < a ∧ a < b ∧ b < c", "v": null, "name": "hi" }, { "t": "(ℝ × ℝ × ℝ) → Prop", "v": null, "name": "P" }, { "t": "(ℝ × ℝ × ℝ) → ℝ", "v": null, "name": "f" }, { "t": "P = fun (x, y, z) => x ≥ 0 ∧ y ≥ 0 ∧ z ≥ 0 ∧ x^b + y^b + z^b = 1", "v": null, "name": "hP" }, { "t": "f = fun (x, y, z) => x^a + y^b + z^c", "v": null, "name": "hf" }, { "t": "(P (answer.1 (a, b, c)) ∧ ∀ x y z : ℝ, P (x, y, z) →\nf (x, y, z) ≤ f (answer.1 (a, b, c))) ∧\n(P (answer.2 (a, b, c)) ∧ ∀ x y z : ℝ, P (x, y, z) →\nf (x, y, z) ≥ f (answer.2 (a, b, c)))", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1975_a3", "tags": [ "algebra" ] }
Let $n = 2m$, where $m$ is an odd integer greater than 1. Let $\theta = e^{2\pi i/n}$. Expression $(1 - \theta)^{-1}$ explicitly as a polynomial in $\theta$ \[ a_k \theta^k + a_{k-1}\theta^{k-1} + \dots + a_1\theta + a_0\], with integer coefficients $a_i$.	$\sum_{j=0}^{(m-1)/2} \theta^{2j+1}$	Show that the solution is the polynomial $0 + \theta + \theta^3 + \dots + \theta^{m-2}$, alternating consecutive coefficients between 0 and 1.	open Polynomial Real Complex	[]	@Eq (Nat → @Polynomial Int Int.instSemiring) answer fun (m_1 : Nat) => @Finset.sum Nat (@Polynomial Int Int.instSemiring) (@NonUnitalNonAssocSemiring.toAddCommMonoid (@Polynomial Int Int.instSemiring) (@NonUnitalNonAssocCommSemiring.toNonUnitalNonAssocSemiring (@Polynomial Int Int.instSemiring) (@NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring (@Polynomial Int Int.instSemiring) (@NonUnitalCommRing.toNonUnitalNonAssocCommRing (@Polynomial Int Int.instSemiring) (@CommRing.toNonUnitalCommRing (@Polynomial Int Int.instSemiring) (@Polynomial.commRing Int Int.instCommRing)))))) (Finset.range (@HDiv.hDiv Nat Nat Nat (@instHDiv Nat Nat.instDiv) (@HSub.hSub Nat Nat Nat (@instHSub Nat instSubNat) m_1 (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))) fun (j : Nat) => @HPow.hPow (@Polynomial Int Int.instSemiring) Nat (@Polynomial Int Int.instSemiring) (@instHPow (@Polynomial Int Int.instSemiring) Nat (@Monoid.toNatPow (@Polynomial Int Int.instSemiring) (@MonoidWithZero.toMonoid (@Polynomial Int Int.instSemiring) (@Semiring.toMonoidWithZero (@Polynomial Int Int.instSemiring) (@Polynomial.semiring Int Int.instSemiring))))) (@Polynomial.X Int Int.instSemiring) (@HAdd.hAdd Nat Nat Nat (@instHAdd Nat instAddNat) (@HMul.hMul Nat Nat Nat (@instHMul Nat instMulNat) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) j) (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))	ℕ → Polynomial ℤ	[ { "t": "ℕ", "v": null, "name": "m" }, { "t": "Odd m ∧ m > 1", "v": null, "name": "hm" }, { "t": "ℂ", "v": null, "name": "θ" }, { "t": "θ = cexp (2 * Real.pi * I / (2 * m))", "v": null, "name": "hθ" }, { "t": "1/(1 - θ) = Polynomial.aeval θ (answer m)", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1975_a4", "tags": [ "algebra" ] }
Let $H$ be a subgroup of the additive group of ordered pairs of integers under componentwise addition. If $H$ is generated by the elements $(3, 8)$, $(4, -1)$, and $(5, 4)$, then $H$ is also generated by two elements $(1, b)$ and $(0, a)$ for some integer $b$ and positive integer $a$. Find $a$.	7	$a$ must equal $7$.	open Polynomial Real Complex	[]	@Eq Int answer (@OfNat.ofNat Int (nat_lit 7) (@instOfNat (nat_lit 7)))	ℤ	[ { "t": "Set (ℤ × ℤ)", "v": null, "name": "H" }, { "t": "H = {h : (ℤ × ℤ) \| ∃ u v w : ℤ, h = (u3 + v4 + w5, u8 + v(-1) + w4)}", "v": null, "name": "hH" }, { "t": "(∃ b : ℤ, H = {h : (ℤ × ℤ) \| ∃ u v : ℤ, h = (u, ub + vanswer)}) ∧ answer > 0", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1975_b1", "tags": [ "abstract_algebra", "number_theory" ] }
Let $s_k (a_1, a_2, \dots, a_n)$ denote the $k$-th elementary symmetric function; that is, the sum of all $k$-fold products of the $a_i$. For example, $s_1 (a_1, \dots, a_n) = \sum_{i=1}^{n} a_i$, and $s_2 (a_1, a_2, a_3) = a_1a_2 + a_2a_3 + a_1a_3$. Find the supremum $M_k$ (which is never attained) of $$\frac{s_k (a_1, a_2, \dots, a_n)}{(s_1 (a_1, a_2, \dots, a_n))^k}$$ across all $n$-tuples $(a_1, a_2, \dots, a_n)$ of positive real numbers with $n \ge k$.	fun k : ℕ => (1: ℝ)/(Nat.factorial k)	The supremum $M_k$ is $\frac{1}{k!}$.	open Polynomial Real Complex Matrix Filter Topology Multiset	[]	@Eq (Nat → Real) answer fun (k : Nat) => @HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)) (@Nat.cast Real Real.instNatCast (Nat.factorial k))	ℕ → ℝ	[ { "t": "∀ k : ℕ, k > 0 → (∀ a : Multiset ℝ, (∀ i ∈ a, i > 0) ∧ card a ≥ k →\n(esymm a k)/(esymm a 1)^k ≤ answer k) ∧\n∀ M : ℝ, M < answer k → (∃ a : Multiset ℝ, (∀ i ∈ a, i > 0) ∧ card a ≥ k ∧\n(esymm a k)/(esymm a 1)^k > M)", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1975_b3", "tags": [ "analysis", "algebra" ] }
Let $C = \{(x, y) \in \mathbb{R}^2 : x^2 + y^2 = 1\}$ denote the unit circle. Does there exist $B \subseteq C$ for which $B$ is topologically closed and contains exactly one point from each pair of diametrically opposite points in $C$?	False	Such $B$ does not exist.	open Polynomial Real Complex Matrix Filter Topology Multiset	[]	@Eq Prop answer False	Prop	[ { "t": "ℝ × ℝ → Prop", "v": null, "name": "P" }, { "t": "P = fun (x, y) => x^2 + y^2 = 1", "v": null, "name": "hP" }, { "t": "(∃ B ⊆ setOf P, IsClosed B ∧ ∀ x y : ℝ, P (x, y) → Xor' ((x, y) ∈ B) ((-x, -y) ∈ B)) ↔ answer", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1975_b4", "tags": [ "analysis" ] }
Find all integer solutions $(p, r, q, s)$ of the equation $\|p^r - q^s\| = 1$, where $p$ and $q$ are prime and $r$ and $s$ are greater than $1$.	{(3, 2, 2, 3), (2, 3, 3, 2)}	The only solutions are $(p, r, q, s) = (3, 2, 2, 3)$ and $(p, r, q, s) = (2, 3, 3, 2)$.	null	[]	@Eq (Set (Prod Nat (Prod Nat (Prod Nat Nat)))) answer (@Insert.insert (Prod Nat (Prod Nat (Prod Nat Nat))) (Set (Prod Nat (Prod Nat (Prod Nat Nat)))) (@Set.instInsert (Prod Nat (Prod Nat (Prod Nat Nat)))) (@Prod.mk Nat (Prod Nat (Prod Nat Nat)) (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3))) (@Prod.mk Nat (Prod Nat Nat) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) (@Prod.mk Nat Nat (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))))) (@Singleton.singleton (Prod Nat (Prod Nat (Prod Nat Nat))) (Set (Prod Nat (Prod Nat (Prod Nat Nat)))) (@Set.instSingletonSet (Prod Nat (Prod Nat (Prod Nat Nat)))) (@Prod.mk Nat (Prod Nat (Prod Nat Nat)) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) (@Prod.mk Nat (Prod Nat Nat) (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3))) (@Prod.mk Nat Nat (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3))) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))))))	Set (ℕ × ℕ × ℕ × ℕ)	[ { "t": "{a : ℕ × ℕ × ℕ × ℕ \| Nat.Prime a.1 ∧ Nat.Prime a.2.2.1 ∧ a.2.1 > 1 ∧ a.2.2.2 > 1 ∧ \|(a.1^a.2.1 : ℤ) - a.2.2.1^a.2.2.2\| = 1} = answer", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1976_a3", "tags": [ "number_theory" ] }
Let $r$ be a real root of $P(x) = x^3 + ax^2 + bx - 1$, where $a$ and $b$ are integers and $P$ is irreducible over the rationals. Suppose that $r + 1$ is a root of $x^3 + cx^2 + dx + 1$, where $c$ and $d$ are also integers. Express another root $s$ of $P$ as a function of $r$ that does not depend on the values of $a$, $b$, $c$, or $d$.	$\left(-\frac{1}{r + 1}, -\frac{r + 1}{r}\right)$	The possible answers are $s = -\frac{1}{r + 1}$ and $s = -\frac{r + 1}{r}$.	open Polynomial	[]	@Eq (Prod (Real → Real) (Real → Real)) answer (@Prod.mk (Real → Real) (Real → Real) (fun (r_1 : Real) => @HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) (@Neg.neg Real Real.instNeg (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))) (@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd) r_1 (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)))) fun (r_1 : Real) => @HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) (@Neg.neg Real Real.instNeg (@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd) r_1 (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)))) r_1)	(ℝ → ℝ) × (ℝ → ℝ)	[ { "t": "ℤ", "v": null, "name": "a" }, { "t": "ℤ", "v": null, "name": "b" }, { "t": "ℤ", "v": null, "name": "c" }, { "t": "ℤ", "v": null, "name": "d" }, { "t": "ℝ", "v": null, "name": "r" }, { "t": "Polynomial ℚ", "v": null, "name": "P" }, { "t": "Polynomial ℚ", "v": null, "name": "Q" }, { "t": "P = Polynomial.X^3 + (Polynomial.C (a : ℚ))Polynomial.X^2 + (Polynomial.C (b : ℚ))Polynomial.X - (Polynomial.C 1) ∧ Polynomial.aeval r P = 0 ∧ Irreducible P", "v": null, "name": "hP" }, { "t": "Q = Polynomial.X^3 + (Polynomial.C (c : ℚ))Polynomial.X^2 + (Polynomial.C (d : ℚ))Polynomial.X + (Polynomial.C 1) ∧ Polynomial.aeval (r + 1) Q = 0", "v": null, "name": "hQ" }, { "t": "∃ s : ℝ, Polynomial.aeval s P = 0 ∧ (s = answer.1 r ∨ s = answer.2 r)", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1976_a4", "tags": [ "algebra" ] }
Find $$\lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^{n}\left(\left\lfloor \frac{2n}{k} \right\rfloor - 2\left\lfloor \frac{n}{k} \right\rfloor\right).$$ Your answer should be in the form $\ln(a) - b$, where $a$ and $b$ are positive integers.	ln(4) - 1	The limit equals $\ln(4) - 1$, so $a = 4$ and $b = 1$.	open Polynomial Filter Topology	[]	@Eq (Prod Nat Nat) answer (@Prod.mk Nat Nat (@OfNat.ofNat Nat (nat_lit 4) (instOfNatNat (nat_lit 4))) (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))	ℕ × ℕ	[ { "t": "Tendsto (fun n : ℕ => ((1 : ℝ)/n)∑ k in Finset.Icc (1 : ℤ) n, (Int.floor ((2n)/k) - 2*Int.floor (n/k))) atTop\n (𝓝 (Real.log answer.1 - answer.2))", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1976_b1", "tags": [ "analysis" ] }
Let $G$ be a group generated by two elements $A$ and $B$; i.e., every element of $G$ can be expressed as a finite word $A^{n_1}B^{n_2} \cdots A^{n_{k-1}}B^{n_k}$, where the $n_i$ can assume any integer values and $A^0 = B^0 = 1$. Further assume that $A^4 = B^7 = ABA^{-1}B = 1$, but $A^2 \ne 1$ and $B \ne 1$. Find the number of elements of $G$ than can be written as $C^2$ for some $C \in G$ and express each such square as a word in $A$ and $B$.	(8, {[(0, 0)], [(2, 0)], [(0, 1)], [(0, 2)], [(0, 3)], [(0, 4)], [(0, 5)], [(0, 6)]})	There are $8$ such squares: $1$, $A^2$, $B$, $B^2$, $B^3$, $B^4$, $B^5$, and $B^6$.	open Polynomial Filter Topology	[]	@Eq (Prod Nat (Set (List (Prod Int Int)))) answer (@Prod.mk Nat (Set (List (Prod Int Int))) (@OfNat.ofNat Nat (nat_lit 8) (instOfNatNat (nat_lit 8))) (@Insert.insert (List (Prod Int Int)) (Set (List (Prod Int Int))) (@Set.instInsert (List (Prod Int Int))) (@List.cons (Prod Int Int) (@Prod.mk Int Int (@OfNat.ofNat Int (nat_lit 0) (@instOfNat (nat_lit 0))) (@OfNat.ofNat Int (nat_lit 0) (@instOfNat (nat_lit 0)))) (@List.nil (Prod Int Int))) (@Insert.insert (List (Prod Int Int)) (Set (List (Prod Int Int))) (@Set.instInsert (List (Prod Int Int))) (@List.cons (Prod Int Int) (@Prod.mk Int Int (@OfNat.ofNat Int (nat_lit 2) (@instOfNat (nat_lit 2))) (@OfNat.ofNat Int (nat_lit 0) (@instOfNat (nat_lit 0)))) (@List.nil (Prod Int Int))) (@Insert.insert (List (Prod Int Int)) (Set (List (Prod Int Int))) (@Set.instInsert (List (Prod Int Int))) (@List.cons (Prod Int Int) (@Prod.mk Int Int (@OfNat.ofNat Int (nat_lit 0) (@instOfNat (nat_lit 0))) (@OfNat.ofNat Int (nat_lit 1) (@instOfNat (nat_lit 1)))) (@List.nil (Prod Int Int))) (@Insert.insert (List (Prod Int Int)) (Set (List (Prod Int Int))) (@Set.instInsert (List (Prod Int Int))) (@List.cons (Prod Int Int) (@Prod.mk Int Int (@OfNat.ofNat Int (nat_lit 0) (@instOfNat (nat_lit 0))) (@OfNat.ofNat Int (nat_lit 2) (@instOfNat (nat_lit 2)))) (@List.nil (Prod Int Int))) (@Insert.insert (List (Prod Int Int)) (Set (List (Prod Int Int))) (@Set.instInsert (List (Prod Int Int))) (@List.cons (Prod Int Int) (@Prod.mk Int Int (@OfNat.ofNat Int (nat_lit 0) (@instOfNat (nat_lit 0))) (@OfNat.ofNat Int (nat_lit 3) (@instOfNat (nat_lit 3)))) (@List.nil (Prod Int Int))) (@Insert.insert (List (Prod Int Int)) (Set (List (Prod Int Int))) (@Set.instInsert (List (Prod Int Int))) (@List.cons (Prod Int Int) (@Prod.mk Int Int (@OfNat.ofNat Int (nat_lit 0) (@instOfNat (nat_lit 0))) (@OfNat.ofNat Int (nat_lit 4) (@instOfNat (nat_lit 4)))) (@List.nil (Prod Int Int))) (@Insert.insert (List (Prod Int Int)) (Set (List (Prod Int Int))) (@Set.instInsert (List (Prod Int Int))) (@List.cons (Prod Int Int) (@Prod.mk Int Int (@OfNat.ofNat Int (nat_lit 0) (@instOfNat (nat_lit 0))) (@OfNat.ofNat Int (nat_lit 5) (@instOfNat (nat_lit 5)))) (@List.nil (Prod Int Int))) (@Singleton.singleton (List (Prod Int Int)) (Set (List (Prod Int Int))) (@Set.instSingletonSet (List (Prod Int Int))) (@List.cons (Prod Int Int) (@Prod.mk Int Int (@OfNat.ofNat Int (nat_lit 0) (@instOfNat (nat_lit 0))) (@OfNat.ofNat Int (nat_lit 6) (@instOfNat (nat_lit 6)))) (@List.nil (Prod Int Int))))))))))))	ℕ × Set (List (ℤ × ℤ))	[ { "t": "Type", "v": null, "name": "G" }, { "t": "Group G", "v": null, "name": null }, { "t": "G", "v": null, "name": "A" }, { "t": "G", "v": null, "name": "B" }, { "t": "List (ℤ × ℤ) → G", "v": null, "name": "word" }, { "t": "word = fun w : List (ℤ × ℤ) => (List.map (fun t : ℤ × ℤ => A^(t.1)B^(t.2)) w).prod", "v": null, "name": "hword" }, { "t": "∀ g : G, ∃ w : List (ℤ × ℤ), g = word w", "v": null, "name": "hG" }, { "t": "A^4 = 1 ∧ A^2 ≠ 1", "v": null, "name": "hA" }, { "t": "B^7 = 1 ∧ B ≠ 1", "v": null, "name": "hB" }, { "t": "ABA^(-(1 : ℤ))*B = 1", "v": null, "name": "h1" }, { "t": "Set G", "v": null, "name": "S" }, { "t": "S = {g : G \| ∃ C : G, C^2 = g}", "v": null, "name": "hS" }, { "t": "S.ncard = answer.1 ∧ S = {word w \| w ∈ answer.2}", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1976_b2", "tags": [ "abstract_algebra" ] }
Find $$\sum_{k=0}^{n} (-1)^k {n \choose k} (x - k)^n.$$	fun n => C (Nat.factorial n)	The sum equals $n!$.	open Polynomial Filter Topology ProbabilityTheory MeasureTheory	[]	@Eq (Nat → @Polynomial Int Int.instSemiring) answer fun (n : Nat) => @DFunLike.coe (@RingHom Int (@Polynomial Int Int.instSemiring) (@Semiring.toNonAssocSemiring Int Int.instSemiring) (@Semiring.toNonAssocSemiring (@Polynomial Int Int.instSemiring) (@Polynomial.semiring Int Int.instSemiring))) Int (fun (x : Int) => @Polynomial Int Int.instSemiring) (@RingHom.instFunLike Int (@Polynomial Int Int.instSemiring) (@Semiring.toNonAssocSemiring Int Int.instSemiring) (@Semiring.toNonAssocSemiring (@Polynomial Int Int.instSemiring) (@Polynomial.semiring Int Int.instSemiring))) (@Polynomial.C Int Int.instSemiring) (@Nat.cast Int instNatCastInt (Nat.factorial n))	ℕ → Polynomial ℤ	[ { "t": "∀ n : ℕ, ∑ k in Finset.range (n + 1), C ((-(1 : ℤ))^k * Nat.choose n k) * (X - (C (k : ℤ)))^n = answer n", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1976_b5", "tags": [ "algebra" ] }
Show that if four distinct points of the curve $y = 2x^4 + 7x^3 + 3x - 5$ are collinear, then their average $x$-coordinate is some constant $k$. Find $k$.	$-7/8$	Prove that $k = -\frac{7}{8}$.	null	[]	@Eq Real answer (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) (@Neg.neg Real Real.instNeg (@OfNat.ofNat Real (nat_lit 7) (@instOfNatAtLeastTwo Real (nat_lit 7) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 5) (instOfNatNat (nat_lit 5))))))) (@OfNat.ofNat Real (nat_lit 8) (@instOfNatAtLeastTwo Real (nat_lit 8) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))))))	ℝ	[ { "t": "ℝ → ℝ", "v": null, "name": "y" }, { "t": "y = fun x ↦ 2 * x ^ 4 + 7 * x ^ 3 + 3 * x - 5", "v": null, "name": "hy" }, { "t": "Finset ℝ", "v": null, "name": "S" }, { "t": "S.card = 4", "v": null, "name": "hS" }, { "t": "Collinear ℝ {P : Fin 2 → ℝ \| P 0 ∈ S ∧ P 1 = y (P 0)}", "v": null, "name": "h_collinear" }, { "t": "(∑ x in S, x) / 4 = answer", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1977_a1", "tags": [ "algebra" ] }
Find all real solutions $(a, b, c, d)$ to the equations $a + b + c = d$, $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{d}$.	$d = a \land b = -c \lor d = b \land a = -c \lor d = c \land a = -b$	Prove that the solutions are $d = a$ and $b = -c$, $d = b$ and $a = -c$, or $d = c$ and $a = -b$, with $a, b, c, d$ nonzero.	null	[]	@Eq (Real → Real → Real → Real → Prop) answer fun (a_1 b_1 c_1 d_1 : Real) => Or (And (@Eq Real d_1 a_1) (@Eq Real b_1 (@Neg.neg Real Real.instNeg c_1))) (Or (And (@Eq Real d_1 b_1) (@Eq Real a_1 (@Neg.neg Real Real.instNeg c_1))) (And (@Eq Real d_1 c_1) (@Eq Real a_1 (@Neg.neg Real Real.instNeg b_1))))	ℝ → ℝ → ℝ → ℝ → Prop	[ { "t": "ℝ", "v": null, "name": "a" }, { "t": "ℝ", "v": null, "name": "b" }, { "t": "ℝ", "v": null, "name": "c" }, { "t": "ℝ", "v": null, "name": "d" }, { "t": "answer a b c d ↔\n a ≠ 0 → b ≠ 0 → c ≠ 0 → d ≠ 0 → (a + b + c = d ∧ 1 / a + 1 / b + 1 / c = 1 / d)", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1977_a2", "tags": [ "algebra" ] }
Let $f, g, h$ be functions $\mathbb{R} \to \mathbb{R}$. Find an expression for $h(x)$ in terms of $f$ and $g$ such that $f(x) = \frac{h(x + 1) + h(x - 1)}{2}$ and $g(x) = \frac{h(x + 4) + h(x - 4)}{2}$.	$h(x) = g(x) - f(x - 3) + f(x - 1) + f(x + 1) - f(x + 3)$	Prove that $h(x) = g(x) - f(x - 3) + f(x - 1) + f(x + 1) - f(x + 3)$ suffices.	null	[]	@Eq ((Real → Real) → (Real → Real) → Real → Real) answer fun (f_1 g_1 : Real → Real) (x : Real) => @HSub.hSub Real Real Real (@instHSub Real Real.instSub) (@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd) (@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd) (@HSub.hSub Real Real Real (@instHSub Real Real.instSub) (g_1 x) (f_1 (@HSub.hSub Real Real Real (@instHSub Real Real.instSub) x (@OfNat.ofNat Real (nat_lit 3) (@instOfNatAtLeastTwo Real (nat_lit 3) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))))))) (f_1 (@HSub.hSub Real Real Real (@instHSub Real Real.instSub) x (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))))) (f_1 (@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd) x (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))))) (f_1 (@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd) x (@OfNat.ofNat Real (nat_lit 3) (@instOfNatAtLeastTwo Real (nat_lit 3) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))))))	(ℝ → ℝ) → (ℝ → ℝ) → (ℝ → ℝ)	[ { "t": "ℝ → ℝ", "v": null, "name": "f" }, { "t": "ℝ → ℝ", "v": null, "name": "g" }, { "t": "ℝ → ℝ", "v": null, "name": "h" }, { "t": "∀ x, f x = (h (x + 1) + h (x - 1)) / 2", "v": null, "name": "hf" }, { "t": "∀ x, g x = (h (x + 4) + h (x - 4)) / 2", "v": null, "name": "hg" }, { "t": "h = answer f g", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1977_a3", "tags": [ "algebra" ] }
Find $\sum_{n=0}^{\infty} \frac{x^{2^n}}{1 - x^{2^{n+1}}}$ as a rational function of $x$ for $x \in (0, 1)$.	$\frac{x}{1 - x}$	Prove that the sum equals $\frac{x}{1 - x}$.	open RingHom Set	[]	@Eq (@RatFunc Real Real.commRing) answer (@HDiv.hDiv (@RatFunc Real Real.commRing) (@RatFunc Real Real.commRing) (@RatFunc Real Real.commRing) (@instHDiv (@RatFunc Real Real.commRing) (@RatFunc.instDiv Real Real.commRing Real.instIsDomain)) (@RatFunc.X Real Real.commRing Real.instIsDomain) (@HSub.hSub (@RatFunc Real Real.commRing) (@RatFunc Real Real.commRing) (@RatFunc Real Real.commRing) (@instHSub (@RatFunc Real Real.commRing) (@RatFunc.instSub Real Real.commRing)) (@OfNat.ofNat (@RatFunc Real Real.commRing) (nat_lit 1) (@One.toOfNat1 (@RatFunc Real Real.commRing) (@RatFunc.instOne Real Real.commRing))) (@RatFunc.X Real Real.commRing Real.instIsDomain)))	RatFunc ℝ	[ { "t": "ℝ", "v": null, "name": "x" }, { "t": "x ∈ Ioo 0 1", "v": null, "name": "hx" }, { "t": "answer.eval (id ℝ) x = ∑' n : ℕ, x ^ 2 ^ n / (1 - x ^ 2 ^ (n + 1))", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1977_a4", "tags": [ "algebra", "analysis" ] }
Let $X$ be the square $[0, 1] \times [0, 1]$, and let $f : X \to \mathbb{R}$ be continuous. If $\int_Y f(x, y) \, dx \, dy = 0$ for all squares $Y$ such that \begin{itemize} \item[(1)] $Y \subseteq X$, \item[(2)] $Y$ has sides parallel to those of $X$, \item[(3)] at least one of $Y$'s sides is contained in the boundary of $X$, \end{itemize} is it true that $f(x, y) = 0$ for all $x, y$?	True	Prove that $f(x,y)$ must be identically zero.	open RingHom Set Nat	[]	@Eq Prop answer True	Prop	[ { "t": "Set (ℝ × ℝ)", "v": null, "name": "X" }, { "t": "X = Set.prod (Icc 0 1) (Icc 0 1)", "v": null, "name": "hX" }, { "t": "(ℝ × ℝ) → ℝ", "v": null, "name": "room" }, { "t": "room = fun (a,b) ↦ min (min a (1 - a)) (min b (1 - b))", "v": null, "name": "hroom" }, { "t": "(∀ f : (ℝ × ℝ) → ℝ, Continuous f → (∀ P ∈ X, ∫ x in (P.1 - room P)..(P.1 + room P), ∫ y in (P.2 - room P)..(P.2 + room P), f (x, y) = 0) → (∀ P ∈ X, f P = 0)) ↔ answer", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1977_a6", "tags": [ "analysis" ] }
Find $\prod_{n=2}^{\infty} \frac{(n^3 - 1)}{(n^3 + 1)}$.	2/3	Prove that the product equals $\frac{2}{3}$.	open RingHom Set Nat Filter Topology	[]	@Eq Real answer (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) (@OfNat.ofNat Real (nat_lit 2) (@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0)))))) (@OfNat.ofNat Real (nat_lit 3) (@instOfNatAtLeastTwo Real (nat_lit 3) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))))	ℝ	[ { "t": "Tendsto (fun N ↦ ∏ n in Finset.Icc (2 : ℤ) N, ((n : ℝ) ^ 3 - 1) / (n ^ 3 + 1)) atTop (𝓝 answer)", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1977_b1", "tags": [ "algebra", "analysis" ] }
An ordered triple $(a, b, c)$ of positive irrational numbers with $a + b + c = 1$ is considered $\textit{balanced}$ if all three elements are less than $\frac{1}{2}$. If a triple is not balanced, we can perform a ``balancing act'' $B$ defined by $B(a, b, c) = (f(a), f(b), f(c))$, where $f(x) = 2x - 1$ if $x > 1/2$ and $f(x) = 2x$ otherwise. Will finitely many iterations of this balancing act always eventually produce a balanced triple?	False	Not necessarily.	open RingHom Set Nat Filter Topology	[]	@Eq Prop answer False	Prop	[ { "t": "ℝ × ℝ × ℝ → Prop", "v": null, "name": "P" }, { "t": "ℝ × ℝ × ℝ → Prop", "v": null, "name": "balanced" }, { "t": "ℝ × ℝ × ℝ → ℝ × ℝ × ℝ", "v": null, "name": "B" }, { "t": "P = fun (a, b, c) => Irrational a ∧ Irrational b ∧ Irrational c ∧ a > 0 ∧ b > 0 ∧ c > 0 ∧ a + b + c = 1", "v": null, "name": "hP" }, { "t": "balanced = fun (a, b, c) => a < 1/2 ∧ b < 1/2 ∧ c < 1/2", "v": null, "name": "hbalanced" }, { "t": "B = fun (a, b, c) => (ite (a > 1/2) (2a - 1) (2a), ite (b > 1/2) (2b - 1) (2b), ite (c > 1/2) (2c - 1) (2c))", "v": null, "name": "hB" }, { "t": "(∀ t : ℝ × ℝ × ℝ, P t → ∃ n : ℕ, balanced (B^[n] t)) ↔ answer", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1977_b3", "tags": [ "algebra" ] }
Let $p(x) = 2(x^6 + 1) + 4(x^5 + x) + 3(x^4 + x^2) + 5x^3$. For $k$ with $0 < k < 5$, let \[ I_k = \int_0^{\infty} \frac{x^k}{p(x)} \, dx. \] For which $k$ is $I_k$ smallest?	2	Show that $I_k$ is smallest for $k = 2$.	open Set Polynomial	[]	@Eq Nat answer (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))	ℕ	[ { "t": "Polynomial ℝ", "v": null, "name": "p" }, { "t": "p = 2 * (Polynomial.X ^ 6 + 1) + 4 * (Polynomial.X ^ 5 + Polynomial.X) + 3 * (Polynomial.X ^ 4 + Polynomial.X ^ 2) + 5 * Polynomial.X ^ 3", "v": null, "name": "hp" }, { "t": "ℕ → ℝ", "v": null, "name": "I" }, { "t": "I = fun k ↦ ∫ x in Ioi 0, x ^ k / p.eval x", "v": null, "name": "hI" }, { "t": "IsLeast {y \| ∃ k ∈ Ioo 0 5, I k = y} answer", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1978_a3", "tags": [ "analysis", "algebra" ] }
Find \[ \sum_{i=1}^{\infty} \sum_{j=1}^{\infty} \frac{1}{i^2j + 2ij + ij^2}. \]	7 / 4	Prove that the sum evaluates to $\frac{7}{4}$.	open Set Real	[]	@Eq Rat answer (@HDiv.hDiv Rat Rat Rat (@instHDiv Rat Rat.instDiv) (@OfNat.ofNat Rat (nat_lit 7) (@Rat.instOfNat (nat_lit 7))) (@OfNat.ofNat Rat (nat_lit 4) (@Rat.instOfNat (nat_lit 4))))	ℚ	[ { "t": "(∑' i : ℕ+, ∑' j : ℕ+, (1 : ℚ) / (i ^ 2 * j + 2 * i * j + i * j ^ 2)) = answer", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1978_b2", "tags": [ "algebra", "analysis" ] }
Find the real polynomial $p(x)$ of degree $4$ with largest possible coefficient of $x^4$ such that $p([-1, 1]) \subseteq [0, 1]$.	$4x^4 - 4x^2 + 1$	Prove that $p(x) = 4x^4 - 4x^2 + 1$.	open Set Real Filter Topology Polynomial	[]	@Eq (@Polynomial Real Real.semiring) answer (@HAdd.hAdd (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@instHAdd (@Polynomial Real Real.semiring) (@Polynomial.add' Real Real.semiring)) (@HSub.hSub (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@instHSub (@Polynomial Real Real.semiring) (@Polynomial.sub Real Real.instRing)) (@HMul.hMul (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@instHMul (@Polynomial Real Real.semiring) (@Polynomial.mul' Real Real.semiring)) (@OfNat.ofNat (@Polynomial Real Real.semiring) (nat_lit 4) (@instOfNatAtLeastTwo (@Polynomial Real Real.semiring) (nat_lit 4) (@Polynomial.instNatCast Real Real.semiring) (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))) (@HPow.hPow (@Polynomial Real Real.semiring) Nat (@Polynomial Real Real.semiring) (@instHPow (@Polynomial Real Real.semiring) Nat (@Monoid.toNatPow (@Polynomial Real Real.semiring) (@MonoidWithZero.toMonoid (@Polynomial Real Real.semiring) (@Semiring.toMonoidWithZero (@Polynomial Real Real.semiring) (@Polynomial.semiring Real Real.semiring))))) (@Polynomial.X Real Real.semiring) (@OfNat.ofNat Nat (nat_lit 4) (instOfNatNat (nat_lit 4))))) (@HMul.hMul (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@instHMul (@Polynomial Real Real.semiring) (@Polynomial.mul' Real Real.semiring)) (@OfNat.ofNat (@Polynomial Real Real.semiring) (nat_lit 4) (@instOfNatAtLeastTwo (@Polynomial Real Real.semiring) (nat_lit 4) (@Polynomial.instNatCast Real Real.semiring) (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))) (@HPow.hPow (@Polynomial Real Real.semiring) Nat (@Polynomial Real Real.semiring) (@instHPow (@Polynomial Real Real.semiring) Nat (@Monoid.toNatPow (@Polynomial Real Real.semiring) (@MonoidWithZero.toMonoid (@Polynomial Real Real.semiring) (@Semiring.toMonoidWithZero (@Polynomial Real Real.semiring) (@Polynomial.semiring Real Real.semiring))))) (@Polynomial.X Real Real.semiring) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))) (@OfNat.ofNat (@Polynomial Real Real.semiring) (nat_lit 1) (@One.toOfNat1 (@Polynomial Real Real.semiring) (@Polynomial.one Real Real.semiring))))	Polynomial ℝ	[ { "t": "Set (Polynomial ℝ)", "v": null, "name": "S" }, { "t": "S = {p : Polynomial ℝ \| p.degree = 4 ∧ ∀ x ∈ Icc (-1 : ℝ) 1, p.eval x ∈ Icc 0 1}", "v": null, "name": "hS" }, { "t": "answer ∈ S ∧ (∀ p ∈ S, p.coeff 4 ≤ answer.coeff 4)", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1978_b5", "tags": [ "algebra" ] }
For which positive integers $n$ and $a_1, a_2, \dots, a_n$ with $\sum_{i = 1}^{n} a_i = 1979$ does $\prod_{i = 1}^{n} a_i$ attain the greatest value?	Multiset.replicate 659 3 + {2}	$n$ equals $660$; all but one of the $a_i$ equal $3$ and the remaining $a_i$ equals $2$.	null	[]	@Eq (Multiset Nat) answer (@HAdd.hAdd (Multiset Nat) (Multiset Nat) (Multiset Nat) (@instHAdd (Multiset Nat) (@Multiset.instAdd Nat)) (@Multiset.replicate Nat (@OfNat.ofNat Nat (nat_lit 659) (instOfNatNat (nat_lit 659))) (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) (@Singleton.singleton Nat (Multiset Nat) (@Multiset.instSingleton Nat) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))	Multiset ℕ	[ { "t": "Multiset ℕ → Prop", "v": null, "name": "P" }, { "t": "∀ a, P a ↔ Multiset.card a > 0 ∧ (∀ i ∈ a, i > 0) ∧ a.sum = 1979", "v": null, "name": "hP" }, { "t": "P answer ∧ ∀ a : Multiset ℕ, P a → answer.prod ≥ a.prod", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1979_a1", "tags": [ "algebra" ] }
For which real numbers $k$ does there exist a continuous function $f : \mathbb{R} \to \mathbb{R}$ such that $f(f(x)) = kx^9$ for all real $x$?	$k \geq 0$	Such a function exists if and only if $k \ge 0$.	null	[]	@Eq (Real → Prop) answer fun (k_1 : Real) => @GE.ge Real Real.instLE k_1 (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero))	ℝ → Prop	[ { "t": "ℝ", "v": null, "name": "k" }, { "t": "answer k ↔ ∃ f : ℝ → ℝ, Continuous f ∧ ∀ x : ℝ, f (f x) = k * x^9", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1979_a2", "tags": [ "analysis", "algebra" ] }
Let $x_1, x_2, x_3, \dots$ be a sequence of nonzero real numbers such that $$x_n = \frac{x_{n-2}x_{n-1}}{2x_{n-2}-x_{n-1}}$$ for all $n \ge 3$. For which real values of $x_1$ and $x_2$ does $x_n$ attain integer values for infinitely many $n$?	fun (a, b) => ∃ m : ℤ, a = m ∧ b = m	We must have $x_1 = x_2 = m$ for some integer $m$.	null	[]	@Eq (Prod Real Real → Prop) answer fun (x_1 : Prod Real Real) => _example.match_1 (fun (x_2 : Prod Real Real) => Prop) x_1 fun (a b : Real) => @Exists Int fun (m : Int) => And (@Eq Real a (@Int.cast Real Real.instIntCast m)) (@Eq Real b (@Int.cast Real Real.instIntCast m))	(ℝ × ℝ) → Prop	[ { "t": "ℕ → ℝ", "v": null, "name": "x" }, { "t": "∀ n : ℕ, x n ≠ 0 ∧ (n ≥ 3 → x n = (x (n - 2))(x (n - 1))/(2(x (n - 2)) - (x (n - 1))))", "v": null, "name": "hx" }, { "t": "(∀ m : ℕ, ∃ n : ℕ, n > m ∧ ∃ a : ℤ, a = x n) ↔ answer (x 1, x 2)", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1979_a3", "tags": [ "algebra" ] }
Let $A$ be a set of $2n$ points in the plane, $n$ colored red and $n$ colored blue, such that no three points in $A$ are collinear. Must there exist $n$ closed straight line segments, each connecting one red and one blue point in $A$, such that no two of the $n$ line segments intersect?	True	Such line segments must exist.	open Set	[]	@Eq Prop answer True	Prop	[ { "t": "Finset (Fin 2 → ℝ) × Finset (Fin 2 → ℝ) → Prop", "v": null, "name": "A" }, { "t": "A = fun (R, B) => R.card = B.card ∧ R ∩ B = ∅ ∧\n ∀ u : Finset (Fin 2 → ℝ), u ⊆ R ∪ B → u.card = 3 → ¬Collinear ℝ (u : Set (Fin 2 → ℝ))", "v": null, "name": "hA" }, { "t": "(Fin 2 → ℝ) × (Fin 2 → ℝ) → ℝ → (Fin 2 → ℝ)", "v": null, "name": "w" }, { "t": "w = fun (P, Q) => fun x : ℝ => fun i : Fin 2 => x * P i + (1 - x) * Q i", "v": null, "name": "hw" }, { "t": "answer ↔\n (∀ R B, A (R, B) →\n ∃ v : Finset ((Fin 2 → ℝ) × (Fin 2 → ℝ)),\n (∀ L ∈ v, ∀ M ∈ v, L ≠ M → ∀ x ∈ Icc 0 1, ∀ y ∈ Icc 0 1,\n Real.sqrt ((w (L.1, L.2) x 0 - w (M.1, M.2) y 0)^2 + (w (L.1, L.2) x 1 - w (M.1, M.2) y 1)^2) ≠ 0) ∧\n v.card = R.card ∧ ∀ L ∈ v, L.1 ∈ R ∧ L.2 ∈ B)", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1979_a4", "tags": [ "geometry", "combinatorics" ] }
If $0 < a < b$, find $$\lim_{t \to 0} \left( \int_{0}^{1}(bx + a(1-x))^t dx \right)^{\frac{1}{t}}$$ in terms of $a$ and $b$.	fun (a, b) => (Real.exp (-1))*(b^b/a^a)^(1/(b-a))	The limit equals $$e^{-1}\left(\frac{b^b}{a^a}\right)^{\frac{1}{b-a}}.$$	open Set Topology Filter	[]	@Eq (Prod Real Real → Real) answer fun (x : Prod Real Real) => _example.match_1 (fun (x_1 : Prod Real Real) => Real) x fun (a b : Real) => @HMul.hMul Real Real Real (@instHMul Real Real.instMul) (Real.exp (@Neg.neg Real Real.instNeg (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)))) (@HPow.hPow Real Real Real (@instHPow Real Real Real.instPow) (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) (@HPow.hPow Real Real Real (@instHPow Real Real Real.instPow) b b) (@HPow.hPow Real Real Real (@instHPow Real Real Real.instPow) a a)) (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)) (@HSub.hSub Real Real Real (@instHSub Real Real.instSub) b a)))	ℝ × ℝ → ℝ	[ { "t": "∀ a b : ℝ, 0 < a ∧ a < b → Tendsto (fun t : ℝ => (∫ x in Icc 0 1, (bx + a(1 - x))^t)^(1/t)) (𝓝[≠] 0) (𝓝 (answer (a, b)))", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1979_b2", "tags": [ "analysis" ] }
Let $F$ be a finite field with $n$ elements, and assume $n$ is odd. Suppose $x^2 + bx + c$ is an irreducible polynomial over $F$. For how many elements $d \in F$ is $x^2 + bx + c + d$ irreducible?	fun n : ℕ ↦ (n - (1 : ℤ)) / 2	Show that there are $\frac{n - 1}{2}$ such elements $d$.	open Set Topology Filter Polynomial	[]	@Eq (Nat → Int) answer fun (n_1 : Nat) => @HDiv.hDiv Int Int Int (@instHDiv Int Int.instDiv) (@HSub.hSub Int Int Int (@instHSub Int Int.instSub) (@Nat.cast Int instNatCastInt n_1) (@OfNat.ofNat Int (nat_lit 1) (@instOfNat (nat_lit 1)))) (@OfNat.ofNat Int (nat_lit 2) (@instOfNat (nat_lit 2)))	ℕ → ℤ	[ { "t": "Type", "v": null, "name": "F" }, { "t": "Field F", "v": null, "name": null }, { "t": "Fintype F", "v": null, "name": null }, { "t": "ℕ", "v": null, "name": "n" }, { "t": "n = Fintype.card F", "v": null, "name": "hn" }, { "t": "Odd n", "v": null, "name": "nodd" }, { "t": "F", "v": null, "name": "b" }, { "t": "F", "v": null, "name": "c" }, { "t": "Polynomial F", "v": null, "name": "p" }, { "t": "p = X ^ 2 + (C b) X + (C c) ∧ Irreducible p", "v": null, "name": "hp" }, { "t": "{d : F \| Irreducible (p + (C d))}.ncard = answer n", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1979_b3", "tags": [ "abstract_algebra" ] }
Let $r$ and $s$ be positive integers. Derive a formula for the number of ordered quadruples $(a,b,c,d)$ of positive integers such that $3^r \cdot 7^s=\text{lcm}[a,b,c]=\text{lcm}[a,b,d]=\text{lcm}[a,c,d]=\text{lcm}[b,c,d]$. The answer should be a function of $r$ and $s$. (Note that $\text{lcm}[x,y,z]$ denotes the least common multiple of $x,y,z$.)	(fun r s : ℕ => (1 + 4 * r + 6 * r ^ 2) * (1 + 4 * s + 6 * s ^ 2))	Show that the number is $(1+4r+6r^2)(1+4s+6s^2)$.	null	[]	@Eq (Nat → Nat → Nat) answer fun (r_1 s_1 : Nat) => @HMul.hMul Nat Nat Nat (@instHMul Nat instMulNat) (@HAdd.hAdd Nat Nat Nat (@instHAdd Nat instAddNat) (@HAdd.hAdd Nat Nat Nat (@instHAdd Nat instAddNat) (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))) (@HMul.hMul Nat Nat Nat (@instHMul Nat instMulNat) (@OfNat.ofNat Nat (nat_lit 4) (instOfNatNat (nat_lit 4))) r_1)) (@HMul.hMul Nat Nat Nat (@instHMul Nat instMulNat) (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) (@HPow.hPow Nat Nat Nat (@instHPow Nat Nat (@Monoid.toNatPow Nat Nat.instMonoid)) r_1 (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))) (@HAdd.hAdd Nat Nat Nat (@instHAdd Nat instAddNat) (@HAdd.hAdd Nat Nat Nat (@instHAdd Nat instAddNat) (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))) (@HMul.hMul Nat Nat Nat (@instHMul Nat instMulNat) (@OfNat.ofNat Nat (nat_lit 4) (instOfNatNat (nat_lit 4))) s_1)) (@HMul.hMul Nat Nat Nat (@instHMul Nat instMulNat) (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) (@HPow.hPow Nat Nat Nat (@instHPow Nat Nat (@Monoid.toNatPow Nat Nat.instMonoid)) s_1 (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))))	ℕ → ℕ → ℕ	[ { "t": "ℕ", "v": null, "name": "r" }, { "t": "ℕ", "v": null, "name": "s" }, { "t": "ℕ → ℕ → ℕ → ℕ → Prop", "v": null, "name": "abcdlcm" }, { "t": "r > 0 ∧ s > 0", "v": null, "name": "rspos" }, { "t": "∀ a b c d : ℕ, abcdlcm a b c d ↔\n (a > 0 ∧ b > 0 ∧ c > 0 ∧ d > 0 ∧\n (3 ^ r * 7 ^ s = Nat.lcm (Nat.lcm a b) c) ∧\n (3 ^ r * 7 ^ s = Nat.lcm (Nat.lcm a b) d) ∧\n (3 ^ r * 7 ^ s = Nat.lcm (Nat.lcm a c) d) ∧\n (3 ^ r * 7 ^ s = Nat.lcm (Nat.lcm b c) d))", "v": null, "name": "habcdlcm" }, { "t": "{h : ℕ × ℕ × ℕ × ℕ \| abcdlcm h.1 h.2.1 h.2.2.1 h.2.2.2}.encard = answer r s", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1980_a2", "tags": [ "number_theory" ] }
Evaluate $\int_0^{\pi/2}\frac{dx}{1+(\tan x)^{\sqrt{2}}}$.	$\pi / 4$	Show that the integral is $\pi/4$.	null	[]	@Eq Real answer (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) Real.pi (@OfNat.ofNat Real (nat_lit 4) (@instOfNatAtLeastTwo Real (nat_lit 4) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))))	ℝ	[ { "t": "answer = ∫ x in Set.Ioo 0 (Real.pi / 2), 1 / (1 + (Real.tan x) ^ (Real.sqrt 2))", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1980_a3", "tags": [ "analysis" ] }
Let $C$ be the class of all real valued continuously differentiable functions $f$ on the interval $0 \leq x \leq 1$ with $f(0)=0$ and $f(1)=1$. Determine the largest real number $u$ such that $u \leq \int_0^1\|f'(x)-f(x)\|\,dx$ for all $f$ in $C$.	$1/e$	Show that $u=1/e$.	null	[]	@Eq Real answer (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)) (Real.exp (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))))	ℝ	[ { "t": "Set (ℝ → ℝ)", "v": null, "name": "C" }, { "t": "C = {f : ℝ → ℝ \| ContDiffOn ℝ 1 f (Set.Icc 0 1) ∧ f 0 = 0 ∧ f 1 = 1}", "v": null, "name": "hC" }, { "t": "IsGreatest {u : ℝ \| ∀ f ∈ C, u ≤ (∫ x in Set.Ioo 0 1, \|deriv f x - f x\|)} answer", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1980_a6", "tags": [ "analysis" ] }
For which real numbers $c$ is $(e^x+e^{-x})/2 \leq e^{cx^2}$ for all real $x$?	$\{c : \mathbb{R} \mid c \geq 1/2\}$	Show that the inequality holds if and only if $c \geq 1/2$.	open Real	[]	@Eq (Set Real) answer (@setOf Real fun (c_1 : Real) => @GE.ge Real Real.instLE c_1 (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)) (@OfNat.ofNat Real (nat_lit 2) (@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))))))))	Set ℝ	[ { "t": "ℝ", "v": null, "name": "c" }, { "t": "∀ x : ℝ, (exp x + exp (-x)) / 2 ≤ exp (c * x ^ 2) ↔ c ∈ answer", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1980_b1", "tags": [ "analysis" ] }
For which real numbers $a$ does the sequence defined by the initial condition $u_0=a$ and the recursion $u_{n+1}=2u_n-n^2$ have $u_n>0$ for all $n \geq 0$? (Express the answer in the simplest form.)	$\{a : \mathbb{R} \mid a \geq 3\}$	Show that $u_n>0$ for all $n \geq 0$ if and only if $a \geq 3$.	null	[]	@Eq (Set Real) answer (@setOf Real fun (a_1 : Real) => @GE.ge Real Real.instLE a_1 (@OfNat.ofNat Real (nat_lit 3) (@instOfNatAtLeastTwo Real (nat_lit 3) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))))	Set ℝ	[ { "t": "ℝ", "v": null, "name": "a" }, { "t": "ℕ → ℝ", "v": null, "name": "u" }, { "t": "u 0 = a ∧ (∀ n : ℕ, u (n + 1) = 2 * u n - n ^ 2)", "v": null, "name": "hu" }, { "t": "(∀ n : ℕ, u n > 0) ↔ a ∈ answer", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1980_b3", "tags": [ "algebra" ] }
A function $f$ is convex on $[0, 1]$ if and only if $$f(su + (1-s)v) \le sf(u) + (1 - s)f(v)$$ for all $s \in [0, 1]$. Let $S_t$ denote the set of all nonnegative increasing convex continuous functions $f : [0, 1] \rightarrow \mathbb{R}$ such that $$f(1) - 2f\left(\frac{2}{3}\right) + f\left(\frac{1}{3}\right) \ge t\left(f\left(\frac{2}{3}\right) - 2f\left(\frac{1}{3}\right) + f(0)\right).$$ For which real numbers $t \ge 0$ is $S_t$ closed under multiplication?	$t \le 1$	$S_t$ is closed under multiplication if and only if $1 \ge t$.	open Set	[]	@Eq (Real → Prop) answer fun (t_1 : Real) => @LE.le Real Real.instLE t_1 (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))	ℝ → Prop	[ { "t": "Set ℝ", "v": null, "name": "T" }, { "t": "T = Icc 0 1", "v": null, "name": "hT" }, { "t": "ℝ → (ℝ → ℝ) → Prop", "v": null, "name": "P" }, { "t": "(ℝ → ℝ) → Prop", "v": null, "name": "IsConvex" }, { "t": "ℝ → Set (ℝ → ℝ)", "v": null, "name": "S" }, { "t": "∀ t f, P t f ↔ f 1 - 2f (2/3) + f (1/3) ≥ t(f (2/3) - 2f (1/3) + f 0)", "v": null, "name": "P_def" }, { "t": "∀ f, IsConvex f ↔ ∀ u ∈ T, ∀ v ∈ T, ∀ s ∈ T, f (su + (1 - s)v) ≤ s(f u) + (1 - s)(f v)", "v": null, "name": "IsConvex_def" }, { "t": "S = fun t : ℝ => {f : ℝ → ℝ \| (∀ x ∈ T, f x ≥ 0) ∧ StrictMonoOn f T ∧ IsConvex f ∧ ContinuousOn f T ∧ P t f}", "v": null, "name": "hS" }, { "t": "ℝ", "v": null, "name": "t" }, { "t": "t ≥ 0", "v": null, "name": "ht" }, { "t": "answer t ↔ (∀ f ∈ S t, ∀ g ∈ S t, f g ∈ S t)", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1980_b5", "tags": [ "analysis", "algebra" ] }
Let $E(n)$ be the greatest integer $k$ such that $5^k$ divides $1^1 2^2 3^3 \cdots n^n$. Find $\lim_{n \rightarrow \infty} \frac{E(n)}{n^2}$.	$\frac{1}{8}$	The limit equals $\frac{1}{8}$.	open Topology Filter Set Polynomial Function	[]	@Eq Real answer (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)) (@OfNat.ofNat Real (nat_lit 8) (@instOfNatAtLeastTwo Real (nat_lit 8) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))))))	ℝ	[ { "t": "ℕ → ℕ → Prop", "v": null, "name": "P" }, { "t": "∀ n k, P n k ↔ 5^k ∣ ∏ m in Finset.Icc 1 n, (m^m : ℤ)", "v": null, "name": "hP" }, { "t": "ℕ → ℕ", "v": null, "name": "E" }, { "t": "∀ n ∈ Ici 1, P n (E n) ∧ ∀ k : ℕ, P n k → k ≤ E n", "v": null, "name": "hE" }, { "t": "Tendsto (fun n : ℕ => ((E n) : ℝ)/n^2) atTop (𝓝 answer)", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1981_a1", "tags": [ "analysis", "number_theory" ] }
Does the limit $$lim_{t \rightarrow \infty}e^{-t}\int_{0}^{t}\int_{0}^{t}\frac{e^x - e^y}{x - y} dx dy$$exist?	False	The limit does not exist.	open Topology Filter Set Polynomial Function	[]	@Eq Prop answer False	Prop	[ { "t": "ℝ → ℝ", "v": null, "name": "f" }, { "t": "f = fun t : ℝ => Real.exp (-t) * ∫ y in (Ico 0 t), ∫ x in (Ico 0 t), (Real.exp x - Real.exp y) / (x - y)", "v": null, "name": "hf" }, { "t": "(∃ L : ℝ, Tendsto f atTop (𝓝 L)) ↔ answer", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1981_a3", "tags": [ "analysis" ] }
Let $P(x)$ be a polynomial with real coefficients; let $$Q(x) = (x^2 + 1)P(x)P'(x) + x((P(x))^2 + (P'(x))^2).$$ Given that $P$ has $n$ distinct real roots all greater than $1$, prove or disprove that $Q$ must have at least $2n - 1$ distinct real roots.	True	$Q(x)$ must have at least $2n - 1$ distinct real roots.	open Topology Filter Set Polynomial Function	[]	@Eq Prop answer True	Prop	[ { "t": "Polynomial ℝ → Polynomial ℝ", "v": null, "name": "Q" }, { "t": "Q = fun P : Polynomial ℝ => (Polynomial.X^2 + 1) * P * (Polynomial.derivative P) + Polynomial.X * (P^2 + (Polynomial.derivative P)^2)", "v": null, "name": "hQ" }, { "t": "Polynomial ℝ → ℝ", "v": null, "name": "n" }, { "t": "n = fun P : Polynomial ℝ => ({x ∈ Ioi 1 \| P.eval x = 0}.ncard : ℝ)", "v": null, "name": "hn" }, { "t": "answer ↔ (∀ P : Polynomial ℝ, {x : ℝ \| (Q P).eval x = 0}.ncard ≥ 2*(n P) - 1)", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1981_a5", "tags": [ "algebra" ] }
Find the value of $$\lim_{n \rightarrow \infty} \frac{1}{n^5}\sum_{h=1}^{n}\sum_{k=1}^{n}(5h^4 - 18h^2k^2 + 5k^4).$$	-1	The limit equals $-1$.	open Topology Filter Set Polynomial Function	[]	@Eq Real answer (@Neg.neg Real Real.instNeg (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)))	ℝ	[ { "t": "ℕ → ℝ", "v": null, "name": "f" }, { "t": "f = fun n : ℕ => ((1 : ℝ)/n^5) * ∑ h in Finset.Icc 1 n, ∑ k in Finset.Icc 1 n, (5(h : ℝ)^4 - 18h^2k^2 + 5k^4)", "v": null, "name": "hf" }, { "t": "Tendsto f atTop (𝓝 answer)", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1981_b1", "tags": [ "analysis" ] }
Determine the minimum value attained by $$(r - 1)^2 + (\frac{s}{r} - 1)^2 + (\frac{t}{s} - 1)^2 + (\frac{4}{t} - 1)^2$$ across all choices of real $r$, $s$, and $t$ that satisfy $1 \le r \le s \le t \le 4$.	12 - 8 * Real.sqrt 2	The minimum is $12 - 8\sqrt{2}$.	open Topology Filter Set Polynomial Function	[]	@Eq Real answer (@HSub.hSub Real Real Real (@instHSub Real Real.instSub) (@OfNat.ofNat Real (nat_lit 12) (@instOfNatAtLeastTwo Real (nat_lit 12) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 10) (instOfNatNat (nat_lit 10)))))) (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) (@OfNat.ofNat Real (nat_lit 8) (@instOfNatAtLeastTwo Real (nat_lit 8) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))))) (Real.sqrt (@OfNat.ofNat Real (nat_lit 2) (@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0)))))))))	ℝ	[ { "t": "ℝ × ℝ × ℝ → Prop", "v": null, "name": "P" }, { "t": "P = fun (r, s, t) => 1 ≤ r ∧ r ≤ s ∧ s ≤ t ∧ t ≤ 4", "v": null, "name": "hP" }, { "t": "ℝ × ℝ × ℝ → ℝ", "v": null, "name": "f" }, { "t": "f = fun (r, s, t) => (r - 1)^2 + (s/r - 1)^2 + (t/s - 1)^2 + (4/t - 1)^2", "v": null, "name": "hf" }, { "t": "IsLeast {y \| ∃ r s t, P (r, s, t) ∧ f (r, s, t) = y} answer", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1981_b2", "tags": [ "algebra" ] }
Let $V$ be a set of $5$ by $7$ matrices, with real entries and with the property that $rA+sB \in V$ whenever $A,B \in V$ and $r$ and $s$ are scalars (i.e., real numbers). \emph{Prove or disprove} the following assertion: If $V$ contains matrices of ranks $0$, $1$, $2$, $4$, and $5$, then it also contains a matrix of rank $3$. [The rank of a nonzero matrix $M$ is the largest $k$ such that the entries of some $k$ rows and some $k$ columns form a $k$ by $k$ matrix with a nonzero determinant.]	False	Show that the assertion is false.	open Topology Filter Set Polynomial Function	[]	@Eq Prop answer False	Prop	[ { "t": "Set (Matrix (Fin 5) (Fin 7) ℝ)", "v": null, "name": "V" }, { "t": "∀ A ∈ V, ∀ B ∈ V, ∀ r s : ℝ, r • A + s • B ∈ V", "v": null, "name": "hVAB" }, { "t": "∃ A ∈ V, A.rank = 0", "v": null, "name": "hVrank0" }, { "t": "∃ A ∈ V, A.rank = 1", "v": null, "name": "hVrank1" }, { "t": "∃ A ∈ V, A.rank = 2", "v": null, "name": "hVrank2" }, { "t": "∃ A ∈ V, A.rank = 4", "v": null, "name": "hVrank4" }, { "t": "∃ A ∈ V, A.rank = 5", "v": null, "name": "hVrank5" }, { "t": "answer ↔ ∃ A ∈ V, A.rank = 3", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1981_b4", "tags": [ "linear_algebra" ] }
Let $B(n)$ be the number of ones in the base two expression for the positive integer $n$. For example, $B(6)=B(110_2)=2$ and $B(15)=B(1111_2)=4$. Determine whether or not $\exp \left(\sum_{n=1}^\infty \frac{B(n)}{n(n+1)}\right)$ is a rational number. Here $\exp(x)$ denotes $e^x$.	True	Show that the expression is a rational number.	open Topology Filter Set Polynomial Function	[]	@Eq Prop answer True	Prop	[ { "t": "List ℕ → ℤ", "v": null, "name": "sumbits" }, { "t": "ℕ → ℤ", "v": null, "name": "B" }, { "t": "∀ bits : List ℕ, sumbits bits = ∑ i : Fin bits.length, (bits[i] : ℤ)", "v": null, "name": "hsumbits" }, { "t": "∀ n > 0, B n = sumbits (Nat.digits 2 n)", "v": null, "name": "hB" }, { "t": "answer ↔ (∃ q : ℚ, Real.exp (∑' n : Set.Ici 1, B n / ((n : ℝ) * ((n : ℝ) + 1))) = q)", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1981_b5", "tags": [ "analysis", "algebra" ] }
Let $B_n(x) = 1^x + 2^x + \dots + n^x$ and let $f(n) = \frac{B_n(\log_n 2)}{(n \log_2 n)^2}$. Does $f(2) + f(3) + f(4) + \dots$ converge?	True	Prove that the series converges.	open Set Function Filter Topology Polynomial Real	[]	@Eq Prop answer True	Prop	[ { "t": "ℕ → ℝ → ℝ", "v": null, "name": "B" }, { "t": "B = fun (n : ℕ) (x : ℝ) ↦ ∑ k in Finset.Icc 1 n, (k : ℝ) ^ x", "v": null, "name": "hB" }, { "t": "ℕ → ℝ", "v": null, "name": "f" }, { "t": "f = fun n ↦ B n (logb n 2) / (n * logb 2 n) ^ 2", "v": null, "name": "hf" }, { "t": "answer ↔ (∃ L : ℝ, Tendsto (fun N ↦ ∑ j in Finset.Icc 2 N, f j) atTop (𝓝 L))", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1982_a2", "tags": [ "algebra" ] }
Evaluate $\int_0^{\infty} \frac{\tan^{-1}(\pi x) - \tan^{-1} x}{x} \, dx$.	$\frac{\pi}{2} \log \pi$	Show that the integral evaluates to $\frac{\pi}{2} \ln \pi$.	open Set Function Filter Topology Polynomial Real	[]	@Eq Real answer (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) Real.pi (@OfNat.ofNat Real (nat_lit 2) (@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))))))) (Real.log Real.pi))	ℝ	[ { "t": "Tendsto (fun t ↦ ∫ x in (0)..t, (arctan (Real.pi * x) - arctan x) / x) atTop (𝓝 answer)", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1982_a3", "tags": [ "analysis" ] }
Let $b$ be a bijection from the positive integers to the positive integers. Also, let $x_1, x_2, x_3, \dots$ be an infinite sequence of real numbers with the following properties: \begin{enumerate} \item $\|x_n\|$ is a strictly decreasing function of $n$; \item $\lim_{n \rightarrow \infty} \|b(n) - n\| \cdot \|x_n\| = 0$; \item $\lim_{n \rightarrow \infty}\sum_{k = 1}^{n} x_k = 1$. \end{enumerate} Prove or disprove: these conditions imply that $$\lim_{n \rightarrow \infty} \sum_{k = 1}^{n} x_{b(k)} = 1.$$	False	The limit need not equal $1$.	open Set Function Filter Topology Polynomial Real	[]	@Eq Prop answer False	Prop	[ { "t": "ℕ → ℕ", "v": null, "name": "b" }, { "t": "ℕ → ℝ", "v": null, "name": "x" }, { "t": "BijOn b (Ici 1) (Ici 1)", "v": null, "name": "h_bij" }, { "t": "StrictAntiOn (fun n : ℕ => \|x n\|) (Ici 1)", "v": null, "name": "h_strict_anti" }, { "t": "Tendsto (fun n : ℕ => \|b n - (n : ℤ)\| * \|x n\|) atTop (𝓝 0)", "v": null, "name": "h_limit" }, { "t": "Tendsto (fun n : ℕ => ∑ k in Finset.Icc 1 n, x k) atTop (𝓝 1)", "v": null, "name": "h_sum_limit" }, { "t": "(Tendsto (fun n : ℕ => ∑ k in Finset.Icc 1 n, x (b k)) atTop (𝓝 1)) ↔ answer", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1982_a6", "tags": [ "analysis" ] }
Let $A(x, y)$ denote the number of points $(m, n)$ with integer coordinates $m$ and $n$ where $m^2 + n^2 \le x^2 + y^2$. Also, let $g = \sum_{k = 0}^{\infty} e^{-k^2}$. Express the value $$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} A(x, y)e^{-x^2 - y^2} dx dy$$ as a polynomial in $g$.	C Real.pi * (2*X - 1)^2	The desired polynomial is $\pi(2g - 1)^2$.	open Set Function Filter Topology Polynomial Real	[]	@Eq (@Polynomial Real Real.semiring) answer (@HMul.hMul (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@instHMul (@Polynomial Real Real.semiring) (@Polynomial.mul' Real Real.semiring)) (@DFunLike.coe (@RingHom Real (@Polynomial Real Real.semiring) (@Semiring.toNonAssocSemiring Real Real.semiring) (@Semiring.toNonAssocSemiring (@Polynomial Real Real.semiring) (@Polynomial.semiring Real Real.semiring))) Real (fun (x : Real) => @Polynomial Real Real.semiring) (@RingHom.instFunLike Real (@Polynomial Real Real.semiring) (@Semiring.toNonAssocSemiring Real Real.semiring) (@Semiring.toNonAssocSemiring (@Polynomial Real Real.semiring) (@Polynomial.semiring Real Real.semiring))) (@Polynomial.C Real Real.semiring) Real.pi) (@HPow.hPow (@Polynomial Real Real.semiring) Nat (@Polynomial Real Real.semiring) (@instHPow (@Polynomial Real Real.semiring) Nat (@Monoid.toNatPow (@Polynomial Real Real.semiring) (@MonoidWithZero.toMonoid (@Polynomial Real Real.semiring) (@Semiring.toMonoidWithZero (@Polynomial Real Real.semiring) (@Polynomial.semiring Real Real.semiring))))) (@HSub.hSub (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@instHSub (@Polynomial Real Real.semiring) (@Polynomial.sub Real Real.instRing)) (@HMul.hMul (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@instHMul (@Polynomial Real Real.semiring) (@Polynomial.mul' Real Real.semiring)) (@OfNat.ofNat (@Polynomial Real Real.semiring) (nat_lit 2) (@instOfNatAtLeastTwo (@Polynomial Real Real.semiring) (nat_lit 2) (@Polynomial.instNatCast Real Real.semiring) (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0)))))) (@Polynomial.X Real Real.semiring)) (@OfNat.ofNat (@Polynomial Real Real.semiring) (nat_lit 1) (@One.toOfNat1 (@Polynomial Real Real.semiring) (@Polynomial.one Real Real.semiring)))) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))	Polynomial ℝ	[ { "t": "ℝ × ℝ → ℕ", "v": null, "name": "A" }, { "t": "ℝ", "v": null, "name": "g" }, { "t": "ℝ", "v": null, "name": "I" }, { "t": "A = fun (x, y) => {a : ℤ × ℤ \| a.1^2 + a.2^2 ≤ x^2 + y^2}.ncard", "v": null, "name": "hA" }, { "t": "g = ∑' k : ℕ, Real.exp (-k^2)", "v": null, "name": "hg" }, { "t": "I = ∫ y : ℝ, ∫ x : ℝ, A (x, y) * Real.exp (-x^2 - y^2)", "v": null, "name": "hI" }, { "t": "I = answer.eval g", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1982_b2", "tags": [ "analysis" ] }
Let $p_n$ denote the probability that $c + d$ will be a perfect square if $c$ and $d$ are selected independently and uniformly at random from $\{1, 2, 3, \dots, n\}$. Express $\lim_{n \rightarrow \infty} p_n \sqrt{n}$ in the form $r(\sqrt{s} - t)$ for integers $s$ and $t$ and rational $r$.	4/3 * (Real.sqrt 2 - 1)	The limit equals $\frac{4}{3}(\sqrt{2} - 1)$.	open Set Function Filter Topology Polynomial Real	[]	@Eq Real answer (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) (@OfNat.ofNat Real (nat_lit 4) (@instOfNatAtLeastTwo Real (nat_lit 4) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))) (@OfNat.ofNat Real (nat_lit 3) (@instOfNatAtLeastTwo Real (nat_lit 3) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))))) (@HSub.hSub Real Real Real (@instHSub Real Real.instSub) (Real.sqrt (@OfNat.ofNat Real (nat_lit 2) (@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))))))) (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))))	ℝ	[ { "t": "ℕ → ℝ", "v": null, "name": "p" }, { "t": "p = fun n : ℕ => ({c : Finset.Icc 1 n × Finset.Icc 1 n \| ∃ m : ℕ, m^2 = c.1 + c.2}.ncard : ℝ) / n^2", "v": null, "name": "hp" }, { "t": "Tendsto (fun n : ℕ => p n * Real.sqrt n) atTop (𝓝 answer)", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1982_b3", "tags": [ "analysis", "number_theory", "probability" ] }
Let $n_1, n_2, \dots, n_s$ be distinct integers such that, for every integer $k$, $n_1n_2\cdots n_s$ divides $(n_1 + k)(n_2 + k) \cdots (n_s + k)$. Prove or provide a counterexample to the following claims: \begin{enumerate} \item For some $i$, $\|n_i\| = 1$. \item If all $n_i$ are positive, then $\{n_1, n_2, \dots, n_s\} = \{1, 2, \dots, s\}$. \end{enumerate}	(True, True)	Both claims are true.	open Set Function Filter Topology Polynomial Real	[]	@Eq (Prod Prop Prop) answer (@Prod.mk Prop Prop True True)	Prop × Prop	[ { "t": "Finset ℤ → Prop", "v": null, "name": "P" }, { "t": "∀ n, P n ↔ n.Nonempty ∧ ∀ k, ∏ i in n, i ∣ ∏ i in n, (i + k)", "v": null, "name": "P_def" }, { "t": "((∀ n, P n → 1 ∈ n ∨ -1 ∈ n) ↔ answer.1) ∧\n ((∀ n, P n → (∀ i ∈ n, 0 < i) → n = Finset.Icc (1 : ℤ) n.card) ↔ answer.2)", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1982_b4", "tags": [ "number_theory" ] }
How many positive integers $n$ are there such that $n$ is an exact divisor of at least one of the numbers $10^{40},20^{30}$?	2301	Show that the desired count is $2301$.	null	[]	@Eq Nat answer (@OfNat.ofNat Nat (nat_lit 2301) (instOfNatNat (nat_lit 2301)))	ℕ	[ { "t": "{n : ℤ \| n > 0 ∧ (n ∣ 10 ^ 40 ∨ n ∣ 20 ^ 30)}.encard = answer", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1983_a1", "tags": [ "number_theory" ] }
Prove or disprove that there exists a positive real number $\alpha$ such that $[\alpha_n] - n$ is even for all integers $n > 0$. (Here $[x]$ denotes the greatest integer less than or equal to $x$.)	True	Prove that such an $\alpha$ exists.	open Nat	[]	@Eq Prop answer True	Prop	[ { "t": "answer ↔ (∃ α : ℝ, α > 0 ∧ ∀ n : ℕ, n > 0 → Even (⌊α ^ n⌋ - n))", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1983_a5", "tags": [ "analysis" ] }
Let $T$ be the triangle with vertices $(0, 0)$, $(a, 0)$, and $(0, a)$. Find $\lim_{a \to \infty} a^4 \exp(-a^3) \int_T \exp(x^3+y^3) \, dx \, dy$.	2 / 9	Show that the integral evaluates to $\frac{2}{9}$.	open Nat Filter Topology Real	[]	@Eq Real answer (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) (@OfNat.ofNat Real (nat_lit 2) (@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0)))))) (@OfNat.ofNat Real (nat_lit 9) (@instOfNatAtLeastTwo Real (nat_lit 9) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 7) (instOfNatNat (nat_lit 7)))))))	ℝ	[ { "t": "ℝ → ℝ", "v": null, "name": "F" }, { "t": "F = fun a ↦ (a ^ 4 / exp (a ^ 3)) * ∫ x in (0)..a, ∫ y in (0)..(a - x), exp (x ^ 3 + y ^ 3)", "v": null, "name": "hF" }, { "t": "Tendsto F atTop (𝓝 answer)", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1983_a6", "tags": [ "analysis" ] }
Let $f(n)$ be the number of ways of representing $n$ as a sum of powers of $2$ with no power being used more than $3$ times. For example, $f(7) = 4$ (the representations are $4 + 2 + 1$, $4 + 1 + 1 + 1$, $2 + 2 + 2 + 1$, $2 + 2 + 1 + 1 + 1$). Can we find a real polynomial $p(x)$ such that $f(n) = [p(n)]$, where $[u]$ denotes the greatest integer less than or equal to $u$?	True	Prove that such a polynomial exists.	open Nat Filter Topology Real	[]	@Eq Prop answer True	Prop	[ { "t": "ℕ+ → ℕ", "v": null, "name": "f" }, { "t": "f = fun (n : ℕ+) ↦\n Set.ncard {M : Multiset ℕ \|\n (∀ m ∈ M, ∃ k : ℕ, m = (2 ^ k : ℤ)) ∧ \n (∀ m ∈ M, M.count m ≤ 3) ∧ \n (M.sum : ℤ) = n}", "v": null, "name": "hf" }, { "t": "answer ↔ (∃ p : Polynomial ℝ, ∀ n : ℕ+, ⌊p.eval (n : ℝ)⌋ = f n)", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1983_b2", "tags": [ "algebra" ] }
Define $\left\lVert x \right\rVert$ as the distance from $x$ to the nearest integer. Find $\lim_{n \to \infty} \frac{1}{n} \int_{1}^{n} \left\lVert \frac{n}{x} \right\rVert \, dx$. You may assume that $\prod_{n=1}^{\infty} \frac{2n}{(2n-1)} \cdot \frac{2n}{(2n+1)} = \frac{\pi}{2}$.	$\log \left(\frac{4}{\pi}\right)$	Show that the limit equals $\ln \left( \frac{4}{\pi} \right)$.	open Nat Filter Topology Real	[]	@Eq Real answer (Real.log (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) (@OfNat.ofNat Real (nat_lit 4) (@instOfNatAtLeastTwo Real (nat_lit 4) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))) Real.pi))	ℝ	[ { "t": "ℝ → ℝ", "v": null, "name": "dist_fun" }, { "t": "dist_fun = fun (x : ℝ) ↦ min (x - ⌊x⌋) (⌈x⌉ - x)", "v": null, "name": "hdist_fun" }, { "t": "Tendsto (fun N ↦ ∏ n in Finset.Icc 1 N, (2 * n / (2 * n - 1)) * (2 * n / (2 * n + 1)) : ℕ → ℝ) atTop (𝓝 (Real.pi / 2))", "v": null, "name": "fact" }, { "t": "Tendsto (fun n ↦ (1 / n) * ∫ x in (1)..n, dist_fun (n / x) : ℕ → ℝ) atTop (𝓝 answer)", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1983_b5", "tags": [ "analysis" ] }
Express $\sum_{k=1}^\infty (6^k/(3^{k+1}-2^{k+1})(3^k-2^k))$ as a rational number.	2	Show that the sum converges to $2$.	null	[]	@Eq Rat answer (@OfNat.ofNat Rat (nat_lit 2) (@Rat.instOfNat (nat_lit 2)))	ℚ	[ { "t": "∑' k : Set.Ici 1, (6 ^ (k : ℕ) / ((3 ^ ((k : ℕ) + 1) - 2 ^ ((k : ℕ) + 1)) * (3 ^ (k : ℕ) - 2 ^ (k : ℕ)))) = answer", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1984_a2", "tags": [ "analysis" ] }
Let $n$ be a positive integer. Let $a,b,x$ be real numbers, with $a \neq b$, and let $M_n$ denote the $2n \times 2n$ matrix whose $(i,j)$ entry $m_{ij}$ is given by \[ m_{ij}=\begin{cases} x & \text{if }i=j, \\ a & \text{if }i \neq j\text{ and }i+j\text{ is even}, \\ b & \text{if }i \neq j\text{ and }i+j\text{ is odd}. \end{cases} \] Thus, for example, $M_2=\begin{pmatrix} x & b & a & b \\ b & x & b & a \\ a & b & x & b \\ b & a & b & x \end{pmatrix}$. Express $\lim_{x \to a} \det M_n/(x-a)^{2n-2}$ as a polynomial in $a$, $b$, and $n$, where $\det M_n$ denotes the determinant of $M_n$.	$(X_2)^2 \cdot ((X_0)^2 - (X_1)^2)$	Show that $\lim_{x \to a} \frac{\det M_n}{(x-a)^{2n-2}}=n^2(a^2-b^2)$.	open Topology Filter	[]	@Eq (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) answer (@HMul.hMul (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@instHMul (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@Distrib.toMul (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@NonUnitalNonAssocSemiring.toDistrib (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@NonUnitalNonAssocCommSemiring.toNonUnitalNonAssocSemiring (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@NonUnitalCommRing.toNonUnitalNonAssocCommRing (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@CommRing.toNonUnitalCommRing (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@MvPolynomial.instCommRingMvPolynomial Real (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real.commRing)))))))) (@HPow.hPow (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) Nat (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@instHPow (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) Nat (@Monoid.toNatPow (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@MonoidWithZero.toMonoid (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@Semiring.toMonoidWithZero (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@CommSemiring.toSemiring (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@MvPolynomial.commSemiring Real (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real.instCommSemiring)))))) (@MvPolynomial.X Real (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real.instCommSemiring (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) (nat_lit 2) (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3))) (@Nat.instNeZeroSucc (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) (nat_lit 2)))) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) (@HSub.hSub (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@instHSub (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@SubNegMonoid.toSub (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@AddGroup.toSubNegMonoid (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@AddGroupWithOne.toAddGroup (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@Ring.toAddGroupWithOne (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@CommRing.toRing (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@MvPolynomial.instCommRingMvPolynomial Real (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real.commRing))))))) (@HPow.hPow (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) Nat (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@instHPow (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) Nat (@Monoid.toNatPow (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@MonoidWithZero.toMonoid (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@Semiring.toMonoidWithZero (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@CommSemiring.toSemiring (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@MvPolynomial.commSemiring Real (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real.instCommSemiring)))))) (@MvPolynomial.X Real (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real.instCommSemiring (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) (nat_lit 0) (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3))) (@Nat.instNeZeroSucc (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) (nat_lit 0)))) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) (@HPow.hPow (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) Nat (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@instHPow (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) Nat (@Monoid.toNatPow (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@MonoidWithZero.toMonoid (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@Semiring.toMonoidWithZero (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@CommSemiring.toSemiring (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@MvPolynomial.commSemiring Real (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real.instCommSemiring)))))) (@MvPolynomial.X Real (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real.instCommSemiring (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) (nat_lit 1) (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3))) (@Nat.instNeZeroSucc (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) (nat_lit 1)))) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))))	MvPolynomial (Fin 3) ℝ	[ { "t": "ℕ", "v": null, "name": "n" }, { "t": "ℝ", "v": null, "name": "a" }, { "t": "ℝ", "v": null, "name": "b" }, { "t": "ℝ → Matrix (Fin (2 * n)) (Fin (2 * n)) ℝ", "v": null, "name": "Mn" }, { "t": "Fin 3 → ℝ", "v": null, "name": "polyabn" }, { "t": "n > 0", "v": null, "name": "npos" }, { "t": "a ≠ b", "v": null, "name": "aneb" }, { "t": "Mn = fun x : ℝ => fun i j : Fin (2 * n) => if i = j then x else if Even (i.1 + j.1) then a else b", "v": null, "name": "hMn" }, { "t": "polyabn 0 = a ∧ polyabn 1 = b ∧ polyabn 2 = n", "v": null, "name": "hpolyabn" }, { "t": "Tendsto (fun x : ℝ => (Mn x).det / (x - a) ^ (2 * n - 2)) (𝓝[≠] a) (𝓝 (MvPolynomial.eval polyabn answer))", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1984_a3", "tags": [ "linear_algebra", "analysis" ] }
Let $R$ be the region consisting of all triples $(x,y,z)$ of nonnegative real numbers satisfying $x+y+z \leq 1$. Let $w=1-x-y-z$. Express the value of the triple integral $\iiint_R x^1y^9z^8w^4\,dx\,dy\,dz$ in the form $a!b!c!d!/n!$, where $a$, $b$, $c$, $d$, and $n$ are positive integers.	(1, 9, 8, 4, 25)	Show that the integral we desire is $1!9!8!4!/25!$.	open Topology Filter Nat	[]	@Eq (Prod Nat (Prod Nat (Prod Nat (Prod Nat Nat)))) answer (@Prod.mk Nat (Prod Nat (Prod Nat (Prod Nat Nat))) (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))) (@Prod.mk Nat (Prod Nat (Prod Nat Nat)) (@OfNat.ofNat Nat (nat_lit 9) (instOfNatNat (nat_lit 9))) (@Prod.mk Nat (Prod Nat Nat) (@OfNat.ofNat Nat (nat_lit 8) (instOfNatNat (nat_lit 8))) (@Prod.mk Nat Nat (@OfNat.ofNat Nat (nat_lit 4) (instOfNatNat (nat_lit 4))) (@OfNat.ofNat Nat (nat_lit 25) (instOfNatNat (nat_lit 25)))))))	ℕ × ℕ × ℕ × ℕ × ℕ	[ { "t": "Set (Fin 3 → ℝ)", "v": null, "name": "R" }, { "t": "(Fin 3 → ℝ) → ℝ", "v": null, "name": "w" }, { "t": "R = {p \| (∀ i : Fin 3, p i ≥ 0) ∧ p 0 + p 1 + p 2 ≤ 1}", "v": null, "name": "hR" }, { "t": "∀ p, w p = 1 - p 0 - p 1 - p 2", "v": null, "name": "hw" }, { "t": "ℕ", "v": null, "name": "a" }, { "t": "ℕ", "v": null, "name": "b" }, { "t": "ℕ", "v": null, "name": "c" }, { "t": "ℕ", "v": null, "name": "d" }, { "t": "ℕ", "v": null, "name": "n" }, { "t": "answer = (a, b, c, d, n)", "v": null, "name": "h_answer" }, { "t": "a > 0 ∧ b > 0 ∧ c > 0 ∧ d > 0 ∧ n > 0", "v": null, "name": "h_pos" }, { "t": "(∫ p in R, (p 0) ^ 1 * (p 1) ^ 9 * (p 2) ^ 8 * (w p) ^ 4 = ((a)! * (b)! * (c)! * (d)! : ℝ) / (n)!)", "v": null, "name": "h_integral" } ]	{ "problem_name": "putnam_1984_a5", "tags": [ "analysis" ] }
Let $n$ be a positive integer, and let $f(n)$ denote the last nonzero digit in the decimal expansion of $n!$. For instance, $f(5)=2$. \begin{enumerate} \item[(a)] Show that if $a_1,a_2,\dots,a_k$ are \emph{distinct} nonnegative integers, then $f(5^{a_1}+5^{a_2}+\dots+5^{a_k})$ depends only on the sum $a_1+a_2+\dots+a_k$. \item[(b)] Assuming part (a), we can define $g(s)=f(5^{a_1}+5^{a_2}+\dots+5^{a_k})$, where $s=a_1+a_2+\dots+a_k$. Find the least positive integer $p$ for which $g(s)=g(s + p)$, for all $s \geq 1$, or else show that no such $p$ exists. \end{enumerate}	4	Show that the least such $p$ is $p=4$.	open Topology Filter Function Nat	[]	@Eq Nat answer (@OfNat.ofNat Nat (nat_lit 4) (instOfNatNat (nat_lit 4)))	ℕ	[ { "t": "ℕ → ℕ", "v": null, "name": "f" }, { "t": "∀ n, some (f n) = (Nat.digits 10 (n !)).find? (fun d ↦ d ≠ 0)", "v": null, "name": "hf" }, { "t": "ℕ → (ℕ → ℕ) → ℕ → Prop", "v": null, "name": "IsPeriodicFrom" }, { "t": "∀ x f p, IsPeriodicFrom x f p ↔ Periodic (f ∘ (· + x)) p", "v": null, "name": "IsPeriodicFrom_def" }, { "t": "ℕ → (ℕ → ℕ) → ℕ → Prop", "v": null, "name": "P" }, { "t": "∀ x g p, P x g p ↔ if p = 0 then (∀ q > 0, ¬ IsPeriodicFrom x g q) else\n IsLeast {q \| 0 < q ∧ IsPeriodicFrom x g q} p", "v": null, "name": "P_def" }, { "t": "∃ g : ℕ → ℕ,\n (∀ᵉ (k > 0) (a : Fin k → ℕ) (ha : Injective a), f (∑ i, 5 ^ (a i)) = g (∑ i, a i)) ∧\n P 1 g answer", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1984_a6", "tags": [ "algebra", "number_theory" ] }
Let $n$ be a positive integer, and define $f(n)=1!+2!+\dots+n!$. Find polynomials $P(x)$ and $Q(x)$ such that $f(n+2)=P(n)f(n+1)+Q(n)f(n)$ for all $n \geq 1$.	$(x + 3, -x - 2)$	Show that we can take $P(x)=x+3$ and $Q(x)=-x-2$.	open Topology Filter Nat	[]	@Eq (Prod (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring)) answer (@Prod.mk (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@HAdd.hAdd (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@instHAdd (@Polynomial Real Real.semiring) (@Polynomial.add' Real Real.semiring)) (@Polynomial.X Real Real.semiring) (@OfNat.ofNat (@Polynomial Real Real.semiring) (nat_lit 3) (@instOfNatAtLeastTwo (@Polynomial Real Real.semiring) (nat_lit 3) (@Polynomial.instNatCast Real Real.semiring) (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))))) (@HSub.hSub (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@instHSub (@Polynomial Real Real.semiring) (@Polynomial.sub Real Real.instRing)) (@Neg.neg (@Polynomial Real Real.semiring) (@Polynomial.neg' Real Real.instRing) (@Polynomial.X Real Real.semiring)) (@OfNat.ofNat (@Polynomial Real Real.semiring) (nat_lit 2) (@instOfNatAtLeastTwo (@Polynomial Real Real.semiring) (nat_lit 2) (@Polynomial.instNatCast Real Real.semiring) (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))))))))	Polynomial ℝ × Polynomial ℝ	[ { "t": "ℕ → ℤ", "v": null, "name": "f" }, { "t": "∀ n > 0, f n = ∑ i : Set.Icc 1 n, ((i)! : ℤ)", "v": null, "name": "hf" }, { "t": "∀ n ≥ 1, f (n + 2) = (answer.1).eval (n : ℝ) * f (n + 1) + (answer.2).eval (n : ℝ) * f n", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1984_b1", "tags": [ "algebra" ] }
Find the minimum value of $(u-v)^2+(\sqrt{2-u^2}-\frac{9}{v})^2$ for $0<u<\sqrt{2}$ and $v>0$.	8	Show that the minimum value is $8$.	open Topology Filter Nat	[]	@Eq Real answer (@OfNat.ofNat Real (nat_lit 8) (@instOfNatAtLeastTwo Real (nat_lit 8) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))))))	ℝ	[ { "t": "ℝ → ℝ → ℝ", "v": null, "name": "f" }, { "t": "∀ u v : ℝ, f u v = (u - v) ^ 2 + (Real.sqrt (2 - u ^ 2) - 9 / v) ^ 2", "v": null, "name": "hf" }, { "t": "IsLeast {y \| ∃ᵉ (u : Set.Ioo 0 √2) (v > 0), f u v = y} answer", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1984_b2", "tags": [ "geometry", "analysis" ] }
Prove or disprove the following statement: If $F$ is a finite set with two or more elements, then there exists a binary operation $$ on F such that for all $x,y,z$ in $F$, \begin{enumerate} \item[(i)] $xz=yz$ implies $x=y$ (right cancellation holds), and \item[(ii)] $x(yz) \neq (xy)*z$ (\emph{no} case of associativity holds). \end{enumerate}	True	Show that the statement is true.	open Topology Filter Nat	[]	@Eq Prop answer True	Prop	[ { "t": "(∀ (F : Type*) (_ : Fintype F), Fintype.card F ≥ 2 → (∃ mul : F → F → F, ∀ x y z : F, (mul x z = mul y z → x = y) ∧ (mul x (mul y z) ≠ mul (mul x y) z))) ↔ answer", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1984_b3", "tags": [ "abstract_algebra" ] }
For each nonnegative integer $k$, let $d(k)$ denote the number of $1$'s in the binary expansion of $k$ (for example, $d(0)=0$ and $d(5)=2$). Let $m$ be a positive integer. Express $\sum_{k=0}^{2^m-1} (-1)^{d(k)}k^m$ in the form $(-1)^ma^{f(m)}(g(m))!$, where $a$ is an integer and $f$ and $g$ are polynomials.	(2, (Polynomial.X * (Polynomial.X - 1)) / 2, Polynomial.X)	Show that $\sum_{k=0}^{2^m-1} (-1)^{d(k)}k^m=(-1)^m2^{m(m-1)/2}m!$.	open Topology Filter Nat	[]	@Eq (Prod Int (Prod (@Polynomial Real Real.semiring) (@Polynomial Nat Nat.instSemiring))) answer (@Prod.mk Int (Prod (@Polynomial Real Real.semiring) (@Polynomial Nat Nat.instSemiring)) (@OfNat.ofNat Int (nat_lit 2) (@instOfNat (nat_lit 2))) (@Prod.mk (@Polynomial Real Real.semiring) (@Polynomial Nat Nat.instSemiring) (@HDiv.hDiv (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@instHDiv (@Polynomial Real Real.semiring) (@Polynomial.instDiv Real Real.field)) (@HMul.hMul (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@instHMul (@Polynomial Real Real.semiring) (@Polynomial.mul' Real Real.semiring)) (@Polynomial.X Real Real.semiring) (@HSub.hSub (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@instHSub (@Polynomial Real Real.semiring) (@Polynomial.sub Real Real.instRing)) (@Polynomial.X Real Real.semiring) (@OfNat.ofNat (@Polynomial Real Real.semiring) (nat_lit 1) (@One.toOfNat1 (@Polynomial Real Real.semiring) (@Polynomial.one Real Real.semiring))))) (@OfNat.ofNat (@Polynomial Real Real.semiring) (nat_lit 2) (@instOfNatAtLeastTwo (@Polynomial Real Real.semiring) (nat_lit 2) (@Polynomial.instNatCast Real Real.semiring) (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))))))) (@Polynomial.X Nat Nat.instSemiring)))	ℤ × Polynomial ℝ × Polynomial ℕ	[ { "t": "ℕ", "v": null, "name": "m" }, { "t": "m > 0", "v": null, "name": "mpos" }, { "t": "ℕ → ℕ", "v": null, "name": "d" }, { "t": "List ℕ → ℕ", "v": null, "name": "sumbits" }, { "t": "∀ bits : List ℕ, sumbits bits = ∑ i : Fin bits.length, bits[i]", "v": null, "name": "hsumbits" }, { "t": "∀ k : ℕ, d k = sumbits (Nat.digits 2 k)", "v": null, "name": "hd" }, { "t": "let (a, f, g) := answer;\n ∑ k : Set.Icc 0 (2 ^ m - 1), (-(1 : ℤ)) ^ (d k) * (k : ℕ) ^ m = (-1) ^ m * (a : ℝ) ^ (f.eval (m : ℝ)) * (g.eval m)!", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1984_b5", "tags": [ "algebra", "analysis" ] }
Determine, with proof, the number of ordered triples $(A_1, A_2, A_3)$ of sets which have the property that \begin{enumerate} \item[(i)] $A_1 \cup A_2 \cup A_3 = \{1,2,3,4,5,6,7,8,9,10\}$, and \item[(ii)] $A_1 \cap A_2 \cap A_3 = \emptyset$. \end{enumerate} Express your answer in the form $2^a 3^b 5^c 7^d$, where $a,b,c,d$ are nonnegative integers.	(10, 10, 0, 0)	Prove that the number of such triples is $2^{10}3^{10}$.	open Set	[]	@Eq (Prod Nat (Prod Nat (Prod Nat Nat))) answer (@Prod.mk Nat (Prod Nat (Prod Nat Nat)) (@OfNat.ofNat Nat (nat_lit 10) (instOfNatNat (nat_lit 10))) (@Prod.mk Nat (Prod Nat Nat) (@OfNat.ofNat Nat (nat_lit 10) (instOfNatNat (nat_lit 10))) (@Prod.mk Nat Nat (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))) (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))))))	ℕ × ℕ × ℕ × ℕ	[ { "t": "let (a, b, c, d) := answer;\n {(A1, A2, A3) : Set ℤ × Set ℤ × Set ℤ \| A1 ∪ A2 ∪ A3 = Icc 1 10 ∧ A1 ∩ A2 ∩ A3 = ∅}.ncard = 2 ^ a * 3 ^ b * 5 ^ c * 7 ^ d", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1985_a1", "tags": [ "algebra" ] }
Let $d$ be a real number. For each integer $m \geq 0$, define a sequence $\{a_m(j)\}$, $j=0,1,2,\dots$ by the condition \begin{align} a_m(0) &= d/2^m, \\ a_m(j+1) &= (a_m(j))^2 + 2a_m(j), \qquad j \geq 0. \end{align} Evaluate $\lim_{n \to \infty} a_n(n)$.	$e^d - 1$	Show that the limit equals $e^d - 1$.	open Set Filter Topology Real	[]	@Eq (Real → Real) answer fun (d_1 : Real) => @HSub.hSub Real Real Real (@instHSub Real Real.instSub) (Real.exp d_1) (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))	ℝ → ℝ	[ { "t": "ℝ", "v": null, "name": "d" }, { "t": "ℕ → ℕ → ℝ", "v": null, "name": "a" }, { "t": "∀ m : ℕ, a m 0 = d / 2 ^ m", "v": null, "name": "ha0" }, { "t": "∀ m : ℕ, ∀ j : ℕ, a m (j + 1) = (a m j) ^ 2 + 2 * a m j", "v": null, "name": "ha" }, { "t": "Tendsto (fun n ↦ a n n) atTop (𝓝 (answer d))", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1985_a3", "tags": [ "analysis" ] }
Define a sequence $\{a_i\}$ by $a_1=3$ and $a_{i+1}=3^{a_i}$ for $i \geq 1$. Which integers between $00$ and $99$ inclusive occur as the last two digits in the decimal expansion of infinitely many $a_i$?	{87}	Prove that the only number that occurs infinitely often is $87$.	open Set Filter Topology Real	[]	@Eq (Set (Fin (@OfNat.ofNat Nat (nat_lit 100) (instOfNatNat (nat_lit 100))))) answer (@Singleton.singleton (Fin (@OfNat.ofNat Nat (nat_lit 100) (instOfNatNat (nat_lit 100)))) (Set (Fin (@OfNat.ofNat Nat (nat_lit 100) (instOfNatNat (nat_lit 100))))) (@Set.instSingletonSet (Fin (@OfNat.ofNat Nat (nat_lit 100) (instOfNatNat (nat_lit 100))))) (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 100) (instOfNatNat (nat_lit 100)))) (nat_lit 87) (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 100) (instOfNatNat (nat_lit 100))) (@Nat.instNeZeroSucc (@OfNat.ofNat Nat (nat_lit 99) (instOfNatNat (nat_lit 99)))) (nat_lit 87))))	Set (Fin 100)	[ { "t": "ℕ → ℕ", "v": null, "name": "a" }, { "t": "a 1 = 3", "v": null, "name": "ha1" }, { "t": "∀ i ≥ 1, a (i + 1) = 3 ^ a i", "v": null, "name": "ha" }, { "t": "{k : Fin 100 \| ∀ N : ℕ, ∃ i ≥ N, a i % 100 = k} = answer", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1985_a4", "tags": [ "number_theory" ] }
Let $I_m = \int_0^{2\pi} \cos(x)\cos(2x)\cdots \cos(mx)\,dx$. For which integers $m$, $1 \leq m \leq 10$ is $I_m \neq 0$?	{3, 4, 7, 8}	Prove that the integers $m$ with $1 \leq m \leq 10$ and $I_m \neq 0$ are $m = 3, 4, 7, 8$.	open Set Filter Topology Real	[]	@Eq (Set Nat) answer (@Insert.insert Nat (Set Nat) (@Set.instInsert Nat) (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3))) (@Insert.insert Nat (Set Nat) (@Set.instInsert Nat) (@OfNat.ofNat Nat (nat_lit 4) (instOfNatNat (nat_lit 4))) (@Insert.insert Nat (Set Nat) (@Set.instInsert Nat) (@OfNat.ofNat Nat (nat_lit 7) (instOfNatNat (nat_lit 7))) (@Singleton.singleton Nat (Set Nat) (@Set.instSingletonSet Nat) (@OfNat.ofNat Nat (nat_lit 8) (instOfNatNat (nat_lit 8)))))))	Set ℕ	[ { "t": "ℕ → ℝ", "v": null, "name": "I" }, { "t": "I = fun (m : ℕ) ↦ ∫ x in (0)..(2 * Real.pi), ∏ k in Finset.Icc 1 m, cos (k * x)", "v": null, "name": "hI" }, { "t": "{m ∈ Finset.Icc 1 10 \| I m ≠ 0} = answer", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1985_a5", "tags": [ "analysis" ] }
If $p(x)= a_0 + a_1 x + \cdots + a_m x^m$ is a polynomial with real coefficients $a_i$, then set \[ \Gamma(p(x)) = a_0^2 + a_1^2 + \cdots + a_m^2. \] Let $F(x) = 3x^2+7x+2$. Find, with proof, a polynomial $g(x)$ with real coefficients such that \begin{enumerate} \item[(i)] $g(0)=1$, and \item[(ii)] $\Gamma(f(x)^n) = \Gamma(g(x)^n)$ \end{enumerate} for every integer $n \geq 1$.	6x^2 + 5x + 1	Show that $g(x) = 6x^2 + 5x + 1$ satisfies the conditions.	open Set Filter Topology Real Polynomial	[]	@Eq (@Polynomial Real Real.semiring) answer (@HAdd.hAdd (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@instHAdd (@Polynomial Real Real.semiring) (@Polynomial.add' Real Real.semiring)) (@HAdd.hAdd (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@instHAdd (@Polynomial Real Real.semiring) (@Polynomial.add' Real Real.semiring)) (@HMul.hMul (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@instHMul (@Polynomial Real Real.semiring) (@Polynomial.mul' Real Real.semiring)) (@OfNat.ofNat (@Polynomial Real Real.semiring) (nat_lit 6) (@instOfNatAtLeastTwo (@Polynomial Real Real.semiring) (nat_lit 6) (@Polynomial.instNatCast Real Real.semiring) (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 4) (instOfNatNat (nat_lit 4)))))) (@HPow.hPow (@Polynomial Real Real.semiring) Nat (@Polynomial Real Real.semiring) (@instHPow (@Polynomial Real Real.semiring) Nat (@Monoid.toNatPow (@Polynomial Real Real.semiring) (@MonoidWithZero.toMonoid (@Polynomial Real Real.semiring) (@Semiring.toMonoidWithZero (@Polynomial Real Real.semiring) (@Polynomial.semiring Real Real.semiring))))) (@Polynomial.X Real Real.semiring) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))) (@HMul.hMul (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@instHMul (@Polynomial Real Real.semiring) (@Polynomial.mul' Real Real.semiring)) (@OfNat.ofNat (@Polynomial Real Real.semiring) (nat_lit 5) (@instOfNatAtLeastTwo (@Polynomial Real Real.semiring) (nat_lit 5) (@Polynomial.instNatCast Real Real.semiring) (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))))) (@Polynomial.X Real Real.semiring))) (@OfNat.ofNat (@Polynomial Real Real.semiring) (nat_lit 1) (@One.toOfNat1 (@Polynomial Real Real.semiring) (@Polynomial.one Real Real.semiring))))	Polynomial ℝ	[ { "t": "Polynomial ℝ → ℝ", "v": null, "name": "Γ" }, { "t": "Polynomial ℝ", "v": null, "name": "f" }, { "t": "Γ = fun p ↦ ∑ k in Finset.range (p.natDegree + 1), coeff p k ^ 2", "v": null, "name": "hΓ" }, { "t": "f = 3 * Polynomial.X ^ 2 + 7 * Polynomial.X + 2", "v": null, "name": "hf" }, { "t": "let g := answer;\n g.eval 0 = 1 ∧ ∀ n : ℕ, n ≥ 1 → Γ (f ^ n) = Γ (g ^ n)", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1985_a6", "tags": [ "algebra" ] }
Let $k$ be the smallest positive integer for which there exist distinct integers $m_1, m_2, m_3, m_4, m_5$ such that the polynomial \[ p(x) = (x-m_1)(x-m_2)(x-m_3)(x-m_4)(x-m_5) \] has exactly $k$ nonzero coefficients. Find, with proof, a set of integers $m_1, m_2, m_3, m_4, m_5$ for which this minimum $k$ is achieved.	fun i : Fin 5 ↦ ↑i - (2 : ℤ)	Show that the minimum $k = 3$ is obtained for $\{m_1, m_2, m_3, m_4, m_5\} = \{-2, -1, 0, 1, 2\}$.	open Set Filter Topology Real Polynomial Function	[]	@Eq (Fin (@OfNat.ofNat Nat (nat_lit 5) (instOfNatNat (nat_lit 5))) → Int) answer fun (i : Fin (@OfNat.ofNat Nat (nat_lit 5) (instOfNatNat (nat_lit 5)))) => @HSub.hSub Int Int Int (@instHSub Int Int.instSub) (@Nat.cast Int instNatCastInt (@Fin.val (@OfNat.ofNat Nat (nat_lit 5) (instOfNatNat (nat_lit 5))) i)) (@OfNat.ofNat Int (nat_lit 2) (@instOfNat (nat_lit 2)))	Fin 5 → ℤ	[ { "t": "(Fin 5 → ℤ) → (Polynomial ℝ)", "v": null, "name": "p" }, { "t": "p = fun m ↦ ∏ i : Fin 5, ((X : Polynomial ℝ) - m i)", "v": null, "name": "hp" }, { "t": "Polynomial ℝ → ℕ", "v": null, "name": "numnzcoeff" }, { "t": "numnzcoeff = fun p ↦ {j ∈ Finset.range (p.natDegree + 1) \| coeff p j ≠ 0}.card", "v": null, "name": "hnumnzcoeff" }, { "t": "(Injective answer ∧ ∀ m : Fin 5 → ℤ, Injective m → numnzcoeff (p answer) ≤ numnzcoeff (p m))", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1985_b1", "tags": [ "algebra" ] }
Define polynomials $f_n(x)$ for $n \geq 0$ by $f_0(x)=1$, $f_n(0)=0$ for $n \geq 1$, and \[ \frac{d}{dx} f_{n+1}(x) = (n+1)f_n(x+1) \] for $n \geq 0$. Find, with proof, the explicit factorization of $f_{100}(1)$ into powers of distinct primes.	99 if $n = 101$, otherwise 0	Show that $f_{100}(1) = 101^{99}$.	open Set Filter Topology Real Polynomial Function	[]	@Eq (Nat → Nat) answer fun (n : Nat) => @ite Nat (@Eq Nat n (@OfNat.ofNat Nat (nat_lit 101) (instOfNatNat (nat_lit 101)))) (instDecidableEqNat n (@OfNat.ofNat Nat (nat_lit 101) (instOfNatNat (nat_lit 101)))) (@OfNat.ofNat Nat (nat_lit 99) (instOfNatNat (nat_lit 99))) (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0)))	ℕ → ℕ	[ { "t": "ℕ -> Polynomial ℕ", "v": null, "name": "f" }, { "t": "f 0 = 1", "v": null, "name": "hf0x" }, { "t": "∀ n ≥ 1, (f n).eval 0 = 0", "v": null, "name": "hfn0" }, { "t": "∀ n : ℕ, derivative (f (n + 1)) = (n + 1) * (Polynomial.comp (f n) (X + 1))", "v": null, "name": "hfderiv" }, { "t": "Nat.factorization ((f 100).eval 1) = answer", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1985_b2", "tags": [ "algebra" ] }
Evaluate $\int_0^\infty t^{-1/2}e^{-1985(t+t^{-1})}\,dt$. You may assume that $\int_{-\infty}^\infty e^{-x^2}\,dx = \sqrt{\pi}$.	$\sqrt{\pi / 1985} \cdot e^{-3970}$	Show that the integral evaluates to $\sqrt{\frac{\pi}{1985}}e^{-3970}$.	open Set Filter Topology Real Polynomial Function	[]	@Eq Real answer (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) (Real.sqrt (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) Real.pi (@OfNat.ofNat Real (nat_lit 1985) (@instOfNatAtLeastTwo Real (nat_lit 1985) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 1983) (instOfNatNat (nat_lit 1983)))))))) (Real.exp (@Neg.neg Real Real.instNeg (@OfNat.ofNat Real (nat_lit 3970) (@instOfNatAtLeastTwo Real (nat_lit 3970) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 3968) (instOfNatNat (nat_lit 3968)))))))))	ℝ	[ { "t": "∫ x in Set.univ, Real.exp (- x ^ 2) = Real.sqrt Real.pi", "v": null, "name": "fact" }, { "t": "∫ t in Set.Ioi 0, t ^ (- (1 : ℝ) / 2) * Real.exp (-1985 * (t + t ^ (-(1 : ℝ)))) = answer", "v": null, "name": "h_answer" } ]	{ "problem_name": "putnam_1985_b5", "tags": [ "analysis" ] }

Find every real-valued function $f$ whose domain is an interval $I$ (finite or infinite) having 0 as a left-hand endpoint, such that for every positive member $x$ of $I$ the average of $f$ over the closed interval $[0, x]$ is equal to the geometric mean of the numbers $f(0)$ and $f(x)$.

the set of functions $f(x) = \frac{a}{(1 - c x)^2}$ where $a \geq 0$

Show that \[ f(x) = \frac{a}{(1 - cx)^2} \begin{cases} \text{for } 0 \le x < \frac{1}{c}, & \text{if } c > 0\\ \text{for } 0 \le x < \infty, & \text{if } c \le 0, \end{cases} \] where $a > 0$.

open MeasureTheory Set

[]

@Eq (Set (Real → Real)) answer (@setOf (Real → Real) fun (f : Real → Real) => @Exists Real fun (a : Real) => @Exists Real fun (c : Real) => And (@GE.ge Real Real.instLE a (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero))) (@Eq (Real → Real) f fun (x : Real) => @HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) a (@HPow.hPow Real Nat Real (@instHPow Real Nat (@Monoid.toNatPow Real Real.instMonoid)) (@HSub.hSub Real Real Real (@instHSub Real Real.instSub) (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)) (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) c x)) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))))

Set (ℝ → ℝ)

[ { "t": "Set ℝ → (ℝ → ℝ) → Prop", "v": null, "name": "P" }, { "t": "∀ s f, P s f ↔ 0 ≤ f ∧ ∀ x ∈ s, ⨍ t in Ico 0 x, f t = √(f 0 * f x)", "v": null, "name": "P_def" }, { "t": "answer = {f : ℝ → ℝ | P (Ioi 0) f ∨ (∃ e > 0, P (Ioo 0 e) f)}", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1962_a2", "tags": [ "analysis" ] }

Evaluate in closed form \[ \sum_{k=1}^n {n \choose k} k^2. \]

$n(n+1)2^{n-2}$

Show that the expression equals $n(n+1)2^{n-2}$.

null

[]

@Eq (Nat → Nat) answer fun (n_1 : Nat) => @HMul.hMul Nat Nat Nat (@instHMul Nat instMulNat) (@HMul.hMul Nat Nat Nat (@instHMul Nat instMulNat) n_1 (@HAdd.hAdd Nat Nat Nat (@instHAdd Nat instAddNat) n_1 (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))) (@HPow.hPow Nat Nat Nat (@instHPow Nat Nat (@Monoid.toNatPow Nat Nat.instMonoid)) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) (@HSub.hSub Nat Nat Nat (@instHSub Nat instSubNat) n_1 (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))

ℕ → ℕ

[ { "t": "ℕ", "v": null, "name": "n" }, { "t": "n ≥ 2", "v": null, "name": "hn" }, { "t": "answer n = ∑ k in Finset.Icc 1 n, Nat.choose n k * k^2", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1962_a5", "tags": [ "algebra", "combinatorics" ] }

Find an integral formula (i.e., a function $z$ such that $y(x) = \int_{1}^{x} z(t) dt$) for the solution of the differential equation $$\delta (\delta - 1) (\delta - 2) \cdots (\delta - n + 1) y = f(x)$$ with the initial conditions $y(1) = y'(1) = \cdots = y^{(n-1)}(1) = 0$, where $n \in \mathbb{N}$, $f$ is continuous for all $x \ge 1$, and $\delta$ denotes $x\frac{d}{dx}$.

$(x - t)^{n-1} \cdot f(t) / ((n-1)! \cdot t^n)$

Show that the solution is $$y(x) = \int_{1}^{x} \frac{(x - t)^{n - 1} f(t)}{(n - 1)!t^n} dt$$.

open Nat Set Topology Filter

[]

@Eq ((Real → Real) → Nat → Real → Real → Real) answer fun (f_1 : Real → Real) (n_1 : Nat) (x t : Real) => @HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) (@HPow.hPow Real Nat Real (@instHPow Real Nat (@Monoid.toNatPow Real Real.instMonoid)) (@HSub.hSub Real Real Real (@instHSub Real Real.instSub) x t) (@HSub.hSub Nat Nat Nat (@instHSub Nat instSubNat) n_1 (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))) (f_1 t)) (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) (@Nat.cast Real Real.instNatCast (Nat.factorial (@HSub.hSub Nat Nat Nat (@instHSub Nat instSubNat) n_1 (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))) (@HPow.hPow Real Nat Real (@instHPow Real Nat (@Monoid.toNatPow Real Real.instMonoid)) t n_1))

(ℝ → ℝ) → ℕ → ℝ → ℝ → ℝ

[ { "t": "ℕ → (ℝ → ℝ) → (ℝ → ℝ)", "v": null, "name": "P" }, { "t": "P 0 = id ∧ ∀ i y, P (i + 1) y = P i (fun x ↦ x * deriv y x - i * y x)", "v": null, "name": "hP" }, { "t": "ℕ", "v": null, "name": "n" }, { "t": "0 < n", "v": null, "name": "hn" }, { "t": "ℝ → ℝ", "v": null, "name": "f" }, { "t": "ℝ → ℝ", "v": null, "name": "y" }, { "t": "ContinuousOn f (Ici 1)", "v": null, "name": "hf" }, { "t": "ContDiffOn ℝ n y (Ici 1)", "v": null, "name": "hy" }, { "t": "∀ x ≥ 1, y x = ∫ t in (1 : ℝ)..x, answer f n x t", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1963_a3", "tags": [ "analysis" ] }

For what integer $a$ does $x^2-x+a$ divide $x^{13}+x+90$?

2

Show that $a=2$.

open Topology Filter Polynomial

[]

@Eq Int answer (@OfNat.ofNat Int (nat_lit 2) (@instOfNat (nat_lit 2)))

ℤ

[ { "t": "ℤ", "v": null, "name": "a" }, { "t": "Polynomial.X^2 - Polynomial.X + (Polynomial.C a) ∣ (Polynomial.X ^ 13 + Polynomial.X + (Polynomial.C 90))", "v": null, "name": "h_div" }, { "t": "answer = a", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1963_b1", "tags": [ "algebra" ] }

Let $S$ be the set of all numbers of the form $2^m3^n$, where $m$ and $n$ are integers, and let $P$ be the set of all positive real numbers. Is $S$ dense in $P$?

True

Show that $S$ is dense in $P$.

open Topology Filter Polynomial

[]

@Eq Prop answer True

Prop

[ { "t": "Set ℝ", "v": null, "name": "S" }, { "t": "S = {2 ^ m * 3 ^ n | (m : ℤ) (n : ℤ)}", "v": null, "name": "hS" }, { "t": "answer = (closure S ⊇ Set.Ioi (0 : ℝ))", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1963_b2", "tags": [ "analysis" ] }

Find every twice-differentiable real-valued function $f$ with domain the set of all real numbers and satisfying the functional equation $(f(x))^2-(f(y))^2=f(x+y)f(x-y)$ for all real numbers $x$ and $y$.

the set of functions of the form $A \sinh(k u)$, $A u$, or $A \sin(k u)$

Show that the solution is the sets of functions $f(u)=A\sinh ku$, $f(u)=Au$, and $f(u)=A\sin ku$ with $A,k \in \mathbb{R}$.

open Topology Filter Polynomial

[]

@Eq (Set (Real → Real)) answer (@Union.union (Set (Real → Real)) (@Set.instUnion (Real → Real)) (@Union.union (Set (Real → Real)) (@Set.instUnion (Real → Real)) (@setOf (Real → Real) fun (x : Real → Real) => @Exists Real fun (A : Real) => @Exists Real fun (k : Real) => @Eq (Real → Real) (fun (u : Real) => @HMul.hMul Real Real Real (@instHMul Real Real.instMul) A (Real.sinh (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) k u))) x) (@setOf (Real → Real) fun (x : Real → Real) => @Exists Real fun (A : Real) => @Eq (Real → Real) (fun (u : Real) => @HMul.hMul Real Real Real (@instHMul Real Real.instMul) A u) x)) (@setOf (Real → Real) fun (x : Real → Real) => @Exists Real fun (A : Real) => @Exists Real fun (k : Real) => @Eq (Real → Real) (fun (u : Real) => @HMul.hMul Real Real Real (@instHMul Real Real.instMul) A (Real.sin (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) k u))) x))

Set (ℝ → ℝ)

[ { "t": "ℝ → ℝ", "v": null, "name": "f" }, { "t": "f ∈ answer", "v": null, "name": "h_answer" }, { "t": "ContDiff ℝ 1 f", "v": null, "name": "h_cont_diff" }, { "t": "Differentiable ℝ (deriv f)", "v": null, "name": "h_diff" }, { "t": "∀ x y : ℝ, (f x) ^ 2 - (f y) ^ 2 = f (x + y) * f (x - y)", "v": null, "name": "h_func_eq" } ]

{ "problem_name": "putnam_1963_b3", "tags": [ "analysis" ] }

Let $\alpha$ be a real number. Find all continuous real-valued functions $f : [0, 1] \to (0, \infty)$ such that \begin{align*} \int_0^1 f(x) dx &= 1, \\ \int_0^1 x f(x) dx &= \alpha, \\ \int_0^1 x^2 f(x) dx &= \alpha^2. \\ \end{align*}

the empty set

Prove that there are no such functions.

open Set

[]

@Eq (Real → Set (Real → Real)) answer fun (x : Real) => @EmptyCollection.emptyCollection (Set (Real → Real)) (@Set.instEmptyCollection (Real → Real))

ℝ → Set (ℝ → ℝ)

[ { "t": "ℝ", "v": null, "name": "α" }, { "t": "answer α = {f : ℝ → ℝ | (∀ x ∈ Icc 0 1, f x > 0) ∧ ContinuousOn f (Icc 0 1) ∧ ∫ x in (0)..1, f x = 1 ∧ ∫ x in (0)..1, x * f x = α ∧ ∫ x in (0)..1, x^2 * f x = α^2}", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1964_a2", "tags": [ "analysis", "algebra" ] }

Let $\triangle ABC$ satisfy $\angle CAB < \angle BCA < \frac{\pi}{2} < \angle ABC$. If the bisector of the external angle at $A$ meets line $BC$ at $P$, the bisector of the external angle at $B$ meets line $CA$ at $Q$, and $AP = BQ = AB$, find $\angle CAB$.

π / 15

Show that the solution is $\angle CAB = \frac{\pi}{15}$.

open EuclideanGeometry Real

[]

@Eq Real answer (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) Real.pi (@OfNat.ofNat Real (nat_lit 15) (@instOfNatAtLeastTwo Real (nat_lit 15) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 13) (instOfNatNat (nat_lit 13)))))))

ℝ

[ { "t": "EuclideanSpace ℝ (Fin 2)", "v": null, "name": "A" }, { "t": "EuclideanSpace ℝ (Fin 2)", "v": null, "name": "B" }, { "t": "EuclideanSpace ℝ (Fin 2)", "v": null, "name": "C" }, { "t": "EuclideanSpace ℝ (Fin 2)", "v": null, "name": "X" }, { "t": "EuclideanSpace ℝ (Fin 2)", "v": null, "name": "Y" }, { "t": "¬Collinear ℝ {A, B, C}", "v": null, "name": "hABC" }, { "t": "∠ C A B < ∠ B C A ∧ ∠ B C A < π/2 ∧ π/2 < ∠ A B C", "v": null, "name": "hangles" }, { "t": "Collinear ℝ {X, B, C} ∧ ∠ X A B = (π - ∠ C A B)/2 ∧ dist A X = dist A B", "v": null, "name": "hX" }, { "t": "Collinear ℝ {Y, C, A} ∧ ∠ Y B C = (π - ∠ A B C)/2 ∧ dist B Y = dist A B", "v": null, "name": "hY" }, { "t": "∠ C A B = answer", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1965_a1", "tags": [ "geometry" ] }

How many orderings of the integers from $1$ to $n$ satisfy the condition that, for every integer $i$ except the first, there exists some earlier integer in the ordering which differs from $i$ by $1$?

$2^{n-1}$

There are $2^{n-1}$ such orderings.

open EuclideanGeometry Topology Filter Complex

[]

@Eq (Nat → Nat) answer fun (n_1 : Nat) => @HPow.hPow Nat Nat Nat (@instHPow Nat Nat (@Monoid.toNatPow Nat Nat.instMonoid)) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) (@HSub.hSub Nat Nat Nat (@instHSub Nat instSubNat) n_1 (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))

ℕ → ℕ

[ { "t": "ℕ", "v": null, "name": "n" }, { "t": "n > 0", "v": null, "name": "npos" }, { "t": "{p ∈ permsOfFinset (Finset.Icc 1 n) | ∀ m ∈ Finset.Icc 2 n, ∃ k ∈ Finset.Ico 1 m, p m = p k + 1 ∨ p m = p k - 1}.card = answer n", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1965_a5", "tags": [ "combinatorics" ] }

Find $$\lim_{n \to \infty} \int_{0}^{1} \int_{0}^{1} \cdots \int_{0}^{1} \cos^2\left(\frac{\pi}{2n}(x_1 + x_2 + \cdots + x_n)\right) dx_1 dx_2 \cdots dx_n.$$

1 / 2

Show that the limit is $\frac{1}{2}$.

open EuclideanGeometry Topology Filter Complex

[]

@Eq Real answer (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)) (@OfNat.ofNat Real (nat_lit 2) (@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0)))))))

ℝ

[ { "t": "Tendsto (fun n : ℕ ↦ ∫ x in {x : Fin (n+1) → ℝ | ∀ k : Fin (n+1), x k ∈ Set.Icc 0 1}, (Real.cos (Real.pi/(2 * (n+1)) * ∑ k : Fin (n+1), x k))^2) atTop (𝓝 answer)", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1965_b1", "tags": [ "analysis" ] }

Consider polynomial forms $ax^2-bx+c$ with integer coefficients which have two distinct zeros in the open interval $0<x<1$. Exhibit with a proof the least positive integer value of $a$ for which such a polynomial exists.

5

Show that the minimum possible value for $a$ is $5$.

open Polynomial

[]

@Eq Nat answer (@OfNat.ofNat Nat (nat_lit 5) (instOfNatNat (nat_lit 5)))

ℕ

[ { "t": "Set ℤ", "v": null, "name": "S" }, { "t": "S = {a | ∃ P : Polynomial ℤ, P.degree = 2 ∧ (∃ z1 z2 : Set.Ioo (0 : ℝ) 1, z1 ≠ z2 ∧ aeval (z1 : ℝ) P = 0 ∧ aeval (z2 : ℝ) P = 0) ∧P.coeff 2 = a ∧ a > 0}", "v": null, "name": "hS" }, { "t": "IsLeast S answer", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1967_a3", "tags": [ "algebra" ] }

Given real numbers $\{a_i\}$ and $\{b_i\}$, ($i=1,2,3,4$), such that $a_1b_2-a_2b_1 \neq 0$. Consider the set of all solutions $(x_1,x_2,x_3,x_4)$ of the simultaneous equations $a_1x_1+a_2x_2+a_3x_3+a_4x_4=0$ and $b_1x_1+b_2x_2+b_3x_3+b_4x_4=0$, for which no $x_i$ ($i=1,2,3,4$) is zero. Each such solution generates a $4$-tuple of plus and minus signs $(\text{signum }x_1,\text{signum }x_2,\text{signum }x_3,\text{signum }x_4)$. Determine, with a proof, the maximum number of distinct $4$-tuples possible.

8

Show that the maximum number of distinct $4$-tuples is eight.

open Nat Topology Filter

[]

@Eq Nat answer (@OfNat.ofNat Nat (nat_lit 8) (instOfNatNat (nat_lit 8)))

ℕ

[ { "t": "Fin 4 → ℝ", "v": null, "name": "a" }, { "t": "Fin 4 → ℝ", "v": null, "name": "b" }, { "t": "a 0 * b 1 - a 1 * b 0 ≠ 0", "v": null, "name": "abneq0" }, { "t": "ℕ", "v": null, "name": "numtuples" }, { "t": "numtuples = {s : Fin 4 → ℝ | ∃ x : Fin 4 → ℝ, (∀ i : Fin 4, x i ≠ 0) ∧ (∑ i : Fin 4, a i * x i) = 0 ∧ (∑ i : Fin 4, b i * x i) = 0 ∧ (∀ i : Fin 4, s i = Real.sign (x i))}.encard", "v": null, "name": "hnumtuples" }, { "t": "numtuples = answer", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1967_a6", "tags": [ "algebra", "geometry" ] }

Let $V$ be the set of all quadratic polynomials with real coefficients such that $|P(x)| \le 1$ for all $x \in [0, 1]$. Find the supremum of $|P'(0)|$ across all $P \in V$.

8

The supremum is $8$.

open Finset Polynomial

[]

@Eq Real answer (@OfNat.ofNat Real (nat_lit 8) (@instOfNatAtLeastTwo Real (nat_lit 8) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))))))

ℝ

[ { "t": "Set ℝ[X]", "v": null, "name": "V" }, { "t": "V = {P : ℝ[X] | P.degree = 2 ∧ ∀ x ∈ Set.Icc 0 1, |P.eval x| ≤ 1}", "v": null, "name": "V_def" }, { "t": "sSup {|(derivative P).eval 0| | P ∈ V} = answer", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1968_a5", "tags": [ "algebra" ] }

Find all polynomials of the form $\sum_{0}^{n} a_{i} x^{n-i}$ with $n \ge 1$ and $a_i = \pm 1$ for all $0 \le i \le n$ whose roots are all real.

{X - 1, -(X - 1), X + 1, -(X + 1), X^2 + X - 1, -(X^2 + X - 1), X^2 - X - 1, -(X^2 - X - 1), X^3 + X^2 - X - 1, -(X^3 + X^2 - X - 1), X^3 - X^2 - X + 1, -(X^3 - X^2 - X + 1)}

The set of such polynomials is $$\{\pm (x - 1), \pm (x + 1), \pm (x^2 + x - 1), \pm (x^2 - x - 1), \pm (x^3 + x^2 - x - 1), \pm (x^3 - x^2 - x + 1)\}.$$

open Finset Polynomial

[]

@Eq (Set (@Polynomial Complex Complex.instSemiring)) answer (@Insert.insert (@Polynomial Complex Complex.instSemiring) (Set (@Polynomial Complex Complex.instSemiring)) (@Set.instInsert (@Polynomial Complex Complex.instSemiring)) (@HSub.hSub (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@instHSub (@Polynomial Complex Complex.instSemiring) (@Polynomial.sub Complex Complex.instRing)) (@Polynomial.X Complex Complex.instSemiring) (@OfNat.ofNat (@Polynomial Complex Complex.instSemiring) (nat_lit 1) (@One.toOfNat1 (@Polynomial Complex Complex.instSemiring) (@Polynomial.one Complex Complex.instSemiring)))) (@Insert.insert (@Polynomial Complex Complex.instSemiring) (Set (@Polynomial Complex Complex.instSemiring)) (@Set.instInsert (@Polynomial Complex Complex.instSemiring)) (@Neg.neg (@Polynomial Complex Complex.instSemiring) (@Polynomial.neg' Complex Complex.instRing) (@HSub.hSub (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@instHSub (@Polynomial Complex Complex.instSemiring) (@Polynomial.sub Complex Complex.instRing)) (@Polynomial.X Complex Complex.instSemiring) (@OfNat.ofNat (@Polynomial Complex Complex.instSemiring) (nat_lit 1) (@One.toOfNat1 (@Polynomial Complex Complex.instSemiring) (@Polynomial.one Complex Complex.instSemiring))))) (@Insert.insert (@Polynomial Complex Complex.instSemiring) (Set (@Polynomial Complex Complex.instSemiring)) (@Set.instInsert (@Polynomial Complex Complex.instSemiring)) (@HAdd.hAdd (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@instHAdd (@Polynomial Complex Complex.instSemiring) (@Polynomial.add' Complex Complex.instSemiring)) (@Polynomial.X Complex Complex.instSemiring) (@OfNat.ofNat (@Polynomial Complex Complex.instSemiring) (nat_lit 1) (@One.toOfNat1 (@Polynomial Complex Complex.instSemiring) (@Polynomial.one Complex Complex.instSemiring)))) (@Insert.insert (@Polynomial Complex Complex.instSemiring) (Set (@Polynomial Complex Complex.instSemiring)) (@Set.instInsert (@Polynomial Complex Complex.instSemiring)) (@Neg.neg (@Polynomial Complex Complex.instSemiring) (@Polynomial.neg' Complex Complex.instRing) (@HAdd.hAdd (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@instHAdd (@Polynomial Complex Complex.instSemiring) (@Polynomial.add' Complex Complex.instSemiring)) (@Polynomial.X Complex Complex.instSemiring) (@OfNat.ofNat (@Polynomial Complex Complex.instSemiring) (nat_lit 1) (@One.toOfNat1 (@Polynomial Complex Complex.instSemiring) (@Polynomial.one Complex Complex.instSemiring))))) (@Insert.insert (@Polynomial Complex Complex.instSemiring) (Set (@Polynomial Complex Complex.instSemiring)) (@Set.instInsert (@Polynomial Complex Complex.instSemiring)) (@HSub.hSub (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@instHSub (@Polynomial Complex Complex.instSemiring) (@Polynomial.sub Complex Complex.instRing)) (@HAdd.hAdd (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@instHAdd (@Polynomial Complex Complex.instSemiring) (@Polynomial.add' Complex Complex.instSemiring)) (@HPow.hPow (@Polynomial Complex Complex.instSemiring) Nat (@Polynomial Complex Complex.instSemiring) (@instHPow (@Polynomial Complex Complex.instSemiring) Nat (@Monoid.toNatPow (@Polynomial Complex Complex.instSemiring) (@MonoidWithZero.toMonoid (@Polynomial Complex Complex.instSemiring) (@Semiring.toMonoidWithZero (@Polynomial Complex Complex.instSemiring) (@Polynomial.semiring Complex Complex.instSemiring))))) (@Polynomial.X Complex Complex.instSemiring) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) (@Polynomial.X Complex Complex.instSemiring)) (@OfNat.ofNat (@Polynomial Complex Complex.instSemiring) (nat_lit 1) (@One.toOfNat1 (@Polynomial Complex Complex.instSemiring) (@Polynomial.one Complex Complex.instSemiring)))) (@Insert.insert (@Polynomial Complex Complex.instSemiring) (Set (@Polynomial Complex Complex.instSemiring)) (@Set.instInsert (@Polynomial Complex Complex.instSemiring)) (@Neg.neg (@Polynomial Complex Complex.instSemiring) (@Polynomial.neg' Complex Complex.instRing) (@HSub.hSub (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@instHSub (@Polynomial Complex Complex.instSemiring) (@Polynomial.sub Complex Complex.instRing)) (@HAdd.hAdd (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@instHAdd (@Polynomial Complex Complex.instSemiring) (@Polynomial.add' Complex Complex.instSemiring)) (@HPow.hPow (@Polynomial Complex Complex.instSemiring) Nat (@Polynomial Complex Complex.instSemiring) (@instHPow (@Polynomial Complex Complex.instSemiring) Nat (@Monoid.toNatPow (@Polynomial Complex Complex.instSemiring) (@MonoidWithZero.toMonoid (@Polynomial Complex Complex.instSemiring) (@Semiring.toMonoidWithZero (@Polynomial Complex Complex.instSemiring) (@Polynomial.semiring Complex Complex.instSemiring))))) (@Polynomial.X Complex Complex.instSemiring) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) (@Polynomial.X Complex Complex.instSemiring)) (@OfNat.ofNat (@Polynomial Complex Complex.instSemiring) (nat_lit 1) (@One.toOfNat1 (@Polynomial Complex Complex.instSemiring) (@Polynomial.one Complex Complex.instSemiring))))) (@Insert.insert (@Polynomial Complex Complex.instSemiring) (Set (@Polynomial Complex Complex.instSemiring)) (@Set.instInsert (@Polynomial Complex Complex.instSemiring)) (@HSub.hSub (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@instHSub (@Polynomial Complex Complex.instSemiring) (@Polynomial.sub Complex Complex.instRing)) (@HSub.hSub (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@instHSub (@Polynomial Complex Complex.instSemiring) (@Polynomial.sub Complex Complex.instRing)) (@HPow.hPow (@Polynomial Complex Complex.instSemiring) Nat (@Polynomial Complex Complex.instSemiring) (@instHPow (@Polynomial Complex Complex.instSemiring) Nat (@Monoid.toNatPow (@Polynomial Complex Complex.instSemiring) (@MonoidWithZero.toMonoid (@Polynomial Complex Complex.instSemiring) (@Semiring.toMonoidWithZero (@Polynomial Complex Complex.instSemiring) (@Polynomial.semiring Complex Complex.instSemiring))))) (@Polynomial.X Complex Complex.instSemiring) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) (@Polynomial.X Complex Complex.instSemiring)) (@OfNat.ofNat (@Polynomial Complex Complex.instSemiring) (nat_lit 1) (@One.toOfNat1 (@Polynomial Complex Complex.instSemiring) (@Polynomial.one Complex Complex.instSemiring)))) (@Insert.insert (@Polynomial Complex Complex.instSemiring) (Set (@Polynomial Complex Complex.instSemiring)) (@Set.instInsert (@Polynomial Complex Complex.instSemiring)) (@Neg.neg (@Polynomial Complex Complex.instSemiring) (@Polynomial.neg' Complex Complex.instRing) (@HSub.hSub (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@instHSub (@Polynomial Complex Complex.instSemiring) (@Polynomial.sub Complex Complex.instRing)) (@HSub.hSub (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@instHSub (@Polynomial Complex Complex.instSemiring) (@Polynomial.sub Complex Complex.instRing)) (@HPow.hPow (@Polynomial Complex Complex.instSemiring) Nat (@Polynomial Complex Complex.instSemiring) (@instHPow (@Polynomial Complex Complex.instSemiring) Nat (@Monoid.toNatPow (@Polynomial Complex Complex.instSemiring) (@MonoidWithZero.toMonoid (@Polynomial Complex Complex.instSemiring) (@Semiring.toMonoidWithZero (@Polynomial Complex Complex.instSemiring) (@Polynomial.semiring Complex Complex.instSemiring))))) (@Polynomial.X Complex Complex.instSemiring) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) (@Polynomial.X Complex Complex.instSemiring)) (@OfNat.ofNat (@Polynomial Complex Complex.instSemiring) (nat_lit 1) (@One.toOfNat1 (@Polynomial Complex Complex.instSemiring) (@Polynomial.one Complex Complex.instSemiring))))) (@Insert.insert (@Polynomial Complex Complex.instSemiring) (Set (@Polynomial Complex Complex.instSemiring)) (@Set.instInsert (@Polynomial Complex Complex.instSemiring)) (@HSub.hSub (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@instHSub (@Polynomial Complex Complex.instSemiring) (@Polynomial.sub Complex Complex.instRing)) (@HSub.hSub (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@instHSub (@Polynomial Complex Complex.instSemiring) (@Polynomial.sub Complex Complex.instRing)) (@HAdd.hAdd (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@instHAdd (@Polynomial Complex Complex.instSemiring) (@Polynomial.add' Complex Complex.instSemiring)) (@HPow.hPow (@Polynomial Complex Complex.instSemiring) Nat (@Polynomial Complex Complex.instSemiring) (@instHPow (@Polynomial Complex Complex.instSemiring) Nat (@Monoid.toNatPow (@Polynomial Complex Complex.instSemiring) (@MonoidWithZero.toMonoid (@Polynomial Complex Complex.instSemiring) (@Semiring.toMonoidWithZero (@Polynomial Complex Complex.instSemiring) (@Polynomial.semiring Complex Complex.instSemiring))))) (@Polynomial.X Complex Complex.instSemiring) (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) (@HPow.hPow (@Polynomial Complex Complex.instSemiring) Nat (@Polynomial Complex Complex.instSemiring) (@instHPow (@Polynomial Complex Complex.instSemiring) Nat (@Monoid.toNatPow (@Polynomial Complex Complex.instSemiring) (@MonoidWithZero.toMonoid (@Polynomial Complex Complex.instSemiring) (@Semiring.toMonoidWithZero (@Polynomial Complex Complex.instSemiring) (@Polynomial.semiring Complex Complex.instSemiring))))) (@Polynomial.X Complex Complex.instSemiring) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))) (@Polynomial.X Complex Complex.instSemiring)) (@OfNat.ofNat (@Polynomial Complex Complex.instSemiring) (nat_lit 1) (@One.toOfNat1 (@Polynomial Complex Complex.instSemiring) (@Polynomial.one Complex Complex.instSemiring)))) (@Insert.insert (@Polynomial Complex Complex.instSemiring) (Set (@Polynomial Complex Complex.instSemiring)) (@Set.instInsert (@Polynomial Complex Complex.instSemiring)) (@Neg.neg (@Polynomial Complex Complex.instSemiring) (@Polynomial.neg' Complex Complex.instRing) (@HSub.hSub (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@instHSub (@Polynomial Complex Complex.instSemiring) (@Polynomial.sub Complex Complex.instRing)) (@HSub.hSub (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@instHSub (@Polynomial Complex Complex.instSemiring) (@Polynomial.sub Complex Complex.instRing)) (@HAdd.hAdd (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@instHAdd (@Polynomial Complex Complex.instSemiring) (@Polynomial.add' Complex Complex.instSemiring)) (@HPow.hPow (@Polynomial Complex Complex.instSemiring) Nat (@Polynomial Complex Complex.instSemiring) (@instHPow (@Polynomial Complex Complex.instSemiring) Nat (@Monoid.toNatPow (@Polynomial Complex Complex.instSemiring) (@MonoidWithZero.toMonoid (@Polynomial Complex Complex.instSemiring) (@Semiring.toMonoidWithZero (@Polynomial Complex Complex.instSemiring) (@Polynomial.semiring Complex Complex.instSemiring))))) (@Polynomial.X Complex Complex.instSemiring) (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) (@HPow.hPow (@Polynomial Complex Complex.instSemiring) Nat (@Polynomial Complex Complex.instSemiring) (@instHPow (@Polynomial Complex Complex.instSemiring) Nat (@Monoid.toNatPow (@Polynomial Complex Complex.instSemiring) (@MonoidWithZero.toMonoid (@Polynomial Complex Complex.instSemiring) (@Semiring.toMonoidWithZero (@Polynomial Complex Complex.instSemiring) (@Polynomial.semiring Complex Complex.instSemiring))))) (@Polynomial.X Complex Complex.instSemiring) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))) (@Polynomial.X Complex Complex.instSemiring)) (@OfNat.ofNat (@Polynomial Complex Complex.instSemiring) (nat_lit 1) (@One.toOfNat1 (@Polynomial Complex Complex.instSemiring) (@Polynomial.one Complex Complex.instSemiring))))) (@Insert.insert (@Polynomial Complex Complex.instSemiring) (Set (@Polynomial Complex Complex.instSemiring)) (@Set.instInsert (@Polynomial Complex Complex.instSemiring)) (@HAdd.hAdd (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@instHAdd (@Polynomial Complex Complex.instSemiring) (@Polynomial.add' Complex Complex.instSemiring)) (@HSub.hSub (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@instHSub (@Polynomial Complex Complex.instSemiring) (@Polynomial.sub Complex Complex.instRing)) (@HSub.hSub (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@instHSub (@Polynomial Complex Complex.instSemiring) (@Polynomial.sub Complex Complex.instRing)) (@HPow.hPow (@Polynomial Complex Complex.instSemiring) Nat (@Polynomial Complex Complex.instSemiring) (@instHPow (@Polynomial Complex Complex.instSemiring) Nat (@Monoid.toNatPow (@Polynomial Complex Complex.instSemiring) (@MonoidWithZero.toMonoid (@Polynomial Complex Complex.instSemiring) (@Semiring.toMonoidWithZero (@Polynomial Complex Complex.instSemiring) (@Polynomial.semiring Complex Complex.instSemiring))))) (@Polynomial.X Complex Complex.instSemiring) (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) (@HPow.hPow (@Polynomial Complex Complex.instSemiring) Nat (@Polynomial Complex Complex.instSemiring) (@instHPow (@Polynomial Complex Complex.instSemiring) Nat (@Monoid.toNatPow (@Polynomial Complex Complex.instSemiring) (@MonoidWithZero.toMonoid (@Polynomial Complex Complex.instSemiring) (@Semiring.toMonoidWithZero (@Polynomial Complex Complex.instSemiring) (@Polynomial.semiring Complex Complex.instSemiring))))) (@Polynomial.X Complex Complex.instSemiring) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))) (@Polynomial.X Complex Complex.instSemiring)) (@OfNat.ofNat (@Polynomial Complex Complex.instSemiring) (nat_lit 1) (@One.toOfNat1 (@Polynomial Complex Complex.instSemiring) (@Polynomial.one Complex Complex.instSemiring)))) (@Singleton.singleton (@Polynomial Complex Complex.instSemiring) (Set (@Polynomial Complex Complex.instSemiring)) (@Set.instSingletonSet (@Polynomial Complex Complex.instSemiring)) (@Neg.neg (@Polynomial Complex Complex.instSemiring) (@Polynomial.neg' Complex Complex.instRing) (@HAdd.hAdd (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@instHAdd (@Polynomial Complex Complex.instSemiring) (@Polynomial.add' Complex Complex.instSemiring)) (@HSub.hSub (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@instHSub (@Polynomial Complex Complex.instSemiring) (@Polynomial.sub Complex Complex.instRing)) (@HSub.hSub (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring) (@instHSub (@Polynomial Complex Complex.instSemiring) (@Polynomial.sub Complex Complex.instRing)) (@HPow.hPow (@Polynomial Complex Complex.instSemiring) Nat (@Polynomial Complex Complex.instSemiring) (@instHPow (@Polynomial Complex Complex.instSemiring) Nat (@Monoid.toNatPow (@Polynomial Complex Complex.instSemiring) (@MonoidWithZero.toMonoid (@Polynomial Complex Complex.instSemiring) (@Semiring.toMonoidWithZero (@Polynomial Complex Complex.instSemiring) (@Polynomial.semiring Complex Complex.instSemiring))))) (@Polynomial.X Complex Complex.instSemiring) (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) (@HPow.hPow (@Polynomial Complex Complex.instSemiring) Nat (@Polynomial Complex Complex.instSemiring) (@instHPow (@Polynomial Complex Complex.instSemiring) Nat (@Monoid.toNatPow (@Polynomial Complex Complex.instSemiring) (@MonoidWithZero.toMonoid (@Polynomial Complex Complex.instSemiring) (@Semiring.toMonoidWithZero (@Polynomial Complex Complex.instSemiring) (@Polynomial.semiring Complex Complex.instSemiring))))) (@Polynomial.X Complex Complex.instSemiring) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))) (@Polynomial.X Complex Complex.instSemiring)) (@OfNat.ofNat (@Polynomial Complex Complex.instSemiring) (nat_lit 1) (@One.toOfNat1 (@Polynomial Complex Complex.instSemiring) (@Polynomial.one Complex Complex.instSemiring)))))))))))))))))

Set ℂ[X]

[ { "t": "{P : ℂ[X] | P.natDegree ≥ 1 ∧ (∀ k ∈ Set.Icc 0 P.natDegree, P.coeff k = 1 ∨ P.coeff k = -1) ∧\n ∀ z : ℂ, P.eval z = 0 → ∃ r : ℝ, r = z} = answer", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1968_a6", "tags": [ "algebra" ] }

Let $p$ be a prime number. Find the number of distinct $2 \times 2$ matrices $$\begin{pmatrix} a & b \\ c & d \end{pmatrix}$$ such that $a, b, c, d \in \{0, 1, ..., p - 1\}$, $a + d \equiv 1 \pmod p$, and $ad - bc \equiv 0 \pmod p$.

$p^2 + p$

There are $p^2 + p$ such matrices.

open Finset Polynomial Topology Filter Metric

[]

@Eq (Nat → Nat) answer fun (p_1 : Nat) => @HAdd.hAdd Nat Nat Nat (@instHAdd Nat instAddNat) (@HPow.hPow Nat Nat Nat (@instHPow Nat Nat (@Monoid.toNatPow Nat Nat.instMonoid)) p_1 (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) p_1

ℕ → ℕ

[ { "t": "ℕ", "v": null, "name": "p" }, { "t": "Nat.Prime p", "v": null, "name": "hp" }, { "t": "{M : Matrix (Fin 2) (Fin 2) (ZMod p) | M 0 0 + M 1 1 = 1 ∧ M 0 0 * M 1 1 - M 0 1 * M 1 0 = 0}.ncard = answer p", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1968_b5", "tags": [ "linear_algebra", "number_theory", "combinatorics" ] }

What are the possible ranges (across all real inputs $x$ and $y$) of a polynomial $f(x, y)$ with real coefficients?

{{x} | x : ℝ} ∪ {Set.Ici x | x : ℝ} ∪ {Set.Iic x | x : ℝ} ∪ {Set.Iio x | x : ℝ} ∪ {Set.Ioi x | x : ℝ} ∪ {Set.univ}

Show that the possibles ranges are a single point, any half-open or half-closed semi-infinite interval, or all real numbers.

open Matrix Filter Topology Set Nat

[]

@Eq (Set (Set Real)) answer (@Union.union (Set (Set Real)) (@Set.instUnion (Set Real)) (@Union.union (Set (Set Real)) (@Set.instUnion (Set Real)) (@Union.union (Set (Set Real)) (@Set.instUnion (Set Real)) (@Union.union (Set (Set Real)) (@Set.instUnion (Set Real)) (@Union.union (Set (Set Real)) (@Set.instUnion (Set Real)) (@setOf (Set Real) fun (x : Set Real) => @Exists Real fun (x_1 : Real) => @Eq (Set Real) (@Singleton.singleton Real (Set Real) (@Set.instSingletonSet Real) x_1) x) (@setOf (Set Real) fun (x : Set Real) => @Exists Real fun (x_1 : Real) => @Eq (Set Real) (@Set.Ici Real Real.instPreorder x_1) x)) (@setOf (Set Real) fun (x : Set Real) => @Exists Real fun (x_1 : Real) => @Eq (Set Real) (@Set.Iic Real Real.instPreorder x_1) x)) (@setOf (Set Real) fun (x : Set Real) => @Exists Real fun (x_1 : Real) => @Eq (Set Real) (@Set.Iio Real Real.instPreorder x_1) x)) (@setOf (Set Real) fun (x : Set Real) => @Exists Real fun (x_1 : Real) => @Eq (Set Real) (@Set.Ioi Real Real.instPreorder x_1) x)) (@Singleton.singleton (Set Real) (Set (Set Real)) (@Set.instSingletonSet (Set Real)) (@Set.univ Real)))

Set (Set ℝ)

[ { "t": "answer = {{z : ℝ | ∃ x : Fin 2 → ℝ, MvPolynomial.eval x f = z} | f : MvPolynomial (Fin 2) ℝ}", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1969_a1", "tags": [ "algebra", "set_theory" ] }

Show that a finite group can not be the union of two of its proper subgroups. Does the statement remain true if 'two' is replaced by 'three'?

False

Show that the statement is no longer true if 'two' is replaced by 'three'.

open Matrix Filter Topology Set Nat

[]

@Eq Prop answer False

Prop

[ { "t": "ℕ → Prop", "v": null, "name": "P" }, { "t": "∀ n, P n ↔ ∀ (G : Type) [Group G] [Finite G],\n ∀ H : Fin n → Subgroup G, (∀ i, H i < ⊤) → ⋃ i, (H i : Set G) < ⊤", "v": null, "name": "P_def" }, { "t": "answer ↔ (P 3)", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1969_b2", "tags": [ "abstract_algebra" ] }

Find the length of the longest possible sequence of equal nonzero digits (in base 10) in which a perfect square can terminate. Also, find the smallest square that attains this length.

(3, 1444)

The maximum attainable length is $3$; the smallest such square is $38^2 = 1444$.

open Metric Set EuclideanGeometry

[]

@Eq (Prod Nat Nat) answer (@Prod.mk Nat Nat (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3))) (@OfNat.ofNat Nat (nat_lit 1444) (instOfNatNat (nat_lit 1444))))

ℕ × ℕ

[ { "t": "ℕ → ℕ", "v": null, "name": "L" }, { "t": "∀ n : ℕ, L n ≤ (Nat.digits 10 n).length ∧\n(∀ k : ℕ, k < L n → (Nat.digits 10 n)[k]! = (Nat.digits 10 n)[0]!) ∧\n(L n ≠ (Nat.digits 10 n).length → (Nat.digits 10 n)[L n]! ≠ (Nat.digits 10 n)[0]!)", "v": null, "name": "hL" }, { "t": "(∃ n : ℕ, (Nat.digits 10 (n^2))[0]! ≠ 0 ∧ L (n^2) = answer.1) ∧\n(∀ n : ℕ, (Nat.digits 10 (n^2))[0]! ≠ 0 → L (n^2) ≤ answer.1) ∧\n(∃ m : ℕ, m^2 = answer.2) ∧\nL (answer.2) = answer.1 ∧\n(Nat.digits 10 answer.2)[0]! ≠ 0 ∧\n∀ n : ℕ, (Nat.digits 10 (n^2))[0]! ≠ 0 ∧ L (n^2) = answer.1 → n^2 ≥ answer.2", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1970_a3", "tags": [ "number_theory" ] }

Evaluate the infinite product $\lim_{n \to \infty} \frac{1}{n^4} \prod_{i = 1}^{2n} (n^2 + i^2)^{1/n}$.

$e^{2 \ln 5 - 4 + 2 \arctan 2}$

Show that the solution is $e^{2 \log(5) - 4 + 2 arctan(2)}$.

open Metric Set EuclideanGeometry Filter Topology

[]

@Eq Real answer (Real.exp (@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd) (@HSub.hSub Real Real Real (@instHSub Real Real.instSub) (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) (@OfNat.ofNat Real (nat_lit 2) (@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0)))))) (Real.log (@OfNat.ofNat Real (nat_lit 5) (@instOfNatAtLeastTwo Real (nat_lit 5) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))))))) (@OfNat.ofNat Real (nat_lit 4) (@instOfNatAtLeastTwo Real (nat_lit 4) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))))) (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) (@OfNat.ofNat Real (nat_lit 2) (@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0)))))) (Real.arctan (@OfNat.ofNat Real (nat_lit 2) (@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))))))))))

ℝ

[ { "t": "Tendsto (fun n => 1/(n^4) * ∏ i in Finset.Icc (1 : ℤ) (2*n), ((n^2 + i^2) : ℝ)^((1 : ℝ)/n)) atTop (𝓝 answer)", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1970_b1", "tags": [ "analysis" ] }

Determine all polynomials $P(x)$ such that $P(x^2 + 1) = (P(x))^2 + 1$ and $P(0) = 0$.

{Polynomial.X}

Show that the only such polynomial is the identity function.

open Set

[]

@Eq (Set (@Polynomial Real Real.semiring)) answer (@Singleton.singleton (@Polynomial Real Real.semiring) (Set (@Polynomial Real Real.semiring)) (@Set.instSingletonSet (@Polynomial Real Real.semiring)) (@Polynomial.X Real Real.semiring))

Set (Polynomial ℝ)

[ { "t": "Polynomial ℝ", "v": null, "name": "P" }, { "t": "∀ P : Polynomial ℝ, P ∈ answer ↔ (P.eval 0 = 0 ∧ (∀ x : ℝ, P.eval (x^2 + 1) = (P.eval x)^2 + 1))", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1971_a2", "tags": [ "algebra" ] }

After each play of a certain game of solitaire, the player receives either $a$ or $b$ points, where $a$ and $b$ are positive integers with $a > b$; scores accumulate from play to play. If there are $35$ unattainable scores, one of which is $58$, find $a$ and $b$.

(11, 8)

Show that the solution is $a = 11$ and $b = 8$.

open Set MvPolynomial

[]

@Eq (Prod Int Int) answer (@Prod.mk Int Int (@OfNat.ofNat Int (nat_lit 11) (@instOfNat (nat_lit 11))) (@OfNat.ofNat Int (nat_lit 8) (@instOfNat (nat_lit 8))))

ℤ × ℤ

[ { "t": "ℤ", "v": null, "name": "a" }, { "t": "ℤ", "v": null, "name": "b" }, { "t": "a > 0 ∧ b > 0 ∧ a > b", "v": null, "name": "hab" }, { "t": "ℤ → ℤ → Prop", "v": null, "name": "pab" }, { "t": "∀ x y, pab x y ↔\n {s : ℕ | ¬∃ m n : ℕ, m*x + n*y = s}.ncard = 35 ∧ \n ¬∃ m n : ℕ, m*x + n*y = 58", "v": null, "name": "hpab" }, { "t": "pab a b ↔ a = answer.1 ∧ b = answer.2", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1971_a5", "tags": [ "number_theory" ] }

Find all functions $F : \mathbb{R} \setminus \{0, 1\} \to \mathbb{R}$ that satisfy $F(x) + F\left(\frac{x - 1}{x}\right) = 1 + x$ for all $x \in \mathbb{R} \setminus \{0, 1\}$.

$\left\{x \mapsto \frac{x^3 - x^2 - 1}{2x(x - 1)}\right\}$

The only such function is $F(x) = \frac{x^3 - x^2 - 1}{2x(x - 1)}$.

open Set MvPolynomial

[]

@Eq (Set (Real → Real)) answer (@Singleton.singleton (Real → Real) (Set (Real → Real)) (@Set.instSingletonSet (Real → Real)) fun (x : Real) => @HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) (@HSub.hSub Real Real Real (@instHSub Real Real.instSub) (@HSub.hSub Real Real Real (@instHSub Real Real.instSub) (@HPow.hPow Real Nat Real (@instHPow Real Nat (@Monoid.toNatPow Real Real.instMonoid)) x (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) (@HPow.hPow Real Nat Real (@instHPow Real Nat (@Monoid.toNatPow Real Real.instMonoid)) x (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))) (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))) (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) (@OfNat.ofNat Real (nat_lit 2) (@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0)))))) x) (@HSub.hSub Real Real Real (@instHSub Real Real.instSub) x (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)))))

Set (ℝ → ℝ)

[ { "t": "Set ℝ", "v": null, "name": "S" }, { "t": "S = univ \\ {0, 1}", "v": null, "name": "hS" }, { "t": "(ℝ → ℝ) → Prop", "v": null, "name": "P" }, { "t": "P = fun (F : ℝ → ℝ) => ∀ x ∈ S, F x + F ((x - 1)/x) = 1 + x", "v": null, "name": "hP" }, { "t": "∀ F ∈ answer, P F ∧ ∀ f : ℝ → ℝ, P f → ∃ F ∈ answer, (∀ x ∈ S, f x = F x)", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1971_b2", "tags": [ "algebra" ] }

We call a function $f$ from $[0,1]$ to the reals to be supercontinuous on $[0,1]$ if the Cesaro-limit exists for the sequence $f(x_1), f(x_2), f(x_3), \dots$ whenever it does for the sequence $x_1, x_2, x_3 \dots$. Find all supercontinuous functions on $[0,1]$.

the set of all linear functions on [0,1]

Show that the solution is the set of affine functions.

open EuclideanGeometry Filter Topology Set

[]

@Eq (Set (Real → Real)) answer (@setOf (Real → Real) fun (f : Real → Real) => @Exists Real fun (A : Real) => @Exists Real fun (B : Real) => ∀ (x : Real), @Membership.mem Real (Set Real) (@Set.instMembership Real) (@Set.Icc Real Real.instPreorder (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))) x → @Eq Real (f x) (@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd) (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) A x) B))

Set (ℝ → ℝ)

[ { "t": "(ℕ → ℝ) → Prop", "v": null, "name": "climit_exists" }, { "t": "(ℝ → ℝ) → Prop", "v": null, "name": "supercontinuous" }, { "t": "∀ x, climit_exists x ↔ ∃ C : ℝ, Tendsto (fun n => (∑ i in Finset.range n, (x i))/(n : ℝ)) atTop (𝓝 C)", "v": null, "name": "hclimit_exists" }, { "t": "∀ f, supercontinuous f ↔ ∀ (x : ℕ → ℝ), (∀ i : ℕ, x i ∈ Icc 0 1) → climit_exists x → climit_exists (fun i => f (x i))", "v": null, "name": "hsupercontinuous" }, { "t": "{f | supercontinuous f} = answer", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1972_a3", "tags": [ "analysis" ] }

Let $x : \mathbb{R} \to \mathbb{R}$ be a twice differentiable function whose second derivative is nonstrictly decreasing. If $x(t) - x(0) = s$, $x'(0) = 0$, and $x'(t) = v$ for some $t > 0$, find the maximum possible value of $t$ in terms of $s$ and $v$.

$2s / v$

Show that the maximum possible time is $t = \frac{2s}{v}$.

open EuclideanGeometry Filter Topology Set MeasureTheory Metric

[]

@Eq (Real → Real → Real) answer fun (s_1 v_1 : Real) => @HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) (@OfNat.ofNat Real (nat_lit 2) (@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0)))))) s_1) v_1

ℝ → ℝ → ℝ

[ { "t": "ℝ", "v": null, "name": "s" }, { "t": "ℝ", "v": null, "name": "v" }, { "t": "s > 0", "v": null, "name": "hs" }, { "t": "v > 0", "v": null, "name": "hv" }, { "t": "ℝ → (ℝ → ℝ) → Prop", "v": null, "name": "valid" }, { "t": "∀ t x, valid t x ↔\n DifferentiableOn ℝ x (Set.Icc 0 t) ∧ DifferentiableOn ℝ (deriv x) (Set.Icc 0 t) ∧\n AntitoneOn (deriv (deriv x)) (Set.Icc 0 t) ∧\n deriv x 0 = 0 ∧ deriv x t = v ∧ x t - x 0 = s", "v": null, "name": "hvalid" }, { "t": "IsGreatest {t | ∃ x : ℝ → ℝ, valid t x} (answer s v)", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1972_b2", "tags": [ "analysis" ] }

Consider an infinite series whose $n$th term is given by $\pm \frac{1}{n}$, where the actual values of the $\pm$ signs repeat in blocks of $8$ (so the $\frac{1}{9}$ term has the same sign as the $\frac{1}{1}$ term, and so on). Call such a sequence balanced if each block contains four $+$ and four $-$ signs. Prove that being balanced is a sufficient condition for the sequence to converge. Is being balanced also necessary for the sequence to converge?

True

Show that the condition is necessary.

open Nat Set MeasureTheory Topology Filter

[]

@Eq Prop answer True

Prop

[ { "t": "List ℝ", "v": null, "name": "L" }, { "t": "L.length = 8 ∧ ∀ i : Fin L.length, L[i] = 1 ∨ L[i] = -1", "v": null, "name": "hL" }, { "t": "ℕ", "v": null, "name": "pluses" }, { "t": "pluses = {i : Fin L.length | L[i] = 1}.ncard", "v": null, "name": "hpluses" }, { "t": "ℕ → ℝ", "v": null, "name": "S" }, { "t": "S = fun n : ℕ ↦ ∑ i in Finset.Icc 1 n, L[i % 8]/i", "v": null, "name": "hS" }, { "t": "answer ↔ ((∃ l : ℝ, Tendsto S atTop (𝓝 l)) → pluses = 4)", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1973_a2", "tags": [ "analysis" ] }

How many zeros does the function $f(x) = 2^x - 1 - x^2$ have on the real line?

3

Show that the solution is 3.

open Nat Set MeasureTheory Topology Filter

[]

@Eq Nat answer (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))

ℕ

[ { "t": "ℝ → ℝ", "v": null, "name": "f" }, { "t": "f = fun x => 2^x - 1 - x^2", "v": null, "name": "hf" }, { "t": "answer = {x : ℝ | f x = 0}.ncard", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1973_a4", "tags": [ "analysis" ] }

Suppose $f$ is a function on $[0,1]$ with continuous derivative satisfying $0 < f'(x) \leq 1$ and $f 0 = 0$. Prove that $\left[\int_0^1 f(x) dx\right]]^2 \geq \int_0^1 (f(x))^3 dx$, and find an example where equality holds.

the identity function $f(x) = x$

Show that one such example where equality holds is the identity function.

open Nat Set MeasureTheory Topology Filter

[]

@Eq (Real → Real) answer fun (x : Real) => x

ℝ → ℝ

[ { "t": "ℝ → ℝ", "v": null, "name": "f" }, { "t": "(ℝ → ℝ) → Prop", "v": null, "name": "hprop" }, { "t": "hprop = fun g => ContDiff ℝ 1 g ∧ (∀ x : ℝ, 0 < deriv g x ∧ deriv g x ≤ 1) ∧ g 0 = 0", "v": null, "name": "hprop_def" }, { "t": "hprop f", "v": null, "name": "hf" }, { "t": "hprop answer ∧ (∫ x in Icc 0 1, answer x)^2 = ∫ x in Icc 0 1, (answer x)^3", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1973_b4", "tags": [ "analysis" ] }

Call a set of positive integers 'conspiratorial' if no three of them are pairwise relatively prime. What is the largest number of elements in any conspiratorial subset of the integers 1 through 16?

11

Show that the answer is 11.

open Set

[]

@Eq Nat answer (@OfNat.ofNat Nat (nat_lit 11) (instOfNatNat (nat_lit 11)))

ℕ

[ { "t": "Set ℤ → Prop", "v": null, "name": "conspiratorial" }, { "t": "∀ S, conspiratorial S ↔ ∀ a ∈ S, ∀ b ∈ S, ∀ c ∈ S, (a > 0 ∧ b > 0 ∧ c > 0) ∧ ((a ≠ b ∧ b ≠ c ∧ a ≠ c) → (Int.gcd a b > 1 ∨ Int.gcd b c > 1 ∨ Int.gcd a c > 1))", "v": null, "name": "hconspiratorial" }, { "t": "IsGreatest {k | ∃ S, S ⊆ Icc 1 16 ∧ conspiratorial S ∧ S.encard = k} answer", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1974_a1", "tags": [ "number_theory" ] }

A well-known theorem asserts that a prime $p > 2$ can be written as the sum of two perfect squres if and only if $p \equiv 1 \bmod 4$. Find which primes $p > 2$ can be written in each of the following forms, using (not necessarily positive) integers $x$ and $y$: (a) $x^2 + 16y^2$, (b) $4x^2 + 4xy + 5y^2$.

({p : ℕ | p.Prime ∧ p ≡ 1 [MOD 8]}, {p : ℕ | p.Prime ∧ p ≡ 5 [MOD 8]})

Show that that the answer to (a) is the set of primes which are $1 \bmod 8$, and the solution to (b) is the set of primes which are $5 \bmod 8$.

open Set

[]

@Eq (Prod (Set Nat) (Set Nat)) answer (@Prod.mk (Set Nat) (Set Nat) (@setOf Nat fun (p_1 : Nat) => And (Nat.Prime p_1) (Nat.ModEq (@OfNat.ofNat Nat (nat_lit 8) (instOfNatNat (nat_lit 8))) p_1 (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))) (@setOf Nat fun (p_1 : Nat) => And (Nat.Prime p_1) (Nat.ModEq (@OfNat.ofNat Nat (nat_lit 8) (instOfNatNat (nat_lit 8))) p_1 (@OfNat.ofNat Nat (nat_lit 5) (instOfNatNat (nat_lit 5))))))

(Set ℕ) × (Set ℕ)

[ { "t": "ℕ", "v": null, "name": "p" }, { "t": "∀ p : ℕ, p.Prime ∧ p > 2 → ((∃ m n : ℤ, p = m^2 + n^2) ↔ p ≡ 1 [MOD 4])", "v": null, "name": "h_assumption" }, { "t": "∀ p : ℕ,\n ((p.Prime ∧ p > 2 ∧ (∃ x y : ℤ, p = x^2 + 16*y^2)) ↔ p ∈ answer.1) ∧\n ((p.Prime ∧ p > 2 ∧ (∃ x y : ℤ, p = 4*x^2 + 4*x*y + 5*y^2)) ↔ p ∈ answer.2)", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1974_a3", "tags": [ "number_theory" ] }

Evaluate in closed form: $\frac{1}{2^{n-1}} \sum_{k < n/2} (n-2k)*{n \choose k}$.

(fun n ↦ (1 : ℚ) / ((2 : ℚ) ^ ((n :ℕ) - 1)) * (n * (n - 1).choose ⌊n / 2⌋₊))

Show that the solution is $\frac{n}{2^{n-1}} * {(n-1) \choose \left[ (n-1)/2 \right]}$.

open Set Nat

[]

@Eq (Nat → Rat) answer fun (n_1 : Nat) => @HMul.hMul Rat Rat Rat (@instHMul Rat Rat.instMul) (@HDiv.hDiv Rat Rat Rat (@instHDiv Rat Rat.instDiv) (@OfNat.ofNat Rat (nat_lit 1) (@Rat.instOfNat (nat_lit 1))) (@HPow.hPow Rat Nat Rat (@instHPow Rat Nat Rat.instPowNat) (@OfNat.ofNat Rat (nat_lit 2) (@Rat.instOfNat (nat_lit 2))) (@HSub.hSub Nat Nat Nat (@instHSub Nat instSubNat) n_1 (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))) (@HMul.hMul Rat Rat Rat (@instHMul Rat Rat.instMul) (@Nat.cast Rat Rat.instNatCast n_1) (@Nat.cast Rat Rat.instNatCast (Nat.choose (@HSub.hSub Nat Nat Nat (@instHSub Nat instSubNat) n_1 (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))) (@Nat.floor Nat Nat.instOrderedSemiring instFloorSemiringNat (@HDiv.hDiv Nat Nat Nat (@instHDiv Nat Nat.instDiv) n_1 (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))))))

ℕ → ℚ

[ { "t": "ℕ", "v": null, "name": "n" }, { "t": "0 < n", "v": null, "name": "hn" }, { "t": "(1 : ℚ) / (2 ^ (n - 1)) * ∑ k in Finset.Icc 0 ⌊n / 2⌋₊, (n - 2 * k) * (n.choose k) = answer n", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1974_a4", "tags": [ "algebra" ] }

Given $n$, let $k(n)$ be the minimal degree of any monic integral polynomial $f$ such that the value of $f(x)$ is divisible by $n$ for every integer $x$. Find the value of $k(1000000)$.

25

Show that the answer is 25.

open Set Nat Polynomial

[]

@Eq Nat answer (@OfNat.ofNat Nat (nat_lit 25) (instOfNatNat (nat_lit 25)))

ℕ

[ { "t": "Polynomial ℤ → Prop", "v": null, "name": "hdivnallx" }, { "t": "hdivnallx = fun f => Monic f ∧ (∀ x : ℤ, (10^6 : ℤ) ∣ f.eval x)", "v": null, "name": "hdivnallx_def" }, { "t": "sInf {d : ℕ | ∃ f : Polynomial ℤ, hdivnallx f ∧ d = f.natDegree} = answer", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1974_a6", "tags": [ "algebra" ] }

For a set with $1000$ elements, how many subsets are there whose candinality is respectively $\equiv 0 \bmod 3, \equiv 1 \bmod 3, \equiv 2 \bmod 3$?

((2^1000 - 1)/3, (2^1000 - 1)/3, 1 + (2^1000 - 1)/3)

Show that there answer is that there are $(2^1000-1)/3$ subsets of cardinality $\equiv 0 \bmod 3$ and $\equiv 1 \bmod 3$, and $1 + (2^1000-1)/3$ subsets of cardinality $\equiv 2 \bmod 3$.

open Set Nat Polynomial Filter Topology

[]

@Eq (Prod Nat (Prod Nat Nat)) answer (@Prod.mk Nat (Prod Nat Nat) (@HDiv.hDiv Nat Nat Nat (@instHDiv Nat Nat.instDiv) (@HSub.hSub Nat Nat Nat (@instHSub Nat instSubNat) (@HPow.hPow Nat Nat Nat (@instHPow Nat Nat (@Monoid.toNatPow Nat Nat.instMonoid)) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) (@OfNat.ofNat Nat (nat_lit 1000) (instOfNatNat (nat_lit 1000)))) (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))) (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) (@Prod.mk Nat Nat (@HDiv.hDiv Nat Nat Nat (@instHDiv Nat Nat.instDiv) (@HSub.hSub Nat Nat Nat (@instHSub Nat instSubNat) (@HPow.hPow Nat Nat Nat (@instHPow Nat Nat (@Monoid.toNatPow Nat Nat.instMonoid)) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) (@OfNat.ofNat Nat (nat_lit 1000) (instOfNatNat (nat_lit 1000)))) (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))) (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) (@HAdd.hAdd Nat Nat Nat (@instHAdd Nat instAddNat) (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))) (@HDiv.hDiv Nat Nat Nat (@instHDiv Nat Nat.instDiv) (@HSub.hSub Nat Nat Nat (@instHSub Nat instSubNat) (@HPow.hPow Nat Nat Nat (@instHPow Nat Nat (@Monoid.toNatPow Nat Nat.instMonoid)) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) (@OfNat.ofNat Nat (nat_lit 1000) (instOfNatNat (nat_lit 1000)))) (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))) (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))))))

ℕ × ℕ × ℕ

[ { "t": "ℤ", "v": null, "name": "n" }, { "t": "n = 1000", "v": null, "name": "hn" }, { "t": "ℕ", "v": null, "name": "count0" }, { "t": "ℕ", "v": null, "name": "count1" }, { "t": "ℕ", "v": null, "name": "count2" }, { "t": "count0 = {S | S ⊆ Finset.Icc 1 n ∧ S.card ≡ 0 [MOD 3]}.ncard", "v": null, "name": "hcount0" }, { "t": "count1 = {S | S ⊆ Finset.Icc 1 n ∧ S.card ≡ 1 [MOD 3]}.ncard", "v": null, "name": "hcount1" }, { "t": "count2 = {S | S ⊆ Finset.Icc 1 n ∧ S.card ≡ 2 [MOD 3]}.ncard", "v": null, "name": "hcount2" }, { "t": "(count0, count1, count2) = answer", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1974_b6", "tags": [ "set_theory" ] }

If an integer $n$ can be written as the sum of two triangular numbers (that is, $n = \frac{a^2 + a}{2} + \frac{b^2 + b}{2}$ for some integers $a$ and $b$), express $4n + 1$ as the sum of the squares of two integers $x$ and $y$, giving $x$ and $y$ in terms of $a$ and $b$. Also, show that if $4n + 1 = x^2 + y^2$ for some integers $x$ and $y$, then $n$ can be written as the sum of two triangular numbers.

(fun (a, b) => a + b + 1, fun (a, b) => a - b)

$x = a + b + 1$ and $y = a - b$ (or vice versa).

open Polynomial

[]

@Eq (Prod (Prod Int Int → Int) (Prod Int Int → Int)) answer (@Prod.mk (Prod Int Int → Int) (Prod Int Int → Int) (fun (x : Prod Int Int) => _example.match_2 (fun (x_1 : Prod Int Int) => Int) x fun (a b : Int) => @HAdd.hAdd Int Int Int (@instHAdd Int Int.instAdd) (@HAdd.hAdd Int Int Int (@instHAdd Int Int.instAdd) a b) (@OfNat.ofNat Int (nat_lit 1) (@instOfNat (nat_lit 1)))) fun (x : Prod Int Int) => _example.match_2 (fun (x_1 : Prod Int Int) => Int) x fun (a b : Int) => @HSub.hSub Int Int Int (@instHSub Int Int.instSub) a b)

((ℤ × ℤ) → ℤ) × ((ℤ × ℤ) → ℤ)

[ { "t": "(ℤ × ℤ × ℤ) → Prop", "v": null, "name": "nab" }, { "t": "(ℤ × ℤ × ℤ) → Prop", "v": null, "name": "nxy" }, { "t": "nab = fun (n, a, b) => n = (a^2 + (a : ℚ))/2 + (b^2 + (b : ℚ))/2", "v": null, "name": "hnab" }, { "t": "nxy = fun (n, x, y) => 4*n + 1 = x^2 + y^2", "v": null, "name": "hnxy" }, { "t": "(∀ n a b : ℤ, nab (n, a, b) → nxy (n, answer.1 (a, b), answer.2 (a, b))) ∧ ∀ n : ℤ, (∃ x y : ℤ, nxy (n, x, y)) → ∃ a b : ℤ, nab (n, a, b)", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1975_a1", "tags": [ "algebra", "number_theory" ] }

For which ordered pairs $(b, c)$ of real numbers do both roots of $z^2 + bz + c$ lie strictly inside the unit disk (i.e., $\{|z| < 1\}$) in the complex plane?

$c < 1 \land c - b > -1 \land c + b > -1$

The desired region is the strict interior of the triangle with vertices $(0, -1)$, $(2, 1)$, and $(-2, 1)$.

open Polynomial

[ { "t": "ℝ", "v": null, "name": "b" }, { "t": "ℝ", "v": null, "name": "c" } ]

@Eq Prop (answer (@Prod.mk Real Real b c)) (And (@LT.lt Real Real.instLT c (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))) (And (@GT.gt Real Real.instLT (@HSub.hSub Real Real Real (@instHSub Real Real.instSub) c b) (@Neg.neg Real Real.instNeg (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)))) (@GT.gt Real Real.instLT (@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd) c b) (@Neg.neg Real Real.instNeg (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))))))

(ℝ × ℝ) → Prop

[ { "t": "(∀ z : ℂ, (Polynomial.X^2 + (Polynomial.C (b : ℂ)) * Polynomial.X + (Polynomial.C (c : ℂ)) : Polynomial ℂ).eval z = 0 → ‖z‖ < 1) ↔ answer (b, c)", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1975_a2", "tags": [ "algebra" ] }

If $a$, $b$, and $c$ are real numbers satisfying $0 < a < b < c$, at what points in the set $$\{(x, y, z) \in \mathbb{R}^3 : x^b + y^b + z^b = 1, x \ge 0, y \ge 0, z \ge 0\}$$ does $f(x, y, z) = x^a + y^b + z^c$ attain its maximum and minimum?

(fun (a, b, c) => ((a/b)^(1/(b - a)), (1 - ((a/b)^(1/(b - a)))^b)^(1/b), 0), fun (a, b, c) => (0, (1 - ((b/c)^(1/(c - b)))^b)^(1/b), (b/c)^(1/(c - b))))

$f$ attains its maximum at $\left(x_0, (1 - x_0^b)^{\frac{1}{b}}, 0\right)$ and its minimum at $\left(0, (1 - z_0^b)^{\frac{1}{b}}, z_0\right)$, where $x_0 = \left(\frac{a}{b}\right)^{\frac{1}{b-a}}$ and $z_0 = \left(\frac{b}{c}\right)^{\frac{1}{c-b}}$.

open Polynomial

[]

@Eq (Prod (Prod Real (Prod Real Real) → Prod Real (Prod Real Real)) (Prod Real (Prod Real Real) → Prod Real (Prod Real Real))) answer (@Prod.mk (Prod Real (Prod Real Real) → Prod Real (Prod Real Real)) (Prod Real (Prod Real Real) → Prod Real (Prod Real Real)) (fun (x : Prod Real (Prod Real Real)) => _example.match_1 (fun (x_1 : Prod Real (Prod Real Real)) => Prod Real (Prod Real Real)) x fun (a_1 b_1 c_1 : Real) => @Prod.mk Real (Prod Real Real) (@HPow.hPow Real Real Real (@instHPow Real Real Real.instPow) (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) a_1 b_1) (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)) (@HSub.hSub Real Real Real (@instHSub Real Real.instSub) b_1 a_1))) (@Prod.mk Real Real (@HPow.hPow Real Real Real (@instHPow Real Real Real.instPow) (@HSub.hSub Real Real Real (@instHSub Real Real.instSub) (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)) (@HPow.hPow Real Real Real (@instHPow Real Real Real.instPow) (@HPow.hPow Real Real Real (@instHPow Real Real Real.instPow) (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) a_1 b_1) (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)) (@HSub.hSub Real Real Real (@instHSub Real Real.instSub) b_1 a_1))) b_1)) (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)) b_1)) (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)))) fun (x : Prod Real (Prod Real Real)) => _example.match_1 (fun (x_1 : Prod Real (Prod Real Real)) => Prod Real (Prod Real Real)) x fun (a_1 b_1 c_1 : Real) => @Prod.mk Real (Prod Real Real) (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) (@Prod.mk Real Real (@HPow.hPow Real Real Real (@instHPow Real Real Real.instPow) (@HSub.hSub Real Real Real (@instHSub Real Real.instSub) (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)) (@HPow.hPow Real Real Real (@instHPow Real Real Real.instPow) (@HPow.hPow Real Real Real (@instHPow Real Real Real.instPow) (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) b_1 c_1) (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)) (@HSub.hSub Real Real Real (@instHSub Real Real.instSub) c_1 b_1))) b_1)) (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)) b_1)) (@HPow.hPow Real Real Real (@instHPow Real Real Real.instPow) (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) b_1 c_1) (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)) (@HSub.hSub Real Real Real (@instHSub Real Real.instSub) c_1 b_1)))))

((ℝ × ℝ × ℝ) → (ℝ × ℝ × ℝ)) × ((ℝ × ℝ × ℝ) → (ℝ × ℝ × ℝ))

[ { "t": "ℝ", "v": null, "name": "a" }, { "t": "ℝ", "v": null, "name": "b" }, { "t": "ℝ", "v": null, "name": "c" }, { "t": "0 < a ∧ a < b ∧ b < c", "v": null, "name": "hi" }, { "t": "(ℝ × ℝ × ℝ) → Prop", "v": null, "name": "P" }, { "t": "(ℝ × ℝ × ℝ) → ℝ", "v": null, "name": "f" }, { "t": "P = fun (x, y, z) => x ≥ 0 ∧ y ≥ 0 ∧ z ≥ 0 ∧ x^b + y^b + z^b = 1", "v": null, "name": "hP" }, { "t": "f = fun (x, y, z) => x^a + y^b + z^c", "v": null, "name": "hf" }, { "t": "(P (answer.1 (a, b, c)) ∧ ∀ x y z : ℝ, P (x, y, z) →\nf (x, y, z) ≤ f (answer.1 (a, b, c))) ∧\n(P (answer.2 (a, b, c)) ∧ ∀ x y z : ℝ, P (x, y, z) →\nf (x, y, z) ≥ f (answer.2 (a, b, c)))", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1975_a3", "tags": [ "algebra" ] }

Let $n = 2m$, where $m$ is an odd integer greater than 1. Let $\theta = e^{2\pi i/n}$. Expression $(1 - \theta)^{-1}$ explicitly as a polynomial in $\theta$ \[ a_k \theta^k + a_{k-1}\theta^{k-1} + \dots + a_1\theta + a_0\], with integer coefficients $a_i$.

$\sum_{j=0}^{(m-1)/2} \theta^{2j+1}$

Show that the solution is the polynomial $0 + \theta + \theta^3 + \dots + \theta^{m-2}$, alternating consecutive coefficients between 0 and 1.

open Polynomial Real Complex

[]

@Eq (Nat → @Polynomial Int Int.instSemiring) answer fun (m_1 : Nat) => @Finset.sum Nat (@Polynomial Int Int.instSemiring) (@NonUnitalNonAssocSemiring.toAddCommMonoid (@Polynomial Int Int.instSemiring) (@NonUnitalNonAssocCommSemiring.toNonUnitalNonAssocSemiring (@Polynomial Int Int.instSemiring) (@NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring (@Polynomial Int Int.instSemiring) (@NonUnitalCommRing.toNonUnitalNonAssocCommRing (@Polynomial Int Int.instSemiring) (@CommRing.toNonUnitalCommRing (@Polynomial Int Int.instSemiring) (@Polynomial.commRing Int Int.instCommRing)))))) (Finset.range (@HDiv.hDiv Nat Nat Nat (@instHDiv Nat Nat.instDiv) (@HSub.hSub Nat Nat Nat (@instHSub Nat instSubNat) m_1 (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))) fun (j : Nat) => @HPow.hPow (@Polynomial Int Int.instSemiring) Nat (@Polynomial Int Int.instSemiring) (@instHPow (@Polynomial Int Int.instSemiring) Nat (@Monoid.toNatPow (@Polynomial Int Int.instSemiring) (@MonoidWithZero.toMonoid (@Polynomial Int Int.instSemiring) (@Semiring.toMonoidWithZero (@Polynomial Int Int.instSemiring) (@Polynomial.semiring Int Int.instSemiring))))) (@Polynomial.X Int Int.instSemiring) (@HAdd.hAdd Nat Nat Nat (@instHAdd Nat instAddNat) (@HMul.hMul Nat Nat Nat (@instHMul Nat instMulNat) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) j) (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))

ℕ → Polynomial ℤ

[ { "t": "ℕ", "v": null, "name": "m" }, { "t": "Odd m ∧ m > 1", "v": null, "name": "hm" }, { "t": "ℂ", "v": null, "name": "θ" }, { "t": "θ = cexp (2 * Real.pi * I / (2 * m))", "v": null, "name": "hθ" }, { "t": "1/(1 - θ) = Polynomial.aeval θ (answer m)", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1975_a4", "tags": [ "algebra" ] }

Let $H$ be a subgroup of the additive group of ordered pairs of integers under componentwise addition. If $H$ is generated by the elements $(3, 8)$, $(4, -1)$, and $(5, 4)$, then $H$ is also generated by two elements $(1, b)$ and $(0, a)$ for some integer $b$ and positive integer $a$. Find $a$.

7

$a$ must equal $7$.

open Polynomial Real Complex

[]

@Eq Int answer (@OfNat.ofNat Int (nat_lit 7) (@instOfNat (nat_lit 7)))

ℤ

[ { "t": "Set (ℤ × ℤ)", "v": null, "name": "H" }, { "t": "H = {h : (ℤ × ℤ) | ∃ u v w : ℤ, h = (u*3 + v*4 + w*5, u*8 + v*(-1) + w*4)}", "v": null, "name": "hH" }, { "t": "(∃ b : ℤ, H = {h : (ℤ × ℤ) | ∃ u v : ℤ, h = (u, u*b + v*answer)}) ∧ answer > 0", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1975_b1", "tags": [ "abstract_algebra", "number_theory" ] }

Let $s_k (a_1, a_2, \dots, a_n)$ denote the $k$-th elementary symmetric function; that is, the sum of all $k$-fold products of the $a_i$. For example, $s_1 (a_1, \dots, a_n) = \sum_{i=1}^{n} a_i$, and $s_2 (a_1, a_2, a_3) = a_1a_2 + a_2a_3 + a_1a_3$. Find the supremum $M_k$ (which is never attained) of $$\frac{s_k (a_1, a_2, \dots, a_n)}{(s_1 (a_1, a_2, \dots, a_n))^k}$$ across all $n$-tuples $(a_1, a_2, \dots, a_n)$ of positive real numbers with $n \ge k$.

fun k : ℕ => (1: ℝ)/(Nat.factorial k)

The supremum $M_k$ is $\frac{1}{k!}$.

open Polynomial Real Complex Matrix Filter Topology Multiset

[]

@Eq (Nat → Real) answer fun (k : Nat) => @HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)) (@Nat.cast Real Real.instNatCast (Nat.factorial k))

ℕ → ℝ

[ { "t": "∀ k : ℕ, k > 0 → (∀ a : Multiset ℝ, (∀ i ∈ a, i > 0) ∧ card a ≥ k →\n(esymm a k)/(esymm a 1)^k ≤ answer k) ∧\n∀ M : ℝ, M < answer k → (∃ a : Multiset ℝ, (∀ i ∈ a, i > 0) ∧ card a ≥ k ∧\n(esymm a k)/(esymm a 1)^k > M)", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1975_b3", "tags": [ "analysis", "algebra" ] }

Let $C = \{(x, y) \in \mathbb{R}^2 : x^2 + y^2 = 1\}$ denote the unit circle. Does there exist $B \subseteq C$ for which $B$ is topologically closed and contains exactly one point from each pair of diametrically opposite points in $C$?

False

Such $B$ does not exist.

open Polynomial Real Complex Matrix Filter Topology Multiset

[]

@Eq Prop answer False

Prop

[ { "t": "ℝ × ℝ → Prop", "v": null, "name": "P" }, { "t": "P = fun (x, y) => x^2 + y^2 = 1", "v": null, "name": "hP" }, { "t": "(∃ B ⊆ setOf P, IsClosed B ∧ ∀ x y : ℝ, P (x, y) → Xor' ((x, y) ∈ B) ((-x, -y) ∈ B)) ↔ answer", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1975_b4", "tags": [ "analysis" ] }

Find all integer solutions $(p, r, q, s)$ of the equation $|p^r - q^s| = 1$, where $p$ and $q$ are prime and $r$ and $s$ are greater than $1$.

{(3, 2, 2, 3), (2, 3, 3, 2)}

The only solutions are $(p, r, q, s) = (3, 2, 2, 3)$ and $(p, r, q, s) = (2, 3, 3, 2)$.

null

[]

@Eq (Set (Prod Nat (Prod Nat (Prod Nat Nat)))) answer (@Insert.insert (Prod Nat (Prod Nat (Prod Nat Nat))) (Set (Prod Nat (Prod Nat (Prod Nat Nat)))) (@Set.instInsert (Prod Nat (Prod Nat (Prod Nat Nat)))) (@Prod.mk Nat (Prod Nat (Prod Nat Nat)) (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3))) (@Prod.mk Nat (Prod Nat Nat) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) (@Prod.mk Nat Nat (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))))) (@Singleton.singleton (Prod Nat (Prod Nat (Prod Nat Nat))) (Set (Prod Nat (Prod Nat (Prod Nat Nat)))) (@Set.instSingletonSet (Prod Nat (Prod Nat (Prod Nat Nat)))) (@Prod.mk Nat (Prod Nat (Prod Nat Nat)) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) (@Prod.mk Nat (Prod Nat Nat) (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3))) (@Prod.mk Nat Nat (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3))) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))))))

Set (ℕ × ℕ × ℕ × ℕ)

[ { "t": "{a : ℕ × ℕ × ℕ × ℕ | Nat.Prime a.1 ∧ Nat.Prime a.2.2.1 ∧ a.2.1 > 1 ∧ a.2.2.2 > 1 ∧ |(a.1^a.2.1 : ℤ) - a.2.2.1^a.2.2.2| = 1} = answer", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1976_a3", "tags": [ "number_theory" ] }

Let $r$ be a real root of $P(x) = x^3 + ax^2 + bx - 1$, where $a$ and $b$ are integers and $P$ is irreducible over the rationals. Suppose that $r + 1$ is a root of $x^3 + cx^2 + dx + 1$, where $c$ and $d$ are also integers. Express another root $s$ of $P$ as a function of $r$ that does not depend on the values of $a$, $b$, $c$, or $d$.

$\left(-\frac{1}{r + 1}, -\frac{r + 1}{r}\right)$

The possible answers are $s = -\frac{1}{r + 1}$ and $s = -\frac{r + 1}{r}$.

open Polynomial

[]

@Eq (Prod (Real → Real) (Real → Real)) answer (@Prod.mk (Real → Real) (Real → Real) (fun (r_1 : Real) => @HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) (@Neg.neg Real Real.instNeg (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))) (@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd) r_1 (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)))) fun (r_1 : Real) => @HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) (@Neg.neg Real Real.instNeg (@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd) r_1 (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)))) r_1)

(ℝ → ℝ) × (ℝ → ℝ)

[ { "t": "ℤ", "v": null, "name": "a" }, { "t": "ℤ", "v": null, "name": "b" }, { "t": "ℤ", "v": null, "name": "c" }, { "t": "ℤ", "v": null, "name": "d" }, { "t": "ℝ", "v": null, "name": "r" }, { "t": "Polynomial ℚ", "v": null, "name": "P" }, { "t": "Polynomial ℚ", "v": null, "name": "Q" }, { "t": "P = Polynomial.X^3 + (Polynomial.C (a : ℚ))*Polynomial.X^2 + (Polynomial.C (b : ℚ))*Polynomial.X - (Polynomial.C 1) ∧ Polynomial.aeval r P = 0 ∧ Irreducible P", "v": null, "name": "hP" }, { "t": "Q = Polynomial.X^3 + (Polynomial.C (c : ℚ))*Polynomial.X^2 + (Polynomial.C (d : ℚ))*Polynomial.X + (Polynomial.C 1) ∧ Polynomial.aeval (r + 1) Q = 0", "v": null, "name": "hQ" }, { "t": "∃ s : ℝ, Polynomial.aeval s P = 0 ∧ (s = answer.1 r ∨ s = answer.2 r)", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1976_a4", "tags": [ "algebra" ] }

Find $$\lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^{n}\left(\left\lfloor \frac{2n}{k} \right\rfloor - 2\left\lfloor \frac{n}{k} \right\rfloor\right).$$ Your answer should be in the form $\ln(a) - b$, where $a$ and $b$ are positive integers.

ln(4) - 1

The limit equals $\ln(4) - 1$, so $a = 4$ and $b = 1$.

open Polynomial Filter Topology

[]

@Eq (Prod Nat Nat) answer (@Prod.mk Nat Nat (@OfNat.ofNat Nat (nat_lit 4) (instOfNatNat (nat_lit 4))) (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))

ℕ × ℕ

[ { "t": "Tendsto (fun n : ℕ => ((1 : ℝ)/n)*∑ k in Finset.Icc (1 : ℤ) n, (Int.floor ((2*n)/k) - 2*Int.floor (n/k))) atTop\n (𝓝 (Real.log answer.1 - answer.2))", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1976_b1", "tags": [ "analysis" ] }

Let $G$ be a group generated by two elements $A$ and $B$; i.e., every element of $G$ can be expressed as a finite word $A^{n_1}B^{n_2} \cdots A^{n_{k-1}}B^{n_k}$, where the $n_i$ can assume any integer values and $A^0 = B^0 = 1$. Further assume that $A^4 = B^7 = ABA^{-1}B = 1$, but $A^2 \ne 1$ and $B \ne 1$. Find the number of elements of $G$ than can be written as $C^2$ for some $C \in G$ and express each such square as a word in $A$ and $B$.

(8, {[(0, 0)], [(2, 0)], [(0, 1)], [(0, 2)], [(0, 3)], [(0, 4)], [(0, 5)], [(0, 6)]})

There are $8$ such squares: $1$, $A^2$, $B$, $B^2$, $B^3$, $B^4$, $B^5$, and $B^6$.

open Polynomial Filter Topology

[]

@Eq (Prod Nat (Set (List (Prod Int Int)))) answer (@Prod.mk Nat (Set (List (Prod Int Int))) (@OfNat.ofNat Nat (nat_lit 8) (instOfNatNat (nat_lit 8))) (@Insert.insert (List (Prod Int Int)) (Set (List (Prod Int Int))) (@Set.instInsert (List (Prod Int Int))) (@List.cons (Prod Int Int) (@Prod.mk Int Int (@OfNat.ofNat Int (nat_lit 0) (@instOfNat (nat_lit 0))) (@OfNat.ofNat Int (nat_lit 0) (@instOfNat (nat_lit 0)))) (@List.nil (Prod Int Int))) (@Insert.insert (List (Prod Int Int)) (Set (List (Prod Int Int))) (@Set.instInsert (List (Prod Int Int))) (@List.cons (Prod Int Int) (@Prod.mk Int Int (@OfNat.ofNat Int (nat_lit 2) (@instOfNat (nat_lit 2))) (@OfNat.ofNat Int (nat_lit 0) (@instOfNat (nat_lit 0)))) (@List.nil (Prod Int Int))) (@Insert.insert (List (Prod Int Int)) (Set (List (Prod Int Int))) (@Set.instInsert (List (Prod Int Int))) (@List.cons (Prod Int Int) (@Prod.mk Int Int (@OfNat.ofNat Int (nat_lit 0) (@instOfNat (nat_lit 0))) (@OfNat.ofNat Int (nat_lit 1) (@instOfNat (nat_lit 1)))) (@List.nil (Prod Int Int))) (@Insert.insert (List (Prod Int Int)) (Set (List (Prod Int Int))) (@Set.instInsert (List (Prod Int Int))) (@List.cons (Prod Int Int) (@Prod.mk Int Int (@OfNat.ofNat Int (nat_lit 0) (@instOfNat (nat_lit 0))) (@OfNat.ofNat Int (nat_lit 2) (@instOfNat (nat_lit 2)))) (@List.nil (Prod Int Int))) (@Insert.insert (List (Prod Int Int)) (Set (List (Prod Int Int))) (@Set.instInsert (List (Prod Int Int))) (@List.cons (Prod Int Int) (@Prod.mk Int Int (@OfNat.ofNat Int (nat_lit 0) (@instOfNat (nat_lit 0))) (@OfNat.ofNat Int (nat_lit 3) (@instOfNat (nat_lit 3)))) (@List.nil (Prod Int Int))) (@Insert.insert (List (Prod Int Int)) (Set (List (Prod Int Int))) (@Set.instInsert (List (Prod Int Int))) (@List.cons (Prod Int Int) (@Prod.mk Int Int (@OfNat.ofNat Int (nat_lit 0) (@instOfNat (nat_lit 0))) (@OfNat.ofNat Int (nat_lit 4) (@instOfNat (nat_lit 4)))) (@List.nil (Prod Int Int))) (@Insert.insert (List (Prod Int Int)) (Set (List (Prod Int Int))) (@Set.instInsert (List (Prod Int Int))) (@List.cons (Prod Int Int) (@Prod.mk Int Int (@OfNat.ofNat Int (nat_lit 0) (@instOfNat (nat_lit 0))) (@OfNat.ofNat Int (nat_lit 5) (@instOfNat (nat_lit 5)))) (@List.nil (Prod Int Int))) (@Singleton.singleton (List (Prod Int Int)) (Set (List (Prod Int Int))) (@Set.instSingletonSet (List (Prod Int Int))) (@List.cons (Prod Int Int) (@Prod.mk Int Int (@OfNat.ofNat Int (nat_lit 0) (@instOfNat (nat_lit 0))) (@OfNat.ofNat Int (nat_lit 6) (@instOfNat (nat_lit 6)))) (@List.nil (Prod Int Int))))))))))))

ℕ × Set (List (ℤ × ℤ))

[ { "t": "Type*", "v": null, "name": "G" }, { "t": "Group G", "v": null, "name": null }, { "t": "G", "v": null, "name": "A" }, { "t": "G", "v": null, "name": "B" }, { "t": "List (ℤ × ℤ) → G", "v": null, "name": "word" }, { "t": "word = fun w : List (ℤ × ℤ) => (List.map (fun t : ℤ × ℤ => A^(t.1)*B^(t.2)) w).prod", "v": null, "name": "hword" }, { "t": "∀ g : G, ∃ w : List (ℤ × ℤ), g = word w", "v": null, "name": "hG" }, { "t": "A^4 = 1 ∧ A^2 ≠ 1", "v": null, "name": "hA" }, { "t": "B^7 = 1 ∧ B ≠ 1", "v": null, "name": "hB" }, { "t": "A*B*A^(-(1 : ℤ))*B = 1", "v": null, "name": "h1" }, { "t": "Set G", "v": null, "name": "S" }, { "t": "S = {g : G | ∃ C : G, C^2 = g}", "v": null, "name": "hS" }, { "t": "S.ncard = answer.1 ∧ S = {word w | w ∈ answer.2}", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1976_b2", "tags": [ "abstract_algebra" ] }

Find $$\sum_{k=0}^{n} (-1)^k {n \choose k} (x - k)^n.$$

fun n => C (Nat.factorial n)

The sum equals $n!$.

open Polynomial Filter Topology ProbabilityTheory MeasureTheory

[]

@Eq (Nat → @Polynomial Int Int.instSemiring) answer fun (n : Nat) => @DFunLike.coe (@RingHom Int (@Polynomial Int Int.instSemiring) (@Semiring.toNonAssocSemiring Int Int.instSemiring) (@Semiring.toNonAssocSemiring (@Polynomial Int Int.instSemiring) (@Polynomial.semiring Int Int.instSemiring))) Int (fun (x : Int) => @Polynomial Int Int.instSemiring) (@RingHom.instFunLike Int (@Polynomial Int Int.instSemiring) (@Semiring.toNonAssocSemiring Int Int.instSemiring) (@Semiring.toNonAssocSemiring (@Polynomial Int Int.instSemiring) (@Polynomial.semiring Int Int.instSemiring))) (@Polynomial.C Int Int.instSemiring) (@Nat.cast Int instNatCastInt (Nat.factorial n))

ℕ → Polynomial ℤ

[ { "t": "∀ n : ℕ, ∑ k in Finset.range (n + 1), C ((-(1 : ℤ))^k * Nat.choose n k) * (X - (C (k : ℤ)))^n = answer n", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1976_b5", "tags": [ "algebra" ] }

Show that if four distinct points of the curve $y = 2x^4 + 7x^3 + 3x - 5$ are collinear, then their average $x$-coordinate is some constant $k$. Find $k$.

$-7/8$

Prove that $k = -\frac{7}{8}$.

null

[]

@Eq Real answer (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) (@Neg.neg Real Real.instNeg (@OfNat.ofNat Real (nat_lit 7) (@instOfNatAtLeastTwo Real (nat_lit 7) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 5) (instOfNatNat (nat_lit 5))))))) (@OfNat.ofNat Real (nat_lit 8) (@instOfNatAtLeastTwo Real (nat_lit 8) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))))))

ℝ

[ { "t": "ℝ → ℝ", "v": null, "name": "y" }, { "t": "y = fun x ↦ 2 * x ^ 4 + 7 * x ^ 3 + 3 * x - 5", "v": null, "name": "hy" }, { "t": "Finset ℝ", "v": null, "name": "S" }, { "t": "S.card = 4", "v": null, "name": "hS" }, { "t": "Collinear ℝ {P : Fin 2 → ℝ | P 0 ∈ S ∧ P 1 = y (P 0)}", "v": null, "name": "h_collinear" }, { "t": "(∑ x in S, x) / 4 = answer", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1977_a1", "tags": [ "algebra" ] }

Find all real solutions $(a, b, c, d)$ to the equations $a + b + c = d$, $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{d}$.

$d = a \land b = -c \lor d = b \land a = -c \lor d = c \land a = -b$

Prove that the solutions are $d = a$ and $b = -c$, $d = b$ and $a = -c$, or $d = c$ and $a = -b$, with $a, b, c, d$ nonzero.

null

[]

@Eq (Real → Real → Real → Real → Prop) answer fun (a_1 b_1 c_1 d_1 : Real) => Or (And (@Eq Real d_1 a_1) (@Eq Real b_1 (@Neg.neg Real Real.instNeg c_1))) (Or (And (@Eq Real d_1 b_1) (@Eq Real a_1 (@Neg.neg Real Real.instNeg c_1))) (And (@Eq Real d_1 c_1) (@Eq Real a_1 (@Neg.neg Real Real.instNeg b_1))))

ℝ → ℝ → ℝ → ℝ → Prop

[ { "t": "ℝ", "v": null, "name": "a" }, { "t": "ℝ", "v": null, "name": "b" }, { "t": "ℝ", "v": null, "name": "c" }, { "t": "ℝ", "v": null, "name": "d" }, { "t": "answer a b c d ↔\n a ≠ 0 → b ≠ 0 → c ≠ 0 → d ≠ 0 → (a + b + c = d ∧ 1 / a + 1 / b + 1 / c = 1 / d)", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1977_a2", "tags": [ "algebra" ] }

Let $f, g, h$ be functions $\mathbb{R} \to \mathbb{R}$. Find an expression for $h(x)$ in terms of $f$ and $g$ such that $f(x) = \frac{h(x + 1) + h(x - 1)}{2}$ and $g(x) = \frac{h(x + 4) + h(x - 4)}{2}$.

$h(x) = g(x) - f(x - 3) + f(x - 1) + f(x + 1) - f(x + 3)$

Prove that $h(x) = g(x) - f(x - 3) + f(x - 1) + f(x + 1) - f(x + 3)$ suffices.

null

[]

@Eq ((Real → Real) → (Real → Real) → Real → Real) answer fun (f_1 g_1 : Real → Real) (x : Real) => @HSub.hSub Real Real Real (@instHSub Real Real.instSub) (@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd) (@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd) (@HSub.hSub Real Real Real (@instHSub Real Real.instSub) (g_1 x) (f_1 (@HSub.hSub Real Real Real (@instHSub Real Real.instSub) x (@OfNat.ofNat Real (nat_lit 3) (@instOfNatAtLeastTwo Real (nat_lit 3) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))))))) (f_1 (@HSub.hSub Real Real Real (@instHSub Real Real.instSub) x (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))))) (f_1 (@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd) x (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))))) (f_1 (@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd) x (@OfNat.ofNat Real (nat_lit 3) (@instOfNatAtLeastTwo Real (nat_lit 3) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))))))

(ℝ → ℝ) → (ℝ → ℝ) → (ℝ → ℝ)

[ { "t": "ℝ → ℝ", "v": null, "name": "f" }, { "t": "ℝ → ℝ", "v": null, "name": "g" }, { "t": "ℝ → ℝ", "v": null, "name": "h" }, { "t": "∀ x, f x = (h (x + 1) + h (x - 1)) / 2", "v": null, "name": "hf" }, { "t": "∀ x, g x = (h (x + 4) + h (x - 4)) / 2", "v": null, "name": "hg" }, { "t": "h = answer f g", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1977_a3", "tags": [ "algebra" ] }

Find $\sum_{n=0}^{\infty} \frac{x^{2^n}}{1 - x^{2^{n+1}}}$ as a rational function of $x$ for $x \in (0, 1)$.

$\frac{x}{1 - x}$

Prove that the sum equals $\frac{x}{1 - x}$.

open RingHom Set

[]

@Eq (@RatFunc Real Real.commRing) answer (@HDiv.hDiv (@RatFunc Real Real.commRing) (@RatFunc Real Real.commRing) (@RatFunc Real Real.commRing) (@instHDiv (@RatFunc Real Real.commRing) (@RatFunc.instDiv Real Real.commRing Real.instIsDomain)) (@RatFunc.X Real Real.commRing Real.instIsDomain) (@HSub.hSub (@RatFunc Real Real.commRing) (@RatFunc Real Real.commRing) (@RatFunc Real Real.commRing) (@instHSub (@RatFunc Real Real.commRing) (@RatFunc.instSub Real Real.commRing)) (@OfNat.ofNat (@RatFunc Real Real.commRing) (nat_lit 1) (@One.toOfNat1 (@RatFunc Real Real.commRing) (@RatFunc.instOne Real Real.commRing))) (@RatFunc.X Real Real.commRing Real.instIsDomain)))

RatFunc ℝ

[ { "t": "ℝ", "v": null, "name": "x" }, { "t": "x ∈ Ioo 0 1", "v": null, "name": "hx" }, { "t": "answer.eval (id ℝ) x = ∑' n : ℕ, x ^ 2 ^ n / (1 - x ^ 2 ^ (n + 1))", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1977_a4", "tags": [ "algebra", "analysis" ] }

Let $X$ be the square $[0, 1] \times [0, 1]$, and let $f : X \to \mathbb{R}$ be continuous. If $\int_Y f(x, y) \, dx \, dy = 0$ for all squares $Y$ such that \begin{itemize} \item[(1)] $Y \subseteq X$, \item[(2)] $Y$ has sides parallel to those of $X$, \item[(3)] at least one of $Y$'s sides is contained in the boundary of $X$, \end{itemize} is it true that $f(x, y) = 0$ for all $x, y$?

True

Prove that $f(x,y)$ must be identically zero.

open RingHom Set Nat

[]

@Eq Prop answer True

Prop

[ { "t": "Set (ℝ × ℝ)", "v": null, "name": "X" }, { "t": "X = Set.prod (Icc 0 1) (Icc 0 1)", "v": null, "name": "hX" }, { "t": "(ℝ × ℝ) → ℝ", "v": null, "name": "room" }, { "t": "room = fun (a,b) ↦ min (min a (1 - a)) (min b (1 - b))", "v": null, "name": "hroom" }, { "t": "(∀ f : (ℝ × ℝ) → ℝ, Continuous f → (∀ P ∈ X, ∫ x in (P.1 - room P)..(P.1 + room P), ∫ y in (P.2 - room P)..(P.2 + room P), f (x, y) = 0) → (∀ P ∈ X, f P = 0)) ↔ answer", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1977_a6", "tags": [ "analysis" ] }

Find $\prod_{n=2}^{\infty} \frac{(n^3 - 1)}{(n^3 + 1)}$.

2/3

Prove that the product equals $\frac{2}{3}$.

open RingHom Set Nat Filter Topology

[]

@Eq Real answer (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) (@OfNat.ofNat Real (nat_lit 2) (@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0)))))) (@OfNat.ofNat Real (nat_lit 3) (@instOfNatAtLeastTwo Real (nat_lit 3) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))))

ℝ

[ { "t": "Tendsto (fun N ↦ ∏ n in Finset.Icc (2 : ℤ) N, ((n : ℝ) ^ 3 - 1) / (n ^ 3 + 1)) atTop (𝓝 answer)", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1977_b1", "tags": [ "algebra", "analysis" ] }

An ordered triple $(a, b, c)$ of positive irrational numbers with $a + b + c = 1$ is considered $\textit{balanced}$ if all three elements are less than $\frac{1}{2}$. If a triple is not balanced, we can perform a ``balancing act'' $B$ defined by $B(a, b, c) = (f(a), f(b), f(c))$, where $f(x) = 2x - 1$ if $x > 1/2$ and $f(x) = 2x$ otherwise. Will finitely many iterations of this balancing act always eventually produce a balanced triple?

False

Not necessarily.

open RingHom Set Nat Filter Topology

[]

@Eq Prop answer False

Prop

[ { "t": "ℝ × ℝ × ℝ → Prop", "v": null, "name": "P" }, { "t": "ℝ × ℝ × ℝ → Prop", "v": null, "name": "balanced" }, { "t": "ℝ × ℝ × ℝ → ℝ × ℝ × ℝ", "v": null, "name": "B" }, { "t": "P = fun (a, b, c) => Irrational a ∧ Irrational b ∧ Irrational c ∧ a > 0 ∧ b > 0 ∧ c > 0 ∧ a + b + c = 1", "v": null, "name": "hP" }, { "t": "balanced = fun (a, b, c) => a < 1/2 ∧ b < 1/2 ∧ c < 1/2", "v": null, "name": "hbalanced" }, { "t": "B = fun (a, b, c) => (ite (a > 1/2) (2*a - 1) (2*a), ite (b > 1/2) (2*b - 1) (2*b), ite (c > 1/2) (2*c - 1) (2*c))", "v": null, "name": "hB" }, { "t": "(∀ t : ℝ × ℝ × ℝ, P t → ∃ n : ℕ, balanced (B^[n] t)) ↔ answer", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1977_b3", "tags": [ "algebra" ] }

Let $p(x) = 2(x^6 + 1) + 4(x^5 + x) + 3(x^4 + x^2) + 5x^3$. For $k$ with $0 < k < 5$, let \[ I_k = \int_0^{\infty} \frac{x^k}{p(x)} \, dx. \] For which $k$ is $I_k$ smallest?

2

Show that $I_k$ is smallest for $k = 2$.

open Set Polynomial

[]

@Eq Nat answer (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))

ℕ

[ { "t": "Polynomial ℝ", "v": null, "name": "p" }, { "t": "p = 2 * (Polynomial.X ^ 6 + 1) + 4 * (Polynomial.X ^ 5 + Polynomial.X) + 3 * (Polynomial.X ^ 4 + Polynomial.X ^ 2) + 5 * Polynomial.X ^ 3", "v": null, "name": "hp" }, { "t": "ℕ → ℝ", "v": null, "name": "I" }, { "t": "I = fun k ↦ ∫ x in Ioi 0, x ^ k / p.eval x", "v": null, "name": "hI" }, { "t": "IsLeast {y | ∃ k ∈ Ioo 0 5, I k = y} answer", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1978_a3", "tags": [ "analysis", "algebra" ] }

Find \[ \sum_{i=1}^{\infty} \sum_{j=1}^{\infty} \frac{1}{i^2j + 2ij + ij^2}. \]

7 / 4

Prove that the sum evaluates to $\frac{7}{4}$.

open Set Real

[]

@Eq Rat answer (@HDiv.hDiv Rat Rat Rat (@instHDiv Rat Rat.instDiv) (@OfNat.ofNat Rat (nat_lit 7) (@Rat.instOfNat (nat_lit 7))) (@OfNat.ofNat Rat (nat_lit 4) (@Rat.instOfNat (nat_lit 4))))

ℚ

[ { "t": "(∑' i : ℕ+, ∑' j : ℕ+, (1 : ℚ) / (i ^ 2 * j + 2 * i * j + i * j ^ 2)) = answer", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1978_b2", "tags": [ "algebra", "analysis" ] }

Find the real polynomial $p(x)$ of degree $4$ with largest possible coefficient of $x^4$ such that $p([-1, 1]) \subseteq [0, 1]$.

$4x^4 - 4x^2 + 1$

Prove that $p(x) = 4x^4 - 4x^2 + 1$.

open Set Real Filter Topology Polynomial

[]

@Eq (@Polynomial Real Real.semiring) answer (@HAdd.hAdd (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@instHAdd (@Polynomial Real Real.semiring) (@Polynomial.add' Real Real.semiring)) (@HSub.hSub (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@instHSub (@Polynomial Real Real.semiring) (@Polynomial.sub Real Real.instRing)) (@HMul.hMul (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@instHMul (@Polynomial Real Real.semiring) (@Polynomial.mul' Real Real.semiring)) (@OfNat.ofNat (@Polynomial Real Real.semiring) (nat_lit 4) (@instOfNatAtLeastTwo (@Polynomial Real Real.semiring) (nat_lit 4) (@Polynomial.instNatCast Real Real.semiring) (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))) (@HPow.hPow (@Polynomial Real Real.semiring) Nat (@Polynomial Real Real.semiring) (@instHPow (@Polynomial Real Real.semiring) Nat (@Monoid.toNatPow (@Polynomial Real Real.semiring) (@MonoidWithZero.toMonoid (@Polynomial Real Real.semiring) (@Semiring.toMonoidWithZero (@Polynomial Real Real.semiring) (@Polynomial.semiring Real Real.semiring))))) (@Polynomial.X Real Real.semiring) (@OfNat.ofNat Nat (nat_lit 4) (instOfNatNat (nat_lit 4))))) (@HMul.hMul (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@instHMul (@Polynomial Real Real.semiring) (@Polynomial.mul' Real Real.semiring)) (@OfNat.ofNat (@Polynomial Real Real.semiring) (nat_lit 4) (@instOfNatAtLeastTwo (@Polynomial Real Real.semiring) (nat_lit 4) (@Polynomial.instNatCast Real Real.semiring) (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))) (@HPow.hPow (@Polynomial Real Real.semiring) Nat (@Polynomial Real Real.semiring) (@instHPow (@Polynomial Real Real.semiring) Nat (@Monoid.toNatPow (@Polynomial Real Real.semiring) (@MonoidWithZero.toMonoid (@Polynomial Real Real.semiring) (@Semiring.toMonoidWithZero (@Polynomial Real Real.semiring) (@Polynomial.semiring Real Real.semiring))))) (@Polynomial.X Real Real.semiring) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))) (@OfNat.ofNat (@Polynomial Real Real.semiring) (nat_lit 1) (@One.toOfNat1 (@Polynomial Real Real.semiring) (@Polynomial.one Real Real.semiring))))

Polynomial ℝ

[ { "t": "Set (Polynomial ℝ)", "v": null, "name": "S" }, { "t": "S = {p : Polynomial ℝ | p.degree = 4 ∧ ∀ x ∈ Icc (-1 : ℝ) 1, p.eval x ∈ Icc 0 1}", "v": null, "name": "hS" }, { "t": "answer ∈ S ∧ (∀ p ∈ S, p.coeff 4 ≤ answer.coeff 4)", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1978_b5", "tags": [ "algebra" ] }

For which positive integers $n$ and $a_1, a_2, \dots, a_n$ with $\sum_{i = 1}^{n} a_i = 1979$ does $\prod_{i = 1}^{n} a_i$ attain the greatest value?

Multiset.replicate 659 3 + {2}

$n$ equals $660$; all but one of the $a_i$ equal $3$ and the remaining $a_i$ equals $2$.

null

[]

@Eq (Multiset Nat) answer (@HAdd.hAdd (Multiset Nat) (Multiset Nat) (Multiset Nat) (@instHAdd (Multiset Nat) (@Multiset.instAdd Nat)) (@Multiset.replicate Nat (@OfNat.ofNat Nat (nat_lit 659) (instOfNatNat (nat_lit 659))) (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) (@Singleton.singleton Nat (Multiset Nat) (@Multiset.instSingleton Nat) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))

Multiset ℕ

[ { "t": "Multiset ℕ → Prop", "v": null, "name": "P" }, { "t": "∀ a, P a ↔ Multiset.card a > 0 ∧ (∀ i ∈ a, i > 0) ∧ a.sum = 1979", "v": null, "name": "hP" }, { "t": "P answer ∧ ∀ a : Multiset ℕ, P a → answer.prod ≥ a.prod", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1979_a1", "tags": [ "algebra" ] }

For which real numbers $k$ does there exist a continuous function $f : \mathbb{R} \to \mathbb{R}$ such that $f(f(x)) = kx^9$ for all real $x$?

$k \geq 0$

Such a function exists if and only if $k \ge 0$.

null

[]

@Eq (Real → Prop) answer fun (k_1 : Real) => @GE.ge Real Real.instLE k_1 (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero))

ℝ → Prop

[ { "t": "ℝ", "v": null, "name": "k" }, { "t": "answer k ↔ ∃ f : ℝ → ℝ, Continuous f ∧ ∀ x : ℝ, f (f x) = k * x^9", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1979_a2", "tags": [ "analysis", "algebra" ] }

Let $x_1, x_2, x_3, \dots$ be a sequence of nonzero real numbers such that $$x_n = \frac{x_{n-2}x_{n-1}}{2x_{n-2}-x_{n-1}}$$ for all $n \ge 3$. For which real values of $x_1$ and $x_2$ does $x_n$ attain integer values for infinitely many $n$?

fun (a, b) => ∃ m : ℤ, a = m ∧ b = m

We must have $x_1 = x_2 = m$ for some integer $m$.

null

[]

@Eq (Prod Real Real → Prop) answer fun (x_1 : Prod Real Real) => _example.match_1 (fun (x_2 : Prod Real Real) => Prop) x_1 fun (a b : Real) => @Exists Int fun (m : Int) => And (@Eq Real a (@Int.cast Real Real.instIntCast m)) (@Eq Real b (@Int.cast Real Real.instIntCast m))

(ℝ × ℝ) → Prop

[ { "t": "ℕ → ℝ", "v": null, "name": "x" }, { "t": "∀ n : ℕ, x n ≠ 0 ∧ (n ≥ 3 → x n = (x (n - 2))*(x (n - 1))/(2*(x (n - 2)) - (x (n - 1))))", "v": null, "name": "hx" }, { "t": "(∀ m : ℕ, ∃ n : ℕ, n > m ∧ ∃ a : ℤ, a = x n) ↔ answer (x 1, x 2)", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1979_a3", "tags": [ "algebra" ] }

Let $A$ be a set of $2n$ points in the plane, $n$ colored red and $n$ colored blue, such that no three points in $A$ are collinear. Must there exist $n$ closed straight line segments, each connecting one red and one blue point in $A$, such that no two of the $n$ line segments intersect?

True

Such line segments must exist.

open Set

[]

@Eq Prop answer True

Prop

[ { "t": "Finset (Fin 2 → ℝ) × Finset (Fin 2 → ℝ) → Prop", "v": null, "name": "A" }, { "t": "A = fun (R, B) => R.card = B.card ∧ R ∩ B = ∅ ∧\n ∀ u : Finset (Fin 2 → ℝ), u ⊆ R ∪ B → u.card = 3 → ¬Collinear ℝ (u : Set (Fin 2 → ℝ))", "v": null, "name": "hA" }, { "t": "(Fin 2 → ℝ) × (Fin 2 → ℝ) → ℝ → (Fin 2 → ℝ)", "v": null, "name": "w" }, { "t": "w = fun (P, Q) => fun x : ℝ => fun i : Fin 2 => x * P i + (1 - x) * Q i", "v": null, "name": "hw" }, { "t": "answer ↔\n (∀ R B, A (R, B) →\n ∃ v : Finset ((Fin 2 → ℝ) × (Fin 2 → ℝ)),\n (∀ L ∈ v, ∀ M ∈ v, L ≠ M → ∀ x ∈ Icc 0 1, ∀ y ∈ Icc 0 1,\n Real.sqrt ((w (L.1, L.2) x 0 - w (M.1, M.2) y 0)^2 + (w (L.1, L.2) x 1 - w (M.1, M.2) y 1)^2) ≠ 0) ∧\n v.card = R.card ∧ ∀ L ∈ v, L.1 ∈ R ∧ L.2 ∈ B)", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1979_a4", "tags": [ "geometry", "combinatorics" ] }

If $0 < a < b$, find $$\lim_{t \to 0} \left( \int_{0}^{1}(bx + a(1-x))^t dx \right)^{\frac{1}{t}}$$ in terms of $a$ and $b$.

fun (a, b) => (Real.exp (-1))*(b^b/a^a)^(1/(b-a))

The limit equals $$e^{-1}\left(\frac{b^b}{a^a}\right)^{\frac{1}{b-a}}.$$

open Set Topology Filter

[]

@Eq (Prod Real Real → Real) answer fun (x : Prod Real Real) => _example.match_1 (fun (x_1 : Prod Real Real) => Real) x fun (a b : Real) => @HMul.hMul Real Real Real (@instHMul Real Real.instMul) (Real.exp (@Neg.neg Real Real.instNeg (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)))) (@HPow.hPow Real Real Real (@instHPow Real Real Real.instPow) (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) (@HPow.hPow Real Real Real (@instHPow Real Real Real.instPow) b b) (@HPow.hPow Real Real Real (@instHPow Real Real Real.instPow) a a)) (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)) (@HSub.hSub Real Real Real (@instHSub Real Real.instSub) b a)))

ℝ × ℝ → ℝ

[ { "t": "∀ a b : ℝ, 0 < a ∧ a < b → Tendsto (fun t : ℝ => (∫ x in Icc 0 1, (b*x + a*(1 - x))^t)^(1/t)) (𝓝[≠] 0) (𝓝 (answer (a, b)))", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1979_b2", "tags": [ "analysis" ] }

Let $F$ be a finite field with $n$ elements, and assume $n$ is odd. Suppose $x^2 + bx + c$ is an irreducible polynomial over $F$. For how many elements $d \in F$ is $x^2 + bx + c + d$ irreducible?

fun n : ℕ ↦ (n - (1 : ℤ)) / 2

Show that there are $\frac{n - 1}{2}$ such elements $d$.

open Set Topology Filter Polynomial

[]

@Eq (Nat → Int) answer fun (n_1 : Nat) => @HDiv.hDiv Int Int Int (@instHDiv Int Int.instDiv) (@HSub.hSub Int Int Int (@instHSub Int Int.instSub) (@Nat.cast Int instNatCastInt n_1) (@OfNat.ofNat Int (nat_lit 1) (@instOfNat (nat_lit 1)))) (@OfNat.ofNat Int (nat_lit 2) (@instOfNat (nat_lit 2)))

ℕ → ℤ

[ { "t": "Type*", "v": null, "name": "F" }, { "t": "Field F", "v": null, "name": null }, { "t": "Fintype F", "v": null, "name": null }, { "t": "ℕ", "v": null, "name": "n" }, { "t": "n = Fintype.card F", "v": null, "name": "hn" }, { "t": "Odd n", "v": null, "name": "nodd" }, { "t": "F", "v": null, "name": "b" }, { "t": "F", "v": null, "name": "c" }, { "t": "Polynomial F", "v": null, "name": "p" }, { "t": "p = X ^ 2 + (C b) * X + (C c) ∧ Irreducible p", "v": null, "name": "hp" }, { "t": "{d : F | Irreducible (p + (C d))}.ncard = answer n", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1979_b3", "tags": [ "abstract_algebra" ] }

Let $r$ and $s$ be positive integers. Derive a formula for the number of ordered quadruples $(a,b,c,d)$ of positive integers such that $3^r \cdot 7^s=\text{lcm}[a,b,c]=\text{lcm}[a,b,d]=\text{lcm}[a,c,d]=\text{lcm}[b,c,d]$. The answer should be a function of $r$ and $s$. (Note that $\text{lcm}[x,y,z]$ denotes the least common multiple of $x,y,z$.)

(fun r s : ℕ => (1 + 4 * r + 6 * r ^ 2) * (1 + 4 * s + 6 * s ^ 2))

Show that the number is $(1+4r+6r^2)(1+4s+6s^2)$.

null

[]

@Eq (Nat → Nat → Nat) answer fun (r_1 s_1 : Nat) => @HMul.hMul Nat Nat Nat (@instHMul Nat instMulNat) (@HAdd.hAdd Nat Nat Nat (@instHAdd Nat instAddNat) (@HAdd.hAdd Nat Nat Nat (@instHAdd Nat instAddNat) (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))) (@HMul.hMul Nat Nat Nat (@instHMul Nat instMulNat) (@OfNat.ofNat Nat (nat_lit 4) (instOfNatNat (nat_lit 4))) r_1)) (@HMul.hMul Nat Nat Nat (@instHMul Nat instMulNat) (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) (@HPow.hPow Nat Nat Nat (@instHPow Nat Nat (@Monoid.toNatPow Nat Nat.instMonoid)) r_1 (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))) (@HAdd.hAdd Nat Nat Nat (@instHAdd Nat instAddNat) (@HAdd.hAdd Nat Nat Nat (@instHAdd Nat instAddNat) (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))) (@HMul.hMul Nat Nat Nat (@instHMul Nat instMulNat) (@OfNat.ofNat Nat (nat_lit 4) (instOfNatNat (nat_lit 4))) s_1)) (@HMul.hMul Nat Nat Nat (@instHMul Nat instMulNat) (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) (@HPow.hPow Nat Nat Nat (@instHPow Nat Nat (@Monoid.toNatPow Nat Nat.instMonoid)) s_1 (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))))

ℕ → ℕ → ℕ

[ { "t": "ℕ", "v": null, "name": "r" }, { "t": "ℕ", "v": null, "name": "s" }, { "t": "ℕ → ℕ → ℕ → ℕ → Prop", "v": null, "name": "abcdlcm" }, { "t": "r > 0 ∧ s > 0", "v": null, "name": "rspos" }, { "t": "∀ a b c d : ℕ, abcdlcm a b c d ↔\n (a > 0 ∧ b > 0 ∧ c > 0 ∧ d > 0 ∧\n (3 ^ r * 7 ^ s = Nat.lcm (Nat.lcm a b) c) ∧\n (3 ^ r * 7 ^ s = Nat.lcm (Nat.lcm a b) d) ∧\n (3 ^ r * 7 ^ s = Nat.lcm (Nat.lcm a c) d) ∧\n (3 ^ r * 7 ^ s = Nat.lcm (Nat.lcm b c) d))", "v": null, "name": "habcdlcm" }, { "t": "{h : ℕ × ℕ × ℕ × ℕ | abcdlcm h.1 h.2.1 h.2.2.1 h.2.2.2}.encard = answer r s", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1980_a2", "tags": [ "number_theory" ] }

Evaluate $\int_0^{\pi/2}\frac{dx}{1+(\tan x)^{\sqrt{2}}}$.

$\pi / 4$

Show that the integral is $\pi/4$.

null

[]

@Eq Real answer (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) Real.pi (@OfNat.ofNat Real (nat_lit 4) (@instOfNatAtLeastTwo Real (nat_lit 4) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))))

ℝ

[ { "t": "answer = ∫ x in Set.Ioo 0 (Real.pi / 2), 1 / (1 + (Real.tan x) ^ (Real.sqrt 2))", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1980_a3", "tags": [ "analysis" ] }

Let $C$ be the class of all real valued continuously differentiable functions $f$ on the interval $0 \leq x \leq 1$ with $f(0)=0$ and $f(1)=1$. Determine the largest real number $u$ such that $u \leq \int_0^1|f'(x)-f(x)|\,dx$ for all $f$ in $C$.

$1/e$

Show that $u=1/e$.

null

[]

@Eq Real answer (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)) (Real.exp (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))))

ℝ

[ { "t": "Set (ℝ → ℝ)", "v": null, "name": "C" }, { "t": "C = {f : ℝ → ℝ | ContDiffOn ℝ 1 f (Set.Icc 0 1) ∧ f 0 = 0 ∧ f 1 = 1}", "v": null, "name": "hC" }, { "t": "IsGreatest {u : ℝ | ∀ f ∈ C, u ≤ (∫ x in Set.Ioo 0 1, |deriv f x - f x|)} answer", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1980_a6", "tags": [ "analysis" ] }

For which real numbers $c$ is $(e^x+e^{-x})/2 \leq e^{cx^2}$ for all real $x$?

$\{c : \mathbb{R} \mid c \geq 1/2\}$

Show that the inequality holds if and only if $c \geq 1/2$.

open Real

[]

@Eq (Set Real) answer (@setOf Real fun (c_1 : Real) => @GE.ge Real Real.instLE c_1 (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)) (@OfNat.ofNat Real (nat_lit 2) (@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))))))))

Set ℝ

[ { "t": "ℝ", "v": null, "name": "c" }, { "t": "∀ x : ℝ, (exp x + exp (-x)) / 2 ≤ exp (c * x ^ 2) ↔ c ∈ answer", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1980_b1", "tags": [ "analysis" ] }

For which real numbers $a$ does the sequence defined by the initial condition $u_0=a$ and the recursion $u_{n+1}=2u_n-n^2$ have $u_n>0$ for all $n \geq 0$? (Express the answer in the simplest form.)

$\{a : \mathbb{R} \mid a \geq 3\}$

Show that $u_n>0$ for all $n \geq 0$ if and only if $a \geq 3$.

null

[]

@Eq (Set Real) answer (@setOf Real fun (a_1 : Real) => @GE.ge Real Real.instLE a_1 (@OfNat.ofNat Real (nat_lit 3) (@instOfNatAtLeastTwo Real (nat_lit 3) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))))

Set ℝ

[ { "t": "ℝ", "v": null, "name": "a" }, { "t": "ℕ → ℝ", "v": null, "name": "u" }, { "t": "u 0 = a ∧ (∀ n : ℕ, u (n + 1) = 2 * u n - n ^ 2)", "v": null, "name": "hu" }, { "t": "(∀ n : ℕ, u n > 0) ↔ a ∈ answer", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1980_b3", "tags": [ "algebra" ] }

A function $f$ is convex on $[0, 1]$ if and only if $$f(su + (1-s)v) \le sf(u) + (1 - s)f(v)$$ for all $s \in [0, 1]$. Let $S_t$ denote the set of all nonnegative increasing convex continuous functions $f : [0, 1] \rightarrow \mathbb{R}$ such that $$f(1) - 2f\left(\frac{2}{3}\right) + f\left(\frac{1}{3}\right) \ge t\left(f\left(\frac{2}{3}\right) - 2f\left(\frac{1}{3}\right) + f(0)\right).$$ For which real numbers $t \ge 0$ is $S_t$ closed under multiplication?

$t \le 1$

$S_t$ is closed under multiplication if and only if $1 \ge t$.

open Set

[]

@Eq (Real → Prop) answer fun (t_1 : Real) => @LE.le Real Real.instLE t_1 (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))

ℝ → Prop

[ { "t": "Set ℝ", "v": null, "name": "T" }, { "t": "T = Icc 0 1", "v": null, "name": "hT" }, { "t": "ℝ → (ℝ → ℝ) → Prop", "v": null, "name": "P" }, { "t": "(ℝ → ℝ) → Prop", "v": null, "name": "IsConvex" }, { "t": "ℝ → Set (ℝ → ℝ)", "v": null, "name": "S" }, { "t": "∀ t f, P t f ↔ f 1 - 2*f (2/3) + f (1/3) ≥ t*(f (2/3) - 2*f (1/3) + f 0)", "v": null, "name": "P_def" }, { "t": "∀ f, IsConvex f ↔ ∀ u ∈ T, ∀ v ∈ T, ∀ s ∈ T, f (s*u + (1 - s)*v) ≤ s*(f u) + (1 - s)*(f v)", "v": null, "name": "IsConvex_def" }, { "t": "S = fun t : ℝ => {f : ℝ → ℝ | (∀ x ∈ T, f x ≥ 0) ∧ StrictMonoOn f T ∧ IsConvex f ∧ ContinuousOn f T ∧ P t f}", "v": null, "name": "hS" }, { "t": "ℝ", "v": null, "name": "t" }, { "t": "t ≥ 0", "v": null, "name": "ht" }, { "t": "answer t ↔ (∀ f ∈ S t, ∀ g ∈ S t, f * g ∈ S t)", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1980_b5", "tags": [ "analysis", "algebra" ] }

Let $E(n)$ be the greatest integer $k$ such that $5^k$ divides $1^1 2^2 3^3 \cdots n^n$. Find $\lim_{n \rightarrow \infty} \frac{E(n)}{n^2}$.

$\frac{1}{8}$

The limit equals $\frac{1}{8}$.

open Topology Filter Set Polynomial Function

[]

@Eq Real answer (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)) (@OfNat.ofNat Real (nat_lit 8) (@instOfNatAtLeastTwo Real (nat_lit 8) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))))))

ℝ

[ { "t": "ℕ → ℕ → Prop", "v": null, "name": "P" }, { "t": "∀ n k, P n k ↔ 5^k ∣ ∏ m in Finset.Icc 1 n, (m^m : ℤ)", "v": null, "name": "hP" }, { "t": "ℕ → ℕ", "v": null, "name": "E" }, { "t": "∀ n ∈ Ici 1, P n (E n) ∧ ∀ k : ℕ, P n k → k ≤ E n", "v": null, "name": "hE" }, { "t": "Tendsto (fun n : ℕ => ((E n) : ℝ)/n^2) atTop (𝓝 answer)", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1981_a1", "tags": [ "analysis", "number_theory" ] }

Does the limit $$lim_{t \rightarrow \infty}e^{-t}\int_{0}^{t}\int_{0}^{t}\frac{e^x - e^y}{x - y} dx dy$$exist?

False

The limit does not exist.

open Topology Filter Set Polynomial Function

[]

@Eq Prop answer False

Prop

[ { "t": "ℝ → ℝ", "v": null, "name": "f" }, { "t": "f = fun t : ℝ => Real.exp (-t) * ∫ y in (Ico 0 t), ∫ x in (Ico 0 t), (Real.exp x - Real.exp y) / (x - y)", "v": null, "name": "hf" }, { "t": "(∃ L : ℝ, Tendsto f atTop (𝓝 L)) ↔ answer", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1981_a3", "tags": [ "analysis" ] }

Let $P(x)$ be a polynomial with real coefficients; let $$Q(x) = (x^2 + 1)P(x)P'(x) + x((P(x))^2 + (P'(x))^2).$$ Given that $P$ has $n$ distinct real roots all greater than $1$, prove or disprove that $Q$ must have at least $2n - 1$ distinct real roots.

True

$Q(x)$ must have at least $2n - 1$ distinct real roots.

open Topology Filter Set Polynomial Function

[]

@Eq Prop answer True

Prop

[ { "t": "Polynomial ℝ → Polynomial ℝ", "v": null, "name": "Q" }, { "t": "Q = fun P : Polynomial ℝ => (Polynomial.X^2 + 1) * P * (Polynomial.derivative P) + Polynomial.X * (P^2 + (Polynomial.derivative P)^2)", "v": null, "name": "hQ" }, { "t": "Polynomial ℝ → ℝ", "v": null, "name": "n" }, { "t": "n = fun P : Polynomial ℝ => ({x ∈ Ioi 1 | P.eval x = 0}.ncard : ℝ)", "v": null, "name": "hn" }, { "t": "answer ↔ (∀ P : Polynomial ℝ, {x : ℝ | (Q P).eval x = 0}.ncard ≥ 2*(n P) - 1)", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1981_a5", "tags": [ "algebra" ] }

Find the value of $$\lim_{n \rightarrow \infty} \frac{1}{n^5}\sum_{h=1}^{n}\sum_{k=1}^{n}(5h^4 - 18h^2k^2 + 5k^4).$$

-1

The limit equals $-1$.

open Topology Filter Set Polynomial Function

[]

@Eq Real answer (@Neg.neg Real Real.instNeg (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)))

ℝ

[ { "t": "ℕ → ℝ", "v": null, "name": "f" }, { "t": "f = fun n : ℕ => ((1 : ℝ)/n^5) * ∑ h in Finset.Icc 1 n, ∑ k in Finset.Icc 1 n, (5*(h : ℝ)^4 - 18*h^2*k^2 + 5*k^4)", "v": null, "name": "hf" }, { "t": "Tendsto f atTop (𝓝 answer)", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1981_b1", "tags": [ "analysis" ] }

Determine the minimum value attained by $$(r - 1)^2 + (\frac{s}{r} - 1)^2 + (\frac{t}{s} - 1)^2 + (\frac{4}{t} - 1)^2$$ across all choices of real $r$, $s$, and $t$ that satisfy $1 \le r \le s \le t \le 4$.

12 - 8 * Real.sqrt 2

The minimum is $12 - 8\sqrt{2}$.

open Topology Filter Set Polynomial Function

[]

@Eq Real answer (@HSub.hSub Real Real Real (@instHSub Real Real.instSub) (@OfNat.ofNat Real (nat_lit 12) (@instOfNatAtLeastTwo Real (nat_lit 12) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 10) (instOfNatNat (nat_lit 10)))))) (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) (@OfNat.ofNat Real (nat_lit 8) (@instOfNatAtLeastTwo Real (nat_lit 8) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))))) (Real.sqrt (@OfNat.ofNat Real (nat_lit 2) (@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0)))))))))

ℝ

[ { "t": "ℝ × ℝ × ℝ → Prop", "v": null, "name": "P" }, { "t": "P = fun (r, s, t) => 1 ≤ r ∧ r ≤ s ∧ s ≤ t ∧ t ≤ 4", "v": null, "name": "hP" }, { "t": "ℝ × ℝ × ℝ → ℝ", "v": null, "name": "f" }, { "t": "f = fun (r, s, t) => (r - 1)^2 + (s/r - 1)^2 + (t/s - 1)^2 + (4/t - 1)^2", "v": null, "name": "hf" }, { "t": "IsLeast {y | ∃ r s t, P (r, s, t) ∧ f (r, s, t) = y} answer", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1981_b2", "tags": [ "algebra" ] }

Let $V$ be a set of $5$ by $7$ matrices, with real entries and with the property that $rA+sB \in V$ whenever $A,B \in V$ and $r$ and $s$ are scalars (i.e., real numbers). \emph{Prove or disprove} the following assertion: If $V$ contains matrices of ranks $0$, $1$, $2$, $4$, and $5$, then it also contains a matrix of rank $3$. [The rank of a nonzero matrix $M$ is the largest $k$ such that the entries of some $k$ rows and some $k$ columns form a $k$ by $k$ matrix with a nonzero determinant.]

False

Show that the assertion is false.

open Topology Filter Set Polynomial Function

[]

@Eq Prop answer False

Prop

[ { "t": "Set (Matrix (Fin 5) (Fin 7) ℝ)", "v": null, "name": "V" }, { "t": "∀ A ∈ V, ∀ B ∈ V, ∀ r s : ℝ, r • A + s • B ∈ V", "v": null, "name": "hVAB" }, { "t": "∃ A ∈ V, A.rank = 0", "v": null, "name": "hVrank0" }, { "t": "∃ A ∈ V, A.rank = 1", "v": null, "name": "hVrank1" }, { "t": "∃ A ∈ V, A.rank = 2", "v": null, "name": "hVrank2" }, { "t": "∃ A ∈ V, A.rank = 4", "v": null, "name": "hVrank4" }, { "t": "∃ A ∈ V, A.rank = 5", "v": null, "name": "hVrank5" }, { "t": "answer ↔ ∃ A ∈ V, A.rank = 3", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1981_b4", "tags": [ "linear_algebra" ] }

Let $B(n)$ be the number of ones in the base two expression for the positive integer $n$. For example, $B(6)=B(110_2)=2$ and $B(15)=B(1111_2)=4$. Determine whether or not $\exp \left(\sum_{n=1}^\infty \frac{B(n)}{n(n+1)}\right)$ is a rational number. Here $\exp(x)$ denotes $e^x$.

True

Show that the expression is a rational number.

open Topology Filter Set Polynomial Function

[]

@Eq Prop answer True

Prop

[ { "t": "List ℕ → ℤ", "v": null, "name": "sumbits" }, { "t": "ℕ → ℤ", "v": null, "name": "B" }, { "t": "∀ bits : List ℕ, sumbits bits = ∑ i : Fin bits.length, (bits[i] : ℤ)", "v": null, "name": "hsumbits" }, { "t": "∀ n > 0, B n = sumbits (Nat.digits 2 n)", "v": null, "name": "hB" }, { "t": "answer ↔ (∃ q : ℚ, Real.exp (∑' n : Set.Ici 1, B n / ((n : ℝ) * ((n : ℝ) + 1))) = q)", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1981_b5", "tags": [ "analysis", "algebra" ] }

Let $B_n(x) = 1^x + 2^x + \dots + n^x$ and let $f(n) = \frac{B_n(\log_n 2)}{(n \log_2 n)^2}$. Does $f(2) + f(3) + f(4) + \dots$ converge?

True

Prove that the series converges.

open Set Function Filter Topology Polynomial Real

[]

@Eq Prop answer True

Prop

[ { "t": "ℕ → ℝ → ℝ", "v": null, "name": "B" }, { "t": "B = fun (n : ℕ) (x : ℝ) ↦ ∑ k in Finset.Icc 1 n, (k : ℝ) ^ x", "v": null, "name": "hB" }, { "t": "ℕ → ℝ", "v": null, "name": "f" }, { "t": "f = fun n ↦ B n (logb n 2) / (n * logb 2 n) ^ 2", "v": null, "name": "hf" }, { "t": "answer ↔ (∃ L : ℝ, Tendsto (fun N ↦ ∑ j in Finset.Icc 2 N, f j) atTop (𝓝 L))", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1982_a2", "tags": [ "algebra" ] }

Evaluate $\int_0^{\infty} \frac{\tan^{-1}(\pi x) - \tan^{-1} x}{x} \, dx$.

$\frac{\pi}{2} \log \pi$

Show that the integral evaluates to $\frac{\pi}{2} \ln \pi$.

open Set Function Filter Topology Polynomial Real

[]

@Eq Real answer (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) Real.pi (@OfNat.ofNat Real (nat_lit 2) (@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))))))) (Real.log Real.pi))

ℝ

[ { "t": "Tendsto (fun t ↦ ∫ x in (0)..t, (arctan (Real.pi * x) - arctan x) / x) atTop (𝓝 answer)", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1982_a3", "tags": [ "analysis" ] }

Let $b$ be a bijection from the positive integers to the positive integers. Also, let $x_1, x_2, x_3, \dots$ be an infinite sequence of real numbers with the following properties: \begin{enumerate} \item $|x_n|$ is a strictly decreasing function of $n$; \item $\lim_{n \rightarrow \infty} |b(n) - n| \cdot |x_n| = 0$; \item $\lim_{n \rightarrow \infty}\sum_{k = 1}^{n} x_k = 1$. \end{enumerate} Prove or disprove: these conditions imply that $$\lim_{n \rightarrow \infty} \sum_{k = 1}^{n} x_{b(k)} = 1.$$

False

The limit need not equal $1$.

open Set Function Filter Topology Polynomial Real

[]

@Eq Prop answer False

Prop

[ { "t": "ℕ → ℕ", "v": null, "name": "b" }, { "t": "ℕ → ℝ", "v": null, "name": "x" }, { "t": "BijOn b (Ici 1) (Ici 1)", "v": null, "name": "h_bij" }, { "t": "StrictAntiOn (fun n : ℕ => |x n|) (Ici 1)", "v": null, "name": "h_strict_anti" }, { "t": "Tendsto (fun n : ℕ => |b n - (n : ℤ)| * |x n|) atTop (𝓝 0)", "v": null, "name": "h_limit" }, { "t": "Tendsto (fun n : ℕ => ∑ k in Finset.Icc 1 n, x k) atTop (𝓝 1)", "v": null, "name": "h_sum_limit" }, { "t": "(Tendsto (fun n : ℕ => ∑ k in Finset.Icc 1 n, x (b k)) atTop (𝓝 1)) ↔ answer", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1982_a6", "tags": [ "analysis" ] }

Let $A(x, y)$ denote the number of points $(m, n)$ with integer coordinates $m$ and $n$ where $m^2 + n^2 \le x^2 + y^2$. Also, let $g = \sum_{k = 0}^{\infty} e^{-k^2}$. Express the value $$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} A(x, y)e^{-x^2 - y^2} dx dy$$ as a polynomial in $g$.

C Real.pi * (2*X - 1)^2

The desired polynomial is $\pi(2g - 1)^2$.

open Set Function Filter Topology Polynomial Real

[]

@Eq (@Polynomial Real Real.semiring) answer (@HMul.hMul (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@instHMul (@Polynomial Real Real.semiring) (@Polynomial.mul' Real Real.semiring)) (@DFunLike.coe (@RingHom Real (@Polynomial Real Real.semiring) (@Semiring.toNonAssocSemiring Real Real.semiring) (@Semiring.toNonAssocSemiring (@Polynomial Real Real.semiring) (@Polynomial.semiring Real Real.semiring))) Real (fun (x : Real) => @Polynomial Real Real.semiring) (@RingHom.instFunLike Real (@Polynomial Real Real.semiring) (@Semiring.toNonAssocSemiring Real Real.semiring) (@Semiring.toNonAssocSemiring (@Polynomial Real Real.semiring) (@Polynomial.semiring Real Real.semiring))) (@Polynomial.C Real Real.semiring) Real.pi) (@HPow.hPow (@Polynomial Real Real.semiring) Nat (@Polynomial Real Real.semiring) (@instHPow (@Polynomial Real Real.semiring) Nat (@Monoid.toNatPow (@Polynomial Real Real.semiring) (@MonoidWithZero.toMonoid (@Polynomial Real Real.semiring) (@Semiring.toMonoidWithZero (@Polynomial Real Real.semiring) (@Polynomial.semiring Real Real.semiring))))) (@HSub.hSub (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@instHSub (@Polynomial Real Real.semiring) (@Polynomial.sub Real Real.instRing)) (@HMul.hMul (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@instHMul (@Polynomial Real Real.semiring) (@Polynomial.mul' Real Real.semiring)) (@OfNat.ofNat (@Polynomial Real Real.semiring) (nat_lit 2) (@instOfNatAtLeastTwo (@Polynomial Real Real.semiring) (nat_lit 2) (@Polynomial.instNatCast Real Real.semiring) (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0)))))) (@Polynomial.X Real Real.semiring)) (@OfNat.ofNat (@Polynomial Real Real.semiring) (nat_lit 1) (@One.toOfNat1 (@Polynomial Real Real.semiring) (@Polynomial.one Real Real.semiring)))) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))

Polynomial ℝ

[ { "t": "ℝ × ℝ → ℕ", "v": null, "name": "A" }, { "t": "ℝ", "v": null, "name": "g" }, { "t": "ℝ", "v": null, "name": "I" }, { "t": "A = fun (x, y) => {a : ℤ × ℤ | a.1^2 + a.2^2 ≤ x^2 + y^2}.ncard", "v": null, "name": "hA" }, { "t": "g = ∑' k : ℕ, Real.exp (-k^2)", "v": null, "name": "hg" }, { "t": "I = ∫ y : ℝ, ∫ x : ℝ, A (x, y) * Real.exp (-x^2 - y^2)", "v": null, "name": "hI" }, { "t": "I = answer.eval g", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1982_b2", "tags": [ "analysis" ] }

Let $p_n$ denote the probability that $c + d$ will be a perfect square if $c$ and $d$ are selected independently and uniformly at random from $\{1, 2, 3, \dots, n\}$. Express $\lim_{n \rightarrow \infty} p_n \sqrt{n}$ in the form $r(\sqrt{s} - t)$ for integers $s$ and $t$ and rational $r$.

4/3 * (Real.sqrt 2 - 1)

The limit equals $\frac{4}{3}(\sqrt{2} - 1)$.

open Set Function Filter Topology Polynomial Real

[]

@Eq Real answer (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) (@OfNat.ofNat Real (nat_lit 4) (@instOfNatAtLeastTwo Real (nat_lit 4) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))) (@OfNat.ofNat Real (nat_lit 3) (@instOfNatAtLeastTwo Real (nat_lit 3) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))))) (@HSub.hSub Real Real Real (@instHSub Real Real.instSub) (Real.sqrt (@OfNat.ofNat Real (nat_lit 2) (@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))))))) (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))))

ℝ

[ { "t": "ℕ → ℝ", "v": null, "name": "p" }, { "t": "p = fun n : ℕ => ({c : Finset.Icc 1 n × Finset.Icc 1 n | ∃ m : ℕ, m^2 = c.1 + c.2}.ncard : ℝ) / n^2", "v": null, "name": "hp" }, { "t": "Tendsto (fun n : ℕ => p n * Real.sqrt n) atTop (𝓝 answer)", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1982_b3", "tags": [ "analysis", "number_theory", "probability" ] }

Let $n_1, n_2, \dots, n_s$ be distinct integers such that, for every integer $k$, $n_1n_2\cdots n_s$ divides $(n_1 + k)(n_2 + k) \cdots (n_s + k)$. Prove or provide a counterexample to the following claims: \begin{enumerate} \item For some $i$, $|n_i| = 1$. \item If all $n_i$ are positive, then $\{n_1, n_2, \dots, n_s\} = \{1, 2, \dots, s\}$. \end{enumerate}

(True, True)

Both claims are true.

open Set Function Filter Topology Polynomial Real

[]

@Eq (Prod Prop Prop) answer (@Prod.mk Prop Prop True True)

Prop × Prop

[ { "t": "Finset ℤ → Prop", "v": null, "name": "P" }, { "t": "∀ n, P n ↔ n.Nonempty ∧ ∀ k, ∏ i in n, i ∣ ∏ i in n, (i + k)", "v": null, "name": "P_def" }, { "t": "((∀ n, P n → 1 ∈ n ∨ -1 ∈ n) ↔ answer.1) ∧\n ((∀ n, P n → (∀ i ∈ n, 0 < i) → n = Finset.Icc (1 : ℤ) n.card) ↔ answer.2)", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1982_b4", "tags": [ "number_theory" ] }

How many positive integers $n$ are there such that $n$ is an exact divisor of at least one of the numbers $10^{40},20^{30}$?

2301

Show that the desired count is $2301$.

null

[]

@Eq Nat answer (@OfNat.ofNat Nat (nat_lit 2301) (instOfNatNat (nat_lit 2301)))

ℕ

[ { "t": "{n : ℤ | n > 0 ∧ (n ∣ 10 ^ 40 ∨ n ∣ 20 ^ 30)}.encard = answer", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1983_a1", "tags": [ "number_theory" ] }

Prove or disprove that there exists a positive real number $\alpha$ such that $[\alpha_n] - n$ is even for all integers $n > 0$. (Here $[x]$ denotes the greatest integer less than or equal to $x$.)

True

Prove that such an $\alpha$ exists.

open Nat

[]

@Eq Prop answer True

Prop

[ { "t": "answer ↔ (∃ α : ℝ, α > 0 ∧ ∀ n : ℕ, n > 0 → Even (⌊α ^ n⌋ - n))", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1983_a5", "tags": [ "analysis" ] }

Let $T$ be the triangle with vertices $(0, 0)$, $(a, 0)$, and $(0, a)$. Find $\lim_{a \to \infty} a^4 \exp(-a^3) \int_T \exp(x^3+y^3) \, dx \, dy$.

2 / 9

Show that the integral evaluates to $\frac{2}{9}$.

open Nat Filter Topology Real

[]

@Eq Real answer (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) (@OfNat.ofNat Real (nat_lit 2) (@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0)))))) (@OfNat.ofNat Real (nat_lit 9) (@instOfNatAtLeastTwo Real (nat_lit 9) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 7) (instOfNatNat (nat_lit 7)))))))

ℝ

[ { "t": "ℝ → ℝ", "v": null, "name": "F" }, { "t": "F = fun a ↦ (a ^ 4 / exp (a ^ 3)) * ∫ x in (0)..a, ∫ y in (0)..(a - x), exp (x ^ 3 + y ^ 3)", "v": null, "name": "hF" }, { "t": "Tendsto F atTop (𝓝 answer)", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1983_a6", "tags": [ "analysis" ] }

Let $f(n)$ be the number of ways of representing $n$ as a sum of powers of $2$ with no power being used more than $3$ times. For example, $f(7) = 4$ (the representations are $4 + 2 + 1$, $4 + 1 + 1 + 1$, $2 + 2 + 2 + 1$, $2 + 2 + 1 + 1 + 1$). Can we find a real polynomial $p(x)$ such that $f(n) = [p(n)]$, where $[u]$ denotes the greatest integer less than or equal to $u$?

True

Prove that such a polynomial exists.

open Nat Filter Topology Real

[]

@Eq Prop answer True

Prop

[ { "t": "ℕ+ → ℕ", "v": null, "name": "f" }, { "t": "f = fun (n : ℕ+) ↦\n Set.ncard {M : Multiset ℕ |\n (∀ m ∈ M, ∃ k : ℕ, m = (2 ^ k : ℤ)) ∧ \n (∀ m ∈ M, M.count m ≤ 3) ∧ \n (M.sum : ℤ) = n}", "v": null, "name": "hf" }, { "t": "answer ↔ (∃ p : Polynomial ℝ, ∀ n : ℕ+, ⌊p.eval (n : ℝ)⌋ = f n)", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1983_b2", "tags": [ "algebra" ] }

Define $\left\lVert x \right\rVert$ as the distance from $x$ to the nearest integer. Find $\lim_{n \to \infty} \frac{1}{n} \int_{1}^{n} \left\lVert \frac{n}{x} \right\rVert \, dx$. You may assume that $\prod_{n=1}^{\infty} \frac{2n}{(2n-1)} \cdot \frac{2n}{(2n+1)} = \frac{\pi}{2}$.

$\log \left(\frac{4}{\pi}\right)$

Show that the limit equals $\ln \left( \frac{4}{\pi} \right)$.

open Nat Filter Topology Real

[]

@Eq Real answer (Real.log (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) (@OfNat.ofNat Real (nat_lit 4) (@instOfNatAtLeastTwo Real (nat_lit 4) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))) Real.pi))

ℝ

[ { "t": "ℝ → ℝ", "v": null, "name": "dist_fun" }, { "t": "dist_fun = fun (x : ℝ) ↦ min (x - ⌊x⌋) (⌈x⌉ - x)", "v": null, "name": "hdist_fun" }, { "t": "Tendsto (fun N ↦ ∏ n in Finset.Icc 1 N, (2 * n / (2 * n - 1)) * (2 * n / (2 * n + 1)) : ℕ → ℝ) atTop (𝓝 (Real.pi / 2))", "v": null, "name": "fact" }, { "t": "Tendsto (fun n ↦ (1 / n) * ∫ x in (1)..n, dist_fun (n / x) : ℕ → ℝ) atTop (𝓝 answer)", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1983_b5", "tags": [ "analysis" ] }

Express $\sum_{k=1}^\infty (6^k/(3^{k+1}-2^{k+1})(3^k-2^k))$ as a rational number.

2

Show that the sum converges to $2$.

null

[]

@Eq Rat answer (@OfNat.ofNat Rat (nat_lit 2) (@Rat.instOfNat (nat_lit 2)))

ℚ

[ { "t": "∑' k : Set.Ici 1, (6 ^ (k : ℕ) / ((3 ^ ((k : ℕ) + 1) - 2 ^ ((k : ℕ) + 1)) * (3 ^ (k : ℕ) - 2 ^ (k : ℕ)))) = answer", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1984_a2", "tags": [ "analysis" ] }

Let $n$ be a positive integer. Let $a,b,x$ be real numbers, with $a \neq b$, and let $M_n$ denote the $2n \times 2n$ matrix whose $(i,j)$ entry $m_{ij}$ is given by \[ m_{ij}=\begin{cases} x & \text{if }i=j, \\ a & \text{if }i \neq j\text{ and }i+j\text{ is even}, \\ b & \text{if }i \neq j\text{ and }i+j\text{ is odd}. \end{cases} \] Thus, for example, $M_2=\begin{pmatrix} x & b & a & b \\ b & x & b & a \\ a & b & x & b \\ b & a & b & x \end{pmatrix}$. Express $\lim_{x \to a} \det M_n/(x-a)^{2n-2}$ as a polynomial in $a$, $b$, and $n$, where $\det M_n$ denotes the determinant of $M_n$.

$(X_2)^2 \cdot ((X_0)^2 - (X_1)^2)$

Show that $\lim_{x \to a} \frac{\det M_n}{(x-a)^{2n-2}}=n^2(a^2-b^2)$.

open Topology Filter

[]

@Eq (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) answer (@HMul.hMul (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@instHMul (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@Distrib.toMul (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@NonUnitalNonAssocSemiring.toDistrib (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@NonUnitalNonAssocCommSemiring.toNonUnitalNonAssocSemiring (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@NonUnitalCommRing.toNonUnitalNonAssocCommRing (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@CommRing.toNonUnitalCommRing (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@MvPolynomial.instCommRingMvPolynomial Real (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real.commRing)))))))) (@HPow.hPow (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) Nat (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@instHPow (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) Nat (@Monoid.toNatPow (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@MonoidWithZero.toMonoid (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@Semiring.toMonoidWithZero (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@CommSemiring.toSemiring (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@MvPolynomial.commSemiring Real (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real.instCommSemiring)))))) (@MvPolynomial.X Real (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real.instCommSemiring (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) (nat_lit 2) (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3))) (@Nat.instNeZeroSucc (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) (nat_lit 2)))) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) (@HSub.hSub (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@instHSub (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@SubNegMonoid.toSub (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@AddGroup.toSubNegMonoid (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@AddGroupWithOne.toAddGroup (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@Ring.toAddGroupWithOne (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@CommRing.toRing (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@MvPolynomial.instCommRingMvPolynomial Real (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real.commRing))))))) (@HPow.hPow (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) Nat (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@instHPow (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) Nat (@Monoid.toNatPow (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@MonoidWithZero.toMonoid (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@Semiring.toMonoidWithZero (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@CommSemiring.toSemiring (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@MvPolynomial.commSemiring Real (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real.instCommSemiring)))))) (@MvPolynomial.X Real (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real.instCommSemiring (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) (nat_lit 0) (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3))) (@Nat.instNeZeroSucc (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) (nat_lit 0)))) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) (@HPow.hPow (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) Nat (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@instHPow (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) Nat (@Monoid.toNatPow (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@MonoidWithZero.toMonoid (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@Semiring.toMonoidWithZero (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@CommSemiring.toSemiring (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) (@MvPolynomial.commSemiring Real (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real.instCommSemiring)))))) (@MvPolynomial.X Real (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real.instCommSemiring (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) (nat_lit 1) (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3))) (@Nat.instNeZeroSucc (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) (nat_lit 1)))) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))))

MvPolynomial (Fin 3) ℝ

[ { "t": "ℕ", "v": null, "name": "n" }, { "t": "ℝ", "v": null, "name": "a" }, { "t": "ℝ", "v": null, "name": "b" }, { "t": "ℝ → Matrix (Fin (2 * n)) (Fin (2 * n)) ℝ", "v": null, "name": "Mn" }, { "t": "Fin 3 → ℝ", "v": null, "name": "polyabn" }, { "t": "n > 0", "v": null, "name": "npos" }, { "t": "a ≠ b", "v": null, "name": "aneb" }, { "t": "Mn = fun x : ℝ => fun i j : Fin (2 * n) => if i = j then x else if Even (i.1 + j.1) then a else b", "v": null, "name": "hMn" }, { "t": "polyabn 0 = a ∧ polyabn 1 = b ∧ polyabn 2 = n", "v": null, "name": "hpolyabn" }, { "t": "Tendsto (fun x : ℝ => (Mn x).det / (x - a) ^ (2 * n - 2)) (𝓝[≠] a) (𝓝 (MvPolynomial.eval polyabn answer))", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1984_a3", "tags": [ "linear_algebra", "analysis" ] }

Let $R$ be the region consisting of all triples $(x,y,z)$ of nonnegative real numbers satisfying $x+y+z \leq 1$. Let $w=1-x-y-z$. Express the value of the triple integral $\iiint_R x^1y^9z^8w^4\,dx\,dy\,dz$ in the form $a!b!c!d!/n!$, where $a$, $b$, $c$, $d$, and $n$ are positive integers.

(1, 9, 8, 4, 25)

Show that the integral we desire is $1!9!8!4!/25!$.

open Topology Filter Nat

[]

@Eq (Prod Nat (Prod Nat (Prod Nat (Prod Nat Nat)))) answer (@Prod.mk Nat (Prod Nat (Prod Nat (Prod Nat Nat))) (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))) (@Prod.mk Nat (Prod Nat (Prod Nat Nat)) (@OfNat.ofNat Nat (nat_lit 9) (instOfNatNat (nat_lit 9))) (@Prod.mk Nat (Prod Nat Nat) (@OfNat.ofNat Nat (nat_lit 8) (instOfNatNat (nat_lit 8))) (@Prod.mk Nat Nat (@OfNat.ofNat Nat (nat_lit 4) (instOfNatNat (nat_lit 4))) (@OfNat.ofNat Nat (nat_lit 25) (instOfNatNat (nat_lit 25)))))))

ℕ × ℕ × ℕ × ℕ × ℕ

[ { "t": "Set (Fin 3 → ℝ)", "v": null, "name": "R" }, { "t": "(Fin 3 → ℝ) → ℝ", "v": null, "name": "w" }, { "t": "R = {p | (∀ i : Fin 3, p i ≥ 0) ∧ p 0 + p 1 + p 2 ≤ 1}", "v": null, "name": "hR" }, { "t": "∀ p, w p = 1 - p 0 - p 1 - p 2", "v": null, "name": "hw" }, { "t": "ℕ", "v": null, "name": "a" }, { "t": "ℕ", "v": null, "name": "b" }, { "t": "ℕ", "v": null, "name": "c" }, { "t": "ℕ", "v": null, "name": "d" }, { "t": "ℕ", "v": null, "name": "n" }, { "t": "answer = (a, b, c, d, n)", "v": null, "name": "h_answer" }, { "t": "a > 0 ∧ b > 0 ∧ c > 0 ∧ d > 0 ∧ n > 0", "v": null, "name": "h_pos" }, { "t": "(∫ p in R, (p 0) ^ 1 * (p 1) ^ 9 * (p 2) ^ 8 * (w p) ^ 4 = ((a)! * (b)! * (c)! * (d)! : ℝ) / (n)!)", "v": null, "name": "h_integral" } ]

{ "problem_name": "putnam_1984_a5", "tags": [ "analysis" ] }

Let $n$ be a positive integer, and let $f(n)$ denote the last nonzero digit in the decimal expansion of $n!$. For instance, $f(5)=2$. \begin{enumerate} \item[(a)] Show that if $a_1,a_2,\dots,a_k$ are \emph{distinct} nonnegative integers, then $f(5^{a_1}+5^{a_2}+\dots+5^{a_k})$ depends only on the sum $a_1+a_2+\dots+a_k$. \item[(b)] Assuming part (a), we can define $g(s)=f(5^{a_1}+5^{a_2}+\dots+5^{a_k})$, where $s=a_1+a_2+\dots+a_k$. Find the least positive integer $p$ for which $g(s)=g(s + p)$, for all $s \geq 1$, or else show that no such $p$ exists. \end{enumerate}

4

Show that the least such $p$ is $p=4$.

open Topology Filter Function Nat

[]

@Eq Nat answer (@OfNat.ofNat Nat (nat_lit 4) (instOfNatNat (nat_lit 4)))

ℕ

[ { "t": "ℕ → ℕ", "v": null, "name": "f" }, { "t": "∀ n, some (f n) = (Nat.digits 10 (n !)).find? (fun d ↦ d ≠ 0)", "v": null, "name": "hf" }, { "t": "ℕ → (ℕ → ℕ) → ℕ → Prop", "v": null, "name": "IsPeriodicFrom" }, { "t": "∀ x f p, IsPeriodicFrom x f p ↔ Periodic (f ∘ (· + x)) p", "v": null, "name": "IsPeriodicFrom_def" }, { "t": "ℕ → (ℕ → ℕ) → ℕ → Prop", "v": null, "name": "P" }, { "t": "∀ x g p, P x g p ↔ if p = 0 then (∀ q > 0, ¬ IsPeriodicFrom x g q) else\n IsLeast {q | 0 < q ∧ IsPeriodicFrom x g q} p", "v": null, "name": "P_def" }, { "t": "∃ g : ℕ → ℕ,\n (∀ᵉ (k > 0) (a : Fin k → ℕ) (ha : Injective a), f (∑ i, 5 ^ (a i)) = g (∑ i, a i)) ∧\n P 1 g answer", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1984_a6", "tags": [ "algebra", "number_theory" ] }

Let $n$ be a positive integer, and define $f(n)=1!+2!+\dots+n!$. Find polynomials $P(x)$ and $Q(x)$ such that $f(n+2)=P(n)f(n+1)+Q(n)f(n)$ for all $n \geq 1$.

$(x + 3, -x - 2)$

Show that we can take $P(x)=x+3$ and $Q(x)=-x-2$.

open Topology Filter Nat

[]

@Eq (Prod (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring)) answer (@Prod.mk (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@HAdd.hAdd (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@instHAdd (@Polynomial Real Real.semiring) (@Polynomial.add' Real Real.semiring)) (@Polynomial.X Real Real.semiring) (@OfNat.ofNat (@Polynomial Real Real.semiring) (nat_lit 3) (@instOfNatAtLeastTwo (@Polynomial Real Real.semiring) (nat_lit 3) (@Polynomial.instNatCast Real Real.semiring) (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))))) (@HSub.hSub (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@instHSub (@Polynomial Real Real.semiring) (@Polynomial.sub Real Real.instRing)) (@Neg.neg (@Polynomial Real Real.semiring) (@Polynomial.neg' Real Real.instRing) (@Polynomial.X Real Real.semiring)) (@OfNat.ofNat (@Polynomial Real Real.semiring) (nat_lit 2) (@instOfNatAtLeastTwo (@Polynomial Real Real.semiring) (nat_lit 2) (@Polynomial.instNatCast Real Real.semiring) (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))))))))

Polynomial ℝ × Polynomial ℝ

[ { "t": "ℕ → ℤ", "v": null, "name": "f" }, { "t": "∀ n > 0, f n = ∑ i : Set.Icc 1 n, ((i)! : ℤ)", "v": null, "name": "hf" }, { "t": "∀ n ≥ 1, f (n + 2) = (answer.1).eval (n : ℝ) * f (n + 1) + (answer.2).eval (n : ℝ) * f n", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1984_b1", "tags": [ "algebra" ] }

Find the minimum value of $(u-v)^2+(\sqrt{2-u^2}-\frac{9}{v})^2$ for $0<u<\sqrt{2}$ and $v>0$.

8

Show that the minimum value is $8$.

open Topology Filter Nat

[]

@Eq Real answer (@OfNat.ofNat Real (nat_lit 8) (@instOfNatAtLeastTwo Real (nat_lit 8) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))))))

ℝ

[ { "t": "ℝ → ℝ → ℝ", "v": null, "name": "f" }, { "t": "∀ u v : ℝ, f u v = (u - v) ^ 2 + (Real.sqrt (2 - u ^ 2) - 9 / v) ^ 2", "v": null, "name": "hf" }, { "t": "IsLeast {y | ∃ᵉ (u : Set.Ioo 0 √2) (v > 0), f u v = y} answer", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1984_b2", "tags": [ "geometry", "analysis" ] }

Prove or disprove the following statement: If $F$ is a finite set with two or more elements, then there exists a binary operation $*$ on F such that for all $x,y,z$ in $F$, \begin{enumerate} \item[(i)] $x*z=y*z$ implies $x=y$ (right cancellation holds), and \item[(ii)] $x*(y*z) \neq (x*y)*z$ (\emph{no} case of associativity holds). \end{enumerate}

True

Show that the statement is true.

open Topology Filter Nat

[]

@Eq Prop answer True

Prop

[ { "t": "(∀ (F : Type*) (_ : Fintype F), Fintype.card F ≥ 2 → (∃ mul : F → F → F, ∀ x y z : F, (mul x z = mul y z → x = y) ∧ (mul x (mul y z) ≠ mul (mul x y) z))) ↔ answer", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1984_b3", "tags": [ "abstract_algebra" ] }

For each nonnegative integer $k$, let $d(k)$ denote the number of $1$'s in the binary expansion of $k$ (for example, $d(0)=0$ and $d(5)=2$). Let $m$ be a positive integer. Express $\sum_{k=0}^{2^m-1} (-1)^{d(k)}k^m$ in the form $(-1)^ma^{f(m)}(g(m))!$, where $a$ is an integer and $f$ and $g$ are polynomials.

(2, (Polynomial.X * (Polynomial.X - 1)) / 2, Polynomial.X)

Show that $\sum_{k=0}^{2^m-1} (-1)^{d(k)}k^m=(-1)^m2^{m(m-1)/2}m!$.

open Topology Filter Nat

[]

@Eq (Prod Int (Prod (@Polynomial Real Real.semiring) (@Polynomial Nat Nat.instSemiring))) answer (@Prod.mk Int (Prod (@Polynomial Real Real.semiring) (@Polynomial Nat Nat.instSemiring)) (@OfNat.ofNat Int (nat_lit 2) (@instOfNat (nat_lit 2))) (@Prod.mk (@Polynomial Real Real.semiring) (@Polynomial Nat Nat.instSemiring) (@HDiv.hDiv (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@instHDiv (@Polynomial Real Real.semiring) (@Polynomial.instDiv Real Real.field)) (@HMul.hMul (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@instHMul (@Polynomial Real Real.semiring) (@Polynomial.mul' Real Real.semiring)) (@Polynomial.X Real Real.semiring) (@HSub.hSub (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@instHSub (@Polynomial Real Real.semiring) (@Polynomial.sub Real Real.instRing)) (@Polynomial.X Real Real.semiring) (@OfNat.ofNat (@Polynomial Real Real.semiring) (nat_lit 1) (@One.toOfNat1 (@Polynomial Real Real.semiring) (@Polynomial.one Real Real.semiring))))) (@OfNat.ofNat (@Polynomial Real Real.semiring) (nat_lit 2) (@instOfNatAtLeastTwo (@Polynomial Real Real.semiring) (nat_lit 2) (@Polynomial.instNatCast Real Real.semiring) (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))))))) (@Polynomial.X Nat Nat.instSemiring)))

ℤ × Polynomial ℝ × Polynomial ℕ

[ { "t": "ℕ", "v": null, "name": "m" }, { "t": "m > 0", "v": null, "name": "mpos" }, { "t": "ℕ → ℕ", "v": null, "name": "d" }, { "t": "List ℕ → ℕ", "v": null, "name": "sumbits" }, { "t": "∀ bits : List ℕ, sumbits bits = ∑ i : Fin bits.length, bits[i]", "v": null, "name": "hsumbits" }, { "t": "∀ k : ℕ, d k = sumbits (Nat.digits 2 k)", "v": null, "name": "hd" }, { "t": "let (a, f, g) := answer;\n ∑ k : Set.Icc 0 (2 ^ m - 1), (-(1 : ℤ)) ^ (d k) * (k : ℕ) ^ m = (-1) ^ m * (a : ℝ) ^ (f.eval (m : ℝ)) * (g.eval m)!", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1984_b5", "tags": [ "algebra", "analysis" ] }

Determine, with proof, the number of ordered triples $(A_1, A_2, A_3)$ of sets which have the property that \begin{enumerate} \item[(i)] $A_1 \cup A_2 \cup A_3 = \{1,2,3,4,5,6,7,8,9,10\}$, and \item[(ii)] $A_1 \cap A_2 \cap A_3 = \emptyset$. \end{enumerate} Express your answer in the form $2^a 3^b 5^c 7^d$, where $a,b,c,d$ are nonnegative integers.

(10, 10, 0, 0)

Prove that the number of such triples is $2^{10}3^{10}$.

open Set

[]

@Eq (Prod Nat (Prod Nat (Prod Nat Nat))) answer (@Prod.mk Nat (Prod Nat (Prod Nat Nat)) (@OfNat.ofNat Nat (nat_lit 10) (instOfNatNat (nat_lit 10))) (@Prod.mk Nat (Prod Nat Nat) (@OfNat.ofNat Nat (nat_lit 10) (instOfNatNat (nat_lit 10))) (@Prod.mk Nat Nat (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))) (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))))))

ℕ × ℕ × ℕ × ℕ

[ { "t": "let (a, b, c, d) := answer;\n {(A1, A2, A3) : Set ℤ × Set ℤ × Set ℤ | A1 ∪ A2 ∪ A3 = Icc 1 10 ∧ A1 ∩ A2 ∩ A3 = ∅}.ncard = 2 ^ a * 3 ^ b * 5 ^ c * 7 ^ d", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1985_a1", "tags": [ "algebra" ] }

Let $d$ be a real number. For each integer $m \geq 0$, define a sequence $\{a_m(j)\}$, $j=0,1,2,\dots$ by the condition \begin{align*} a_m(0) &= d/2^m, \\ a_m(j+1) &= (a_m(j))^2 + 2a_m(j), \qquad j \geq 0. \end{align*} Evaluate $\lim_{n \to \infty} a_n(n)$.

$e^d - 1$

Show that the limit equals $e^d - 1$.

open Set Filter Topology Real

[]

@Eq (Real → Real) answer fun (d_1 : Real) => @HSub.hSub Real Real Real (@instHSub Real Real.instSub) (Real.exp d_1) (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))

ℝ → ℝ

[ { "t": "ℝ", "v": null, "name": "d" }, { "t": "ℕ → ℕ → ℝ", "v": null, "name": "a" }, { "t": "∀ m : ℕ, a m 0 = d / 2 ^ m", "v": null, "name": "ha0" }, { "t": "∀ m : ℕ, ∀ j : ℕ, a m (j + 1) = (a m j) ^ 2 + 2 * a m j", "v": null, "name": "ha" }, { "t": "Tendsto (fun n ↦ a n n) atTop (𝓝 (answer d))", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1985_a3", "tags": [ "analysis" ] }

Define a sequence $\{a_i\}$ by $a_1=3$ and $a_{i+1}=3^{a_i}$ for $i \geq 1$. Which integers between $00$ and $99$ inclusive occur as the last two digits in the decimal expansion of infinitely many $a_i$?

{87}

Prove that the only number that occurs infinitely often is $87$.

open Set Filter Topology Real

[]

@Eq (Set (Fin (@OfNat.ofNat Nat (nat_lit 100) (instOfNatNat (nat_lit 100))))) answer (@Singleton.singleton (Fin (@OfNat.ofNat Nat (nat_lit 100) (instOfNatNat (nat_lit 100)))) (Set (Fin (@OfNat.ofNat Nat (nat_lit 100) (instOfNatNat (nat_lit 100))))) (@Set.instSingletonSet (Fin (@OfNat.ofNat Nat (nat_lit 100) (instOfNatNat (nat_lit 100))))) (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 100) (instOfNatNat (nat_lit 100)))) (nat_lit 87) (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 100) (instOfNatNat (nat_lit 100))) (@Nat.instNeZeroSucc (@OfNat.ofNat Nat (nat_lit 99) (instOfNatNat (nat_lit 99)))) (nat_lit 87))))

Set (Fin 100)

[ { "t": "ℕ → ℕ", "v": null, "name": "a" }, { "t": "a 1 = 3", "v": null, "name": "ha1" }, { "t": "∀ i ≥ 1, a (i + 1) = 3 ^ a i", "v": null, "name": "ha" }, { "t": "{k : Fin 100 | ∀ N : ℕ, ∃ i ≥ N, a i % 100 = k} = answer", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1985_a4", "tags": [ "number_theory" ] }

Let $I_m = \int_0^{2\pi} \cos(x)\cos(2x)\cdots \cos(mx)\,dx$. For which integers $m$, $1 \leq m \leq 10$ is $I_m \neq 0$?

{3, 4, 7, 8}

Prove that the integers $m$ with $1 \leq m \leq 10$ and $I_m \neq 0$ are $m = 3, 4, 7, 8$.

open Set Filter Topology Real

[]

@Eq (Set Nat) answer (@Insert.insert Nat (Set Nat) (@Set.instInsert Nat) (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3))) (@Insert.insert Nat (Set Nat) (@Set.instInsert Nat) (@OfNat.ofNat Nat (nat_lit 4) (instOfNatNat (nat_lit 4))) (@Insert.insert Nat (Set Nat) (@Set.instInsert Nat) (@OfNat.ofNat Nat (nat_lit 7) (instOfNatNat (nat_lit 7))) (@Singleton.singleton Nat (Set Nat) (@Set.instSingletonSet Nat) (@OfNat.ofNat Nat (nat_lit 8) (instOfNatNat (nat_lit 8)))))))

Set ℕ

[ { "t": "ℕ → ℝ", "v": null, "name": "I" }, { "t": "I = fun (m : ℕ) ↦ ∫ x in (0)..(2 * Real.pi), ∏ k in Finset.Icc 1 m, cos (k * x)", "v": null, "name": "hI" }, { "t": "{m ∈ Finset.Icc 1 10 | I m ≠ 0} = answer", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1985_a5", "tags": [ "analysis" ] }

If $p(x)= a_0 + a_1 x + \cdots + a_m x^m$ is a polynomial with real coefficients $a_i$, then set \[ \Gamma(p(x)) = a_0^2 + a_1^2 + \cdots + a_m^2. \] Let $F(x) = 3x^2+7x+2$. Find, with proof, a polynomial $g(x)$ with real coefficients such that \begin{enumerate} \item[(i)] $g(0)=1$, and \item[(ii)] $\Gamma(f(x)^n) = \Gamma(g(x)^n)$ \end{enumerate} for every integer $n \geq 1$.

6x^2 + 5x + 1

Show that $g(x) = 6x^2 + 5x + 1$ satisfies the conditions.

open Set Filter Topology Real Polynomial

[]

@Eq (@Polynomial Real Real.semiring) answer (@HAdd.hAdd (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@instHAdd (@Polynomial Real Real.semiring) (@Polynomial.add' Real Real.semiring)) (@HAdd.hAdd (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@instHAdd (@Polynomial Real Real.semiring) (@Polynomial.add' Real Real.semiring)) (@HMul.hMul (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@instHMul (@Polynomial Real Real.semiring) (@Polynomial.mul' Real Real.semiring)) (@OfNat.ofNat (@Polynomial Real Real.semiring) (nat_lit 6) (@instOfNatAtLeastTwo (@Polynomial Real Real.semiring) (nat_lit 6) (@Polynomial.instNatCast Real Real.semiring) (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 4) (instOfNatNat (nat_lit 4)))))) (@HPow.hPow (@Polynomial Real Real.semiring) Nat (@Polynomial Real Real.semiring) (@instHPow (@Polynomial Real Real.semiring) Nat (@Monoid.toNatPow (@Polynomial Real Real.semiring) (@MonoidWithZero.toMonoid (@Polynomial Real Real.semiring) (@Semiring.toMonoidWithZero (@Polynomial Real Real.semiring) (@Polynomial.semiring Real Real.semiring))))) (@Polynomial.X Real Real.semiring) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))) (@HMul.hMul (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@instHMul (@Polynomial Real Real.semiring) (@Polynomial.mul' Real Real.semiring)) (@OfNat.ofNat (@Polynomial Real Real.semiring) (nat_lit 5) (@instOfNatAtLeastTwo (@Polynomial Real Real.semiring) (nat_lit 5) (@Polynomial.instNatCast Real Real.semiring) (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))))) (@Polynomial.X Real Real.semiring))) (@OfNat.ofNat (@Polynomial Real Real.semiring) (nat_lit 1) (@One.toOfNat1 (@Polynomial Real Real.semiring) (@Polynomial.one Real Real.semiring))))

Polynomial ℝ

[ { "t": "Polynomial ℝ → ℝ", "v": null, "name": "Γ" }, { "t": "Polynomial ℝ", "v": null, "name": "f" }, { "t": "Γ = fun p ↦ ∑ k in Finset.range (p.natDegree + 1), coeff p k ^ 2", "v": null, "name": "hΓ" }, { "t": "f = 3 * Polynomial.X ^ 2 + 7 * Polynomial.X + 2", "v": null, "name": "hf" }, { "t": "let g := answer;\n g.eval 0 = 1 ∧ ∀ n : ℕ, n ≥ 1 → Γ (f ^ n) = Γ (g ^ n)", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1985_a6", "tags": [ "algebra" ] }

Let $k$ be the smallest positive integer for which there exist distinct integers $m_1, m_2, m_3, m_4, m_5$ such that the polynomial \[ p(x) = (x-m_1)(x-m_2)(x-m_3)(x-m_4)(x-m_5) \] has exactly $k$ nonzero coefficients. Find, with proof, a set of integers $m_1, m_2, m_3, m_4, m_5$ for which this minimum $k$ is achieved.

fun i : Fin 5 ↦ ↑i - (2 : ℤ)

Show that the minimum $k = 3$ is obtained for $\{m_1, m_2, m_3, m_4, m_5\} = \{-2, -1, 0, 1, 2\}$.

open Set Filter Topology Real Polynomial Function

[]

@Eq (Fin (@OfNat.ofNat Nat (nat_lit 5) (instOfNatNat (nat_lit 5))) → Int) answer fun (i : Fin (@OfNat.ofNat Nat (nat_lit 5) (instOfNatNat (nat_lit 5)))) => @HSub.hSub Int Int Int (@instHSub Int Int.instSub) (@Nat.cast Int instNatCastInt (@Fin.val (@OfNat.ofNat Nat (nat_lit 5) (instOfNatNat (nat_lit 5))) i)) (@OfNat.ofNat Int (nat_lit 2) (@instOfNat (nat_lit 2)))

Fin 5 → ℤ

[ { "t": "(Fin 5 → ℤ) → (Polynomial ℝ)", "v": null, "name": "p" }, { "t": "p = fun m ↦ ∏ i : Fin 5, ((X : Polynomial ℝ) - m i)", "v": null, "name": "hp" }, { "t": "Polynomial ℝ → ℕ", "v": null, "name": "numnzcoeff" }, { "t": "numnzcoeff = fun p ↦ {j ∈ Finset.range (p.natDegree + 1) | coeff p j ≠ 0}.card", "v": null, "name": "hnumnzcoeff" }, { "t": "(Injective answer ∧ ∀ m : Fin 5 → ℤ, Injective m → numnzcoeff (p answer) ≤ numnzcoeff (p m))", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1985_b1", "tags": [ "algebra" ] }

Define polynomials $f_n(x)$ for $n \geq 0$ by $f_0(x)=1$, $f_n(0)=0$ for $n \geq 1$, and \[ \frac{d}{dx} f_{n+1}(x) = (n+1)f_n(x+1) \] for $n \geq 0$. Find, with proof, the explicit factorization of $f_{100}(1)$ into powers of distinct primes.

99 if $n = 101$, otherwise 0

Show that $f_{100}(1) = 101^{99}$.

open Set Filter Topology Real Polynomial Function

[]

@Eq (Nat → Nat) answer fun (n : Nat) => @ite Nat (@Eq Nat n (@OfNat.ofNat Nat (nat_lit 101) (instOfNatNat (nat_lit 101)))) (instDecidableEqNat n (@OfNat.ofNat Nat (nat_lit 101) (instOfNatNat (nat_lit 101)))) (@OfNat.ofNat Nat (nat_lit 99) (instOfNatNat (nat_lit 99))) (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0)))

ℕ → ℕ

[ { "t": "ℕ -> Polynomial ℕ", "v": null, "name": "f" }, { "t": "f 0 = 1", "v": null, "name": "hf0x" }, { "t": "∀ n ≥ 1, (f n).eval 0 = 0", "v": null, "name": "hfn0" }, { "t": "∀ n : ℕ, derivative (f (n + 1)) = (n + 1) * (Polynomial.comp (f n) (X + 1))", "v": null, "name": "hfderiv" }, { "t": "Nat.factorization ((f 100).eval 1) = answer", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1985_b2", "tags": [ "algebra" ] }

Evaluate $\int_0^\infty t^{-1/2}e^{-1985(t+t^{-1})}\,dt$. You may assume that $\int_{-\infty}^\infty e^{-x^2}\,dx = \sqrt{\pi}$.

$\sqrt{\pi / 1985} \cdot e^{-3970}$

Show that the integral evaluates to $\sqrt{\frac{\pi}{1985}}e^{-3970}$.

open Set Filter Topology Real Polynomial Function

[]

@Eq Real answer (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) (Real.sqrt (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) Real.pi (@OfNat.ofNat Real (nat_lit 1985) (@instOfNatAtLeastTwo Real (nat_lit 1985) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 1983) (instOfNatNat (nat_lit 1983)))))))) (Real.exp (@Neg.neg Real Real.instNeg (@OfNat.ofNat Real (nat_lit 3970) (@instOfNatAtLeastTwo Real (nat_lit 3970) Real.instNatCast (@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 3968) (instOfNatNat (nat_lit 3968)))))))))

ℝ

[ { "t": "∫ x in Set.univ, Real.exp (- x ^ 2) = Real.sqrt Real.pi", "v": null, "name": "fact" }, { "t": "∫ t in Set.Ioi 0, t ^ (- (1 : ℝ) / 2) * Real.exp (-1985 * (t + t ^ (-(1 : ℝ)))) = answer", "v": null, "name": "h_answer" } ]

{ "problem_name": "putnam_1985_b5", "tags": [ "analysis" ] }

Datasets:

purewhite42
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putnambench_solving

Dataset Card for PutnamBench-Solving

Contribution

Benchmark Details

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Dataset Structure

References

Collection including purewhite42/putnambench_solving

Formal Problem-Solving