name
stringlengths 7
15
| uuid
stringlengths 36
36
| split
stringclasses 1
value | lean4_statement
stringlengths 75
2.06k
| full_text_no_abbrv
stringlengths 223
2.05k
| full_text
stringlengths 226
2.1k
| lean4_proof
stringclasses 1
value | informal_statement
stringlengths 4
898
| informal_solution
stringlengths 4
282
| tags
sequencelengths 0
3
| coq_statement
stringclasses 1
value | isabelle_statement
stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|---|
putnam_2022_b1 | 0a162932-31ec-5537-8931-76c768859a2e | train | theorem putnam_2022_b1
(P : Polynomial β€)
(b : β β β)
(Pconst : P.coeff 0 = 0)
(Podd : Odd (P.coeff 1))
(hB : β x : β, HasSum (fun i => b i * x ^ i) (Real.exp (aeval x P))) :
β k : β, b k β 0 :=
sorry | import Mathlib
open Polynomial
/--
Suppose that $P(x)=a_1x+a_2x^2+\cdots+a_nx^n$ is a polynomial with integer coefficients, with $a_1$ odd. Suppose that $e^{P(x)}=b_0+b_1x+b_2x^2+\dots$ for all $x$. Prove that $b_k$ is nonzero for all $k \geq 0$.
-/
theorem putnam_2022_b1
(P : Polynomial β€)
(b : β β β)
(Pconst : P.coeff 0 = 0)
(Podd : Odd (P.coeff 1))
(hB : β x : β, HasSum (fun i => b i * x ^ i) (Real.exp (aeval x P))) :
β k : β, b k β 0 := by
| import Mathlib
open Polynomial
/--
Suppose that $P(x)=a_1x+a_2x^2+\cdots+a_nx^n$ is a polynomial with integer coefficients, with $a_1$ odd. Suppose that $e^{P(x)}=b_0+b_1x+b_2x^2+\dots$ for all $x$. Prove that $b_k$ is nonzero for all $k \geq 0$.
-/
theorem putnam_2022_b1
(P : Polynomial β€)
(b : β β β)
(Pconst : P.coeff 0 = 0)
(Podd : Odd (P.coeff 1))
(hB : β x : β, HasSum (fun i => b i * x ^ i) (Real.exp (aeval x P))) :
β k : β, b k β 0 :=
sorry
| Suppose that $P(x)=a_1x+a_2x^2+\cdots+a_nx^n$ is a polynomial with integer coefficients, with $a_1$ odd. Suppose that $e^{P(x)}=b_0+b_1x+b_2x^2+\dots$ for all $x$. Prove that $b_k$ is nonzero for all $k \geq 0$. | null | [
"analysis",
"algebra"
] | null | null |
|
putnam_1979_b2 | 43dceb56-b7c4-5924-92b3-0712a5daff68 | train | abbrev putnam_1979_b2_solution : β Γ β β β := sorry
-- fun (a, b) => (Real.exp (-1))*(b^b/a^a)^(1/(b-a))
/--
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 $b$.
-/
theorem putnam_1979_b2
: β a b : β, 0 < a β§ a < b β Tendsto (fun t : β => (β« x in Icc 0 1, (b*x + a*(1 - x))^t)^(1/t)) (π[β ] 0) (π (putnam_1979_b2_solution (a, b))) :=
sorry | import Mathlib
open Set Topology Filter
-- fun (a, b) => (Real.exp (-1))*(b^b/a^a)^(1/(b-a))
/--
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 $b$.
-/
theorem putnam_1979_b2
: β a b : β, 0 < a β§ a < b β Tendsto (fun t : β => (β« x in Icc 0 1, (b*x + a*(1 - x))^t)^(1/t)) (π[β ] 0) (π (putnam_1979_b2_solution (a, b))) := by
| import Mathlib
open Set Topology Filter
noncomputable abbrev putnam_1979_b2_solution : β Γ β β β := sorry
-- fun (a, b) => (Real.exp (-1))*(b^b/a^a)^(1/(b-a))
/--
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 $b$.
-/
theorem putnam_1979_b2
: β a b : β, 0 < a β§ a < b β Tendsto (fun t : β => (β« x in Icc 0 1, (b*x + a*(1 - x))^t)^(1/t)) (π[β ] 0) (π (putnam_1979_b2_solution (a, b))) :=
sorry
| 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 $b$. | The limit equals $$e^{-1}\left(\frac{b^b}{a^a}\right)^{\frac{1}{b-a}}.$$ | [
"analysis"
] | null | null |
|
putnam_1990_b1 | 715a79e8-e37f-5f36-831c-6ea7a98a53d2 | train | abbrev putnam_1990_b1_solution : Set (β β β) := sorry
-- {fun x : β => (Real.sqrt 1990) * Real.exp x, fun x : β => -(Real.sqrt 1990) * Real.exp x}
/--
Find all real-valued continuously differentiable functions $f$ on the real line such that for all $x$, $(f(x))^2=\int_0^x [(f(t))^2+(f'(t))^2]\,dt+1990$.
-/
theorem putnam_1990_b1
(P : (β β β) β Prop)
(P_def : β f, P f β β x,
(f x) ^ 2 = (β« t in (0 : β)..x, (f t) ^ 2 + (deriv f t) ^ 2) + 1990)
(f : β β β) :
(ContDiff β 1 f β§ P f) β f β putnam_1990_b1_solution :=
sorry | import Mathlib
open Filter Topology Nat
-- {fun x : β => (Real.sqrt 1990) * Real.exp x, fun x : β => -(Real.sqrt 1990) * Real.exp x}
/--
Find all real-valued continuously differentiable functions $f$ on the real line such that for all $x$, $(f(x))^2=\int_0^x [(f(t))^2+(f'(t))^2]\,dt+1990$.
-/
theorem putnam_1990_b1
(P : (β β β) β Prop)
(P_def : β f, P f β β x,
(f x) ^ 2 = (β« t in (0 : β)..x, (f t) ^ 2 + (deriv f t) ^ 2) + 1990)
(f : β β β) :
(ContDiff β 1 f β§ P f) β f β putnam_1990_b1_solution := by
| import Mathlib
open Filter Topology Nat
abbrev putnam_1990_b1_solution : Set (β β β) := sorry
-- {fun x : β => (Real.sqrt 1990) * Real.exp x, fun x : β => -(Real.sqrt 1990) * Real.exp x}
/--
Find all real-valued continuously differentiable functions $f$ on the real line such that for all $x$, $(f(x))^2=\int_0^x [(f(t))^2+(f'(t))^2]\,dt+1990$.
-/
theorem putnam_1990_b1
(P : (β β β) β Prop)
(P_def : β f, P f β β x,
(f x) ^ 2 = (β« t in (0 : β)..x, (f t) ^ 2 + (deriv f t) ^ 2) + 1990)
(f : β β β) :
(ContDiff β 1 f β§ P f) β f β putnam_1990_b1_solution :=
sorry
| Find all real-valued continuously differentiable functions $f$ on the real line such that for all $x$, $(f(x))^2=\int_0^x [(f(t))^2+(f'(t))^2]\,dt+1990$. | Show that there are two such functions, namely $f(x)=\sqrt{1990}e^x$, and $f(x)=-\sqrt{1990}e^x$. | [
"analysis"
] | null | null |
|
putnam_1964_b6 | b959f8d9-c5b8-5ed9-9f55-753078197239 | train | theorem putnam_1964_b6
(D : Set (EuclideanSpace β (Fin 2)))
(hD : D = {v : EuclideanSpace β (Fin 2) | dist 0 v β€ 1})
(cong : Set (EuclideanSpace β (Fin 2)) β Set (EuclideanSpace β (Fin 2)) β Prop)
(hcong : β A B, cong A B β β f : (EuclideanSpace β (Fin 2)) β (EuclideanSpace β (Fin 2)), B = f '' A β§ β v w : EuclideanSpace β (Fin 2), dist v w = dist (f v) (f w))
: (Β¬β A B : Set (Fin 2 β β), cong A B β§ A β© B = β
β§ A βͺ B = D) :=
sorry | import Mathlib
open Set Function Filter Topology
/--
Let $D$ be the unit disk in the plane. Show that we cannot find congruent sets $A, B$ with $A \cap B = \emptyset$ and $A \cup B = D$.
-/
theorem putnam_1964_b6
(D : Set (EuclideanSpace β (Fin 2)))
(hD : D = {v : EuclideanSpace β (Fin 2) | dist 0 v β€ 1})
(cong : Set (EuclideanSpace β (Fin 2)) β Set (EuclideanSpace β (Fin 2)) β Prop)
(hcong : β A B, cong A B β β f : (EuclideanSpace β (Fin 2)) β (EuclideanSpace β (Fin 2)), B = f '' A β§ β v w : EuclideanSpace β (Fin 2), dist v w = dist (f v) (f w))
: (Β¬β A B : Set (Fin 2 β β), cong A B β§ A β© B = β
β§ A βͺ B = D) := by
| import Mathlib
open Set Function Filter Topology
/--
Let $D$ be the unit disk in the plane. Show that we cannot find congruent sets $A, B$ with $A \cap B = \emptyset$ and $A \cup B = D$.
-/
theorem putnam_1964_b6
(D : Set (EuclideanSpace β (Fin 2)))
(hD : D = {v : EuclideanSpace β (Fin 2) | dist 0 v β€ 1})
(cong : Set (EuclideanSpace β (Fin 2)) β Set (EuclideanSpace β (Fin 2)) β Prop)
(hcong : β A B, cong A B β β f : (EuclideanSpace β (Fin 2)) β (EuclideanSpace β (Fin 2)), B = f '' A β§ β v w : EuclideanSpace β (Fin 2), dist v w = dist (f v) (f w))
: (Β¬β A B : Set (Fin 2 β β), cong A B β§ A β© B = β
β§ A βͺ B = D) :=
sorry
| Let $D$ be the unit disk in the plane. Show that we cannot find congruent sets $A, B$ with $A \cap B = \emptyset$ and $A \cup B = D$. | null | [
"geometry"
] | null | null |
|
putnam_1977_a5 | 1d6a5585-c2dc-587b-a954-e586bfb216da | train | theorem putnam_1977_a5
(p m n : β)
(hp : Nat.Prime p)
(hmgen : m β₯ n)
: (choose (p * m) (p * n) β‘ choose m n [MOD p]) :=
sorry | import Mathlib
open RingHom Set Nat
/--
Let $p$ be a prime and $m \geq n$ be non-negative integers. Show that $\binom{pm}{pn} = \binom{m}{n} \pmod p$, where $\binom{m}{n}$ is the binomial coefficient.
-/
theorem putnam_1977_a5
(p m n : β)
(hp : Nat.Prime p)
(hmgen : m β₯ n)
: (choose (p * m) (p * n) β‘ choose m n [MOD p]) := by
| import Mathlib
open RingHom Set Nat
/--
Let $p$ be a prime and $m \geq n$ be non-negative integers. Show that $\binom{pm}{pn} = \binom{m}{n} \pmod p$, where $\binom{m}{n}$ is the binomial coefficient.
-/
theorem putnam_1977_a5
(p m n : β)
(hp : Nat.Prime p)
(hmgen : m β₯ n)
: (choose (p * m) (p * n) β‘ choose m n [MOD p]) :=
sorry
| Let $p$ be a prime and $m \geq n$ be non-negative integers. Show that $\binom{pm}{pn} = \binom{m}{n} \pmod p$, where $\binom{m}{n}$ is the binomial coefficient. | null | [
"algebra",
"number_theory"
] | null | null |
|
putnam_2001_a1 | ebad0c32-b841-509f-a9a5-bc6bf3f39640 | train | theorem putnam_2001_a1
(S : Type*)
[Mul S]
(hS : β a b : S, (a * b) * a = b)
: β a b : S, a * (b * a) = b :=
sorry | import Mathlib
open Topology Filter
/--
Consider a set $S$ and a binary operation $*$, i.e., for each $a,b\in S$, $a*b\in S$. Assume $(a*b)*a=b$ for all $a,b\in S$. Prove that $a*(b*a)=b$ for all $a,b\in S$.
-/
theorem putnam_2001_a1
(S : Type*)
[Mul S]
(hS : β a b : S, (a * b) * a = b)
: β a b : S, a * (b * a) = b := by
| import Mathlib
open Topology Filter
/--
Consider a set $S$ and a binary operation $*$, i.e., for each $a,b\in S$, $a*b\in S$. Assume $(a*b)*a=b$ for all $a,b\in S$. Prove that $a*(b*a)=b$ for all $a,b\in S$.
-/
theorem putnam_2001_a1
(S : Type*)
[Mul S]
(hS : β a b : S, (a * b) * a = b)
: β a b : S, a * (b * a) = b :=
sorry
| Consider a set $S$ and a binary operation $*$, i.e., for each $a,b\in S$, $a*b\in S$. Assume $(a*b)*a=b$ for all $a,b\in S$. Prove that $a*(b*a)=b$ for all $a,b\in S$. | null | [
"abstract_algebra"
] | null | null |
|
putnam_1992_a5 | c800428c-6194-5432-969a-403fb0f86e81 | train | theorem putnam_1992_a5
(a : β β β)
(ha : a = fun n β¦ ite (Even {i | (digits 2 n).get i = 1}.ncard) 0 1) :
Β¬β k > 0, β m > 0, β j β€ m - 1,
a (k + j) = a (k + m + j) β§ a (k + m + j) = a (k + 2 * m + j) :=
sorry | import Mathlib
open Topology Filter Nat Function
/--
For each positive integer $n$, let $a_n = 0$ (or $1$) if the number of $1$'s in the binary representation of $n$ is even (or odd), respectively. Show that there do not exist positive integers $k$ and $m$ such that
\[
a_{k+j} = a_{k+m+j} = a_{k+2m+j},
\]
for $0 \leq j \leq m-1$.
-/
theorem putnam_1992_a5
(a : β β β)
(ha : a = fun n β¦ ite (Even {i | (digits 2 n).get i = 1}.ncard) 0 1) :
Β¬β k > 0, β m > 0, β j β€ m - 1,
a (k + j) = a (k + m + j) β§ a (k + m + j) = a (k + 2 * m + j) := by
| import Mathlib
open Topology Filter Nat Function
/--
For each positive integer $n$, let $a_n = 0$ (or $1$) if the number of $1$'s in the binary representation of $n$ is even (or odd), respectively. Show that there do not exist positive integers $k$ and $m$ such that
\[
a_{k+j} = a_{k+m+j} = a_{k+2m+j},
\]
for $0 \leq j \leq m-1$.
-/
theorem putnam_1992_a5
(a : β β β)
(ha : a = fun n β¦ ite (Even {i | (digits 2 n).get i = 1}.ncard) 0 1) :
Β¬β k > 0, β m > 0, β j β€ m - 1,
a (k + j) = a (k + m + j) β§ a (k + m + j) = a (k + 2 * m + j) :=
sorry
| For each positive integer $n$, let $a_n = 0$ (or $1$) if the number of $1$'s in the binary representation of $n$ is even (or odd), respectively. Show that there do not exist positive integers $k$ and $m$ such that
\[
a_{k+j} = a_{k+m+j} = a_{k+2m+j},
\]
for $0 \leq j \leq m-1$. | null | [
"algebra"
] | null | null |
|
putnam_1998_b5 | 7e7dcc74-533b-5947-bb0b-3374e385d154 | train | abbrev putnam_1998_b5_solution : β := sorry
-- 1
/--
Let $N$ be the positive integer with 1998 decimal digits, all of them 1; that is, \[N=1111\cdots 11.\] Find the thousandth digit after the decimal point of $\sqrt N$.
-/
theorem putnam_1998_b5
(N : β)
(hN : N = β i in Finset.range 1998, 10^i)
: putnam_1998_b5_solution = (Nat.floor (10^1000 * Real.sqrt N)) % 10 :=
sorry | import Mathlib
open Set Function Metric
-- 1
/--
Let $N$ be the positive integer with 1998 decimal digits, all of them 1; that is, \[N=1111\cdots 11.\] Find the thousandth digit after the decimal point of $\sqrt N$.
-/
theorem putnam_1998_b5
(N : β)
(hN : N = β i in Finset.range 1998, 10^i)
: putnam_1998_b5_solution = (Nat.floor (10^1000 * Real.sqrt N)) % 10 := by
| import Mathlib
open Set Function Metric
abbrev putnam_1998_b5_solution : β := sorry
-- 1
/--
Let $N$ be the positive integer with 1998 decimal digits, all of them 1; that is, \[N=1111\cdots 11.\] Find the thousandth digit after the decimal point of $\sqrt N$.
-/
theorem putnam_1998_b5
(N : β)
(hN : N = β i in Finset.range 1998, 10^i)
: putnam_1998_b5_solution = (Nat.floor (10^1000 * Real.sqrt N)) % 10 :=
sorry
| Let $N$ be the positive integer with 1998 decimal digits, all of them 1; that is, \[N=1111\cdots 11.\] Find the thousandth digit after the decimal point of $\sqrt N$. | Show that the thousandth digit is 1. | [
"number_theory"
] | null | null |
|
putnam_1974_a4 | 034d1284-8d0e-5bd2-8876-d8bc215abec4 | train | abbrev putnam_1974_a4_solution : β β β := sorry
-- (fun n β¦ (1 : β) / (2 ^ (n - 1)) * (n * (n - 1).choose βn / 2ββ))
/--
Evaluate in closed form: $\frac{1}{2^{n-1}} \sum_{k < n/2} (n-2k)*{n \choose k}$.
-/
theorem putnam_1974_a4
(n : β)
(hn : 0 < n) :
(1 : β) / (2 ^ (n - 1)) * β k in Finset.Icc 0 βn / 2ββ, (n - 2 * k) * (n.choose k) =
putnam_1974_a4_solution n :=
sorry | import Mathlib
open Set Nat
-- (fun n β¦ (1 : β) / (2 ^ (n - 1)) * (n * (n - 1).choose βn / 2ββ))
/--
Evaluate in closed form: $\frac{1}{2^{n-1}} \sum_{k < n/2} (n-2k)*{n \choose k}$.
-/
theorem putnam_1974_a4
(n : β)
(hn : 0 < n) :
(1 : β) / (2 ^ (n - 1)) * β k in Finset.Icc 0 βn / 2ββ, (n - 2 * k) * (n.choose k) =
putnam_1974_a4_solution n := by
| import Mathlib
open Set Nat
noncomputable abbrev putnam_1974_a4_solution : β β β := sorry
-- (fun n β¦ (1 : β) / (2 ^ (n - 1)) * (n * (n - 1).choose βn / 2ββ))
/--
Evaluate in closed form: $\frac{1}{2^{n-1}} \sum_{k < n/2} (n-2k)*{n \choose k}$.
-/
theorem putnam_1974_a4
(n : β)
(hn : 0 < n) :
(1 : β) / (2 ^ (n - 1)) * β k in Finset.Icc 0 βn / 2ββ, (n - 2 * k) * (n.choose k) =
putnam_1974_a4_solution n :=
sorry
| Evaluate in closed form: $\frac{1}{2^{n-1}} \sum_{k < n/2} (n-2k)*{n \choose k}$. | Show that the solution is $\frac{n}{2^{n-1}} * {(n-1) \choose \left[ (n-1)/2 \right]}$. | [
"algebra"
] | null | null |
|
putnam_1999_b3 | 2a72d4f6-0cf0-5ab0-9d9a-ec013caa2eb4 | train | abbrev putnam_1999_b3_solution : β := sorry
-- 3
/--
Let $A=\{(x,y):0\leq x,y<1\}$. For $(x,y)\in A$, let \[S(x,y) = \sum_{\frac{1}{2}\leq \frac{m}{n}\leq 2} x^m y^n,\] where the sum ranges over all pairs $(m,n)$ of positive integers satisfying the indicated inequalities. Evaluate \[\lim_{(x,y)\rightarrow (1,1), (x,y)\in A} (1-xy^2)(1-x^2y)S(x,y).\]
-/
theorem putnam_1999_b3
(A : Set (β Γ β))
(hA : A = {xy | 0 β€ xy.1 β§ xy.1 < 1 β§ 0 β€ xy.2 β§ xy.2 < 1})
(S : β β β β β)
(hS : S = fun x y => β' m : β, β' n : β, if (m > 0 β§ n > 0 β§ 1/2 β€ m/n β§ m/n β€ 2) then x^m * y^n else 0)
: Tendsto (fun xy : (β Γ β) => (1 - xy.1 * xy.2^2) * (1 - xy.1^2 * xy.2) * (S xy.1 xy.2)) (π[A] β¨1,1β©) (π putnam_1999_b3_solution) :=
sorry | import Mathlib
open Filter Topology Metric
-- 3
/--
Let $A=\{(x,y):0\leq x,y<1\}$. For $(x,y)\in A$, let \[S(x,y) = \sum_{\frac{1}{2}\leq \frac{m}{n}\leq 2} x^m y^n,\] where the sum ranges over all pairs $(m,n)$ of positive integers satisfying the indicated inequalities. Evaluate \[\lim_{(x,y)\rightarrow (1,1), (x,y)\in A} (1-xy^2)(1-x^2y)S(x,y).\]
-/
theorem putnam_1999_b3
(A : Set (β Γ β))
(hA : A = {xy | 0 β€ xy.1 β§ xy.1 < 1 β§ 0 β€ xy.2 β§ xy.2 < 1})
(S : β β β β β)
(hS : S = fun x y => β' m : β, β' n : β, if (m > 0 β§ n > 0 β§ 1/2 β€ m/n β§ m/n β€ 2) then x^m * y^n else 0)
: Tendsto (fun xy : (β Γ β) => (1 - xy.1 * xy.2^2) * (1 - xy.1^2 * xy.2) * (S xy.1 xy.2)) (π[A] β¨1,1β©) (π putnam_1999_b3_solution) := by
| import Mathlib
open Filter Topology Metric
abbrev putnam_1999_b3_solution : β := sorry
-- 3
/--
Let $A=\{(x,y):0\leq x,y<1\}$. For $(x,y)\in A$, let \[S(x,y) = \sum_{\frac{1}{2}\leq \frac{m}{n}\leq 2} x^m y^n,\] where the sum ranges over all pairs $(m,n)$ of positive integers satisfying the indicated inequalities. Evaluate \[\lim_{(x,y)\rightarrow (1,1), (x,y)\in A} (1-xy^2)(1-x^2y)S(x,y).\]
-/
theorem putnam_1999_b3
(A : Set (β Γ β))
(hA : A = {xy | 0 β€ xy.1 β§ xy.1 < 1 β§ 0 β€ xy.2 β§ xy.2 < 1})
(S : β β β β β)
(hS : S = fun x y => β' m : β, β' n : β, if (m > 0 β§ n > 0 β§ 1/2 β€ m/n β§ m/n β€ 2) then x^m * y^n else 0)
: Tendsto (fun xy : (β Γ β) => (1 - xy.1 * xy.2^2) * (1 - xy.1^2 * xy.2) * (S xy.1 xy.2)) (π[A] β¨1,1β©) (π putnam_1999_b3_solution) :=
sorry
| Let $A=\{(x,y):0\leq x,y<1\}$. For $(x,y)\in A$, let \[S(x,y) = \sum_{\frac{1}{2}\leq \frac{m}{n}\leq 2} x^m y^n,\] where the sum ranges over all pairs $(m,n)$ of positive integers satisfying the indicated inequalities. Evaluate \[\lim_{(x,y)\rightarrow (1,1), (x,y)\in A} (1-xy^2)(1-x^2y)S(x,y).\] | Show that the answer is 3. | [
"algebra"
] | null | null |
|
putnam_1977_a4 | 4968e0fb-e488-5088-ba26-fef9d050f0a7 | train | abbrev putnam_1977_a4_solution : RatFunc β := sorry
-- RatFunc.X / (1 - RatFunc.X)
/--
Find $\sum_{n=0}^{\infty} \frac{x^{2^n}}{1 - x^{2^{n+1}}}$ as a rational function of $x$ for $x \in (0, 1)$.
-/
theorem putnam_1977_a4 :
β x β Ioo 0 1,
putnam_1977_a4_solution.eval (id β) x = β' n : β, x ^ 2 ^ n / (1 - x ^ 2 ^ (n + 1)) :=
sorry | import Mathlib
open RingHom Set
-- RatFunc.X / (1 - RatFunc.X)
/--
Find $\sum_{n=0}^{\infty} \frac{x^{2^n}}{1 - x^{2^{n+1}}}$ as a rational function of $x$ for $x \in (0, 1)$.
-/
theorem putnam_1977_a4 :
β x β Ioo 0 1,
putnam_1977_a4_solution.eval (id β) x = β' n : β, x ^ 2 ^ n / (1 - x ^ 2 ^ (n + 1)) := by
| import Mathlib
open RingHom Set
noncomputable abbrev putnam_1977_a4_solution : RatFunc β := sorry
-- RatFunc.X / (1 - RatFunc.X)
/--
Find $\sum_{n=0}^{\infty} \frac{x^{2^n}}{1 - x^{2^{n+1}}}$ as a rational function of $x$ for $x \in (0, 1)$.
-/
theorem putnam_1977_a4 :
β x β Ioo 0 1,
putnam_1977_a4_solution.eval (id β) x = β' n : β, x ^ 2 ^ n / (1 - x ^ 2 ^ (n + 1)) :=
sorry
| Find $\sum_{n=0}^{\infty} \frac{x^{2^n}}{1 - x^{2^{n+1}}}$ as a rational function of $x$ for $x \in (0, 1)$. | Prove that the sum equals $\frac{x}{1 - x}$. | [
"algebra",
"analysis"
] | null | null |
|
putnam_2012_a6 | 21cc9756-17f7-5d54-a411-73a189584e9e | train | abbrev putnam_2012_a6_solution : Prop := sorry
-- True
/--
Let $f(x,y)$ be a continuous, real-valued function on $\mathbb{R}^2$. Suppose that, for every rectangular region $R$ of area $1$, the double integral of $f(x,y)$ over $R$ equals $0$.
Must $f(x,y)$ be identically $0$?
-/
theorem putnam_2012_a6
(p : ((β Γ β) β β) β Prop)
(hp : β f, p f β
Continuous f β§
β x1 x2 y1 y2 : β, x2 > x1 β y2 > y1
β (x2 - x1) * (y2 - y1) = 1 β β« x in x1..x2, β« y in y1..y2, f (x, y) = 0)
: ((β f x y, p f β f (x, y) = 0) β putnam_2012_a6_solution) :=
sorry | import Mathlib
open Matrix Function
-- Note: this formalization differs from the original problem wording in only allowing axis-aligned rectangles. The problem is solvable given this weaker hypothesis.
-- True
/--
Let $f(x,y)$ be a continuous, real-valued function on $\mathbb{R}^2$. Suppose that, for every rectangular region $R$ of area $1$, the double integral of $f(x,y)$ over $R$ equals $0$.
Must $f(x,y)$ be identically $0$?
-/
theorem putnam_2012_a6
(p : ((β Γ β) β β) β Prop)
(hp : β f, p f β
Continuous f β§
β x1 x2 y1 y2 : β, x2 > x1 β y2 > y1
β (x2 - x1) * (y2 - y1) = 1 β β« x in x1..x2, β« y in y1..y2, f (x, y) = 0)
: ((β f x y, p f β f (x, y) = 0) β putnam_2012_a6_solution) := by
| import Mathlib
open Matrix Function
-- Note: this formalization differs from the original problem wording in only allowing axis-aligned rectangles. The problem is solvable given this weaker hypothesis.
abbrev putnam_2012_a6_solution : Prop := sorry
-- True
/--
Let $f(x,y)$ be a continuous, real-valued function on $\mathbb{R}^2$. Suppose that, for every rectangular region $R$ of area $1$, the double integral of $f(x,y)$ over $R$ equals $0$.
Must $f(x,y)$ be identically $0$?
-/
theorem putnam_2012_a6
(p : ((β Γ β) β β) β Prop)
(hp : β f, p f β
Continuous f β§
β x1 x2 y1 y2 : β, x2 > x1 β y2 > y1
β (x2 - x1) * (y2 - y1) = 1 β β« x in x1..x2, β« y in y1..y2, f (x, y) = 0)
: ((β f x y, p f β f (x, y) = 0) β putnam_2012_a6_solution) :=
sorry
| Let $f(x,y)$ be a continuous, real-valued function on $\mathbb{R}^2$. Suppose that, for every rectangular region $R$ of area $1$, the double integral of $f(x,y)$ over $R$ equals $0$.
Must $f(x,y)$ be identically $0$? | Prove that $f(x,y)$ must be identically $0$. | [
"analysis"
] | null | null |
|
putnam_1978_a6 | dfd0faab-74f5-5b30-8344-f98238d7d79f | train | theorem putnam_1978_a6
(S : Finset (EuclideanSpace β (Fin 2)))
(n : β)
(hn : n = S.card)
(npos : n > 0)
: ({pair : Set (EuclideanSpace β (Fin 2)) | β P β S, β Q β S, pair = {P, Q} β§ dist P Q = 1}.ncard < 2 * (n : β) ^ ((3 : β) / 2)) :=
sorry | import Mathlib
open Set Real
/--
Given $n$ distinct points in the plane, prove that fewer than $2n^{3/2}$ pairs of these points are a distance of $1$ apart.
-/
theorem putnam_1978_a6
(S : Finset (EuclideanSpace β (Fin 2)))
(n : β)
(hn : n = S.card)
(npos : n > 0)
: ({pair : Set (EuclideanSpace β (Fin 2)) | β P β S, β Q β S, pair = {P, Q} β§ dist P Q = 1}.ncard < 2 * (n : β) ^ ((3 : β) / 2)) := by
| import Mathlib
open Set Real
/--
Given $n$ distinct points in the plane, prove that fewer than $2n^{3/2}$ pairs of these points are a distance of $1$ apart.
-/
theorem putnam_1978_a6
(S : Finset (EuclideanSpace β (Fin 2)))
(n : β)
(hn : n = S.card)
(npos : n > 0)
: ({pair : Set (EuclideanSpace β (Fin 2)) | β P β S, β Q β S, pair = {P, Q} β§ dist P Q = 1}.ncard < 2 * (n : β) ^ ((3 : β) / 2)) :=
sorry
| Given $n$ distinct points in the plane, prove that fewer than $2n^{3/2}$ pairs of these points are a distance of $1$ apart. | null | [
"geometry",
"combinatorics"
] | null | null |
|
putnam_2010_a1 | 6f593012-f770-5642-9906-b1a3b5a86bf7 | train | abbrev putnam_2010_a1_solution : β β β := sorry
-- (fun n : β => Nat.ceil ((n : β) / 2))
/--
Given a positive integer $n$, what is the largest $k$ such that the numbers $1,2,\dots,n$ can be put into $k$ boxes so that the sum of the numbers in each box is the same? [When $n=8$, the example $\{1,2,3,6\},\{4,8\},\{5,7\}$ shows that the largest $k$ is \emph{at least} $3$.]
-/
theorem putnam_2010_a1
(n : β)
(kboxes : β β Prop)
(npos : n > 0)
(hkboxes : β k : β, kboxes k =
(β boxes : Finset.Icc 1 n β Fin k, β i j : Fin k,
β x in Finset.univ.filter (boxes Β· = i), (x : β) =
β x in Finset.univ.filter (boxes Β· = j), (x : β))) :
IsGreatest kboxes (putnam_2010_a1_solution n) :=
sorry | import Mathlib
-- (fun n : β => Nat.ceil ((n : β) / 2))
/--
Given a positive integer $n$, what is the largest $k$ such that the numbers $1,2,\dots,n$ can be put into $k$ boxes so that the sum of the numbers in each box is the same? [When $n=8$, the example $\{1,2,3,6\},\{4,8\},\{5,7\}$ shows that the largest $k$ is \emph{at least} $3$.]
-/
theorem putnam_2010_a1
(n : β)
(kboxes : β β Prop)
(npos : n > 0)
(hkboxes : β k : β, kboxes k =
(β boxes : Finset.Icc 1 n β Fin k, β i j : Fin k,
β x in Finset.univ.filter (boxes Β· = i), (x : β) =
β x in Finset.univ.filter (boxes Β· = j), (x : β))) :
IsGreatest kboxes (putnam_2010_a1_solution n) := by
| import Mathlib
noncomputable abbrev putnam_2010_a1_solution : β β β := sorry
-- (fun n : β => Nat.ceil ((n : β) / 2))
/--
Given a positive integer $n$, what is the largest $k$ such that the numbers $1,2,\dots,n$ can be put into $k$ boxes so that the sum of the numbers in each box is the same? [When $n=8$, the example $\{1,2,3,6\},\{4,8\},\{5,7\}$ shows that the largest $k$ is \emph{at least} $3$.]
-/
theorem putnam_2010_a1
(n : β)
(kboxes : β β Prop)
(npos : n > 0)
(hkboxes : β k : β, kboxes k =
(β boxes : Finset.Icc 1 n β Fin k, β i j : Fin k,
β x in Finset.univ.filter (boxes Β· = i), (x : β) =
β x in Finset.univ.filter (boxes Β· = j), (x : β))) :
IsGreatest kboxes (putnam_2010_a1_solution n) :=
sorry
| Given a positive integer $n$, what is the largest $k$ such that the numbers $1,2,\dots,n$ can be put into $k$ boxes so that the sum of the numbers in each box is the same? [When $n=8$, the example $\{1,2,3,6\},\{4,8\},\{5,7\}$ shows that the largest $k$ is \emph{at least} $3$.] | Show that the largest such $k$ is $\lceil \frac{n}{2} \rceil$. | [
"algebra"
] | null | null |
|
putnam_1972_a5 | 2b1b56c2-f1c6-58b5-b399-21f9bd3f79d4 | train | theorem putnam_1972_a5
(n : β)
(hn : n > 1)
: Β¬((n : β€) β£ 2^n - 1) :=
sorry | import Mathlib
open EuclideanGeometry Filter Topology Set
/--
Show that if $n$ is an integer greater than $1$, then $n$ does not divide $2^n - 1$.
-/
theorem putnam_1972_a5
(n : β)
(hn : n > 1)
: Β¬((n : β€) β£ 2^n - 1) := by
| import Mathlib
open EuclideanGeometry Filter Topology Set
/--
Show that if $n$ is an integer greater than $1$, then $n$ does not divide $2^n - 1$.
-/
theorem putnam_1972_a5
(n : β)
(hn : n > 1)
: Β¬((n : β€) β£ 2^n - 1) :=
sorry
| Show that if $n$ is an integer greater than $1$, then $n$ does not divide $2^n - 1$. | null | [
"number_theory"
] | null | null |
|
putnam_1971_a2 | 6b5a1953-92c9-57c3-9376-cb4fbd79d624 | train | abbrev putnam_1971_a2_solution : Set (Polynomial β) := sorry
-- {Polynomial.X}
/--
Determine all polynomials $P(x)$ such that $P(x^2 + 1) = (P(x))^2 + 1$ and $P(0) = 0$.
-/
theorem putnam_1971_a2
(P : Polynomial β) :
(P.eval 0 = 0 β§ (β x : β, P.eval (x^2 + 1) = (P.eval x)^2 + 1)) β P β putnam_1971_a2_solution :=
sorry | import Mathlib
open Set
-- {Polynomial.X}
/--
Determine all polynomials $P(x)$ such that $P(x^2 + 1) = (P(x))^2 + 1$ and $P(0) = 0$.
-/
theorem putnam_1971_a2
(P : Polynomial β) :
(P.eval 0 = 0 β§ (β x : β, P.eval (x^2 + 1) = (P.eval x)^2 + 1)) β P β putnam_1971_a2_solution := by
| import Mathlib
open Set
abbrev putnam_1971_a2_solution : Set (Polynomial β) := sorry
-- {Polynomial.X}
/--
Determine all polynomials $P(x)$ such that $P(x^2 + 1) = (P(x))^2 + 1$ and $P(0) = 0$.
-/
theorem putnam_1971_a2
(P : Polynomial β) :
(P.eval 0 = 0 β§ (β x : β, P.eval (x^2 + 1) = (P.eval x)^2 + 1)) β P β putnam_1971_a2_solution :=
sorry
| Determine all polynomials $P(x)$ such that $P(x^2 + 1) = (P(x))^2 + 1$ and $P(0) = 0$. | Show that the only such polynomial is the identity function. | [
"algebra"
] | null | null |
|
putnam_1968_b2 | 1a400940-171a-566c-939c-a8d9beeec997 | train | theorem putnam_1968_b2
{G : Type*}
[Group G]
(hG : Finite G)
(A : Set G)
(hA : A.ncard > (Nat.card G : β)/2)
: β g : G, β x β A, β y β A, g = x * y :=
sorry | import Mathlib
open Finset Polynomial
/--
Let $G$ be a finite group (with a multiplicative operation), and $A$ be a subset of $G$ that contains more than half of $G$'s elements. Prove that every element of $G$ can be expressed as the product of two elements of $A$.
-/
theorem putnam_1968_b2
{G : Type*}
[Group G]
(hG : Finite G)
(A : Set G)
(hA : A.ncard > (Nat.card G : β)/2)
: β g : G, β x β A, β y β A, g = x * y := by
| import Mathlib
open Finset Polynomial
/--
Let $G$ be a finite group (with a multiplicative operation), and $A$ be a subset of $G$ that contains more than half of $G$'s elements. Prove that every element of $G$ can be expressed as the product of two elements of $A$.
-/
theorem putnam_1968_b2
{G : Type*}
[Group G]
(hG : Finite G)
(A : Set G)
(hA : A.ncard > (Nat.card G : β)/2)
: β g : G, β x β A, β y β A, g = x * y :=
sorry
| Let $G$ be a finite group (with a multiplicative operation), and $A$ be a subset of $G$ that contains more than half of $G$'s elements. Prove that every element of $G$ can be expressed as the product of two elements of $A$. | null | [
"abstract_algebra"
] | null | null |
|
putnam_1971_b6 | 82b8c24c-f1c2-5883-bf42-3ff0ee616fec | train | theorem putnam_1971_b6
(Ξ΄ : β€ β β€)
(hΞ΄ : Ξ΄ = fun n => sSup {t | Odd t β§ t β£ n})
: β x : β€, x > 0 β |β i in Finset.Icc 1 x, (Ξ΄ i)/(i : β) - 2*x/3| < 1 :=
sorry | import Mathlib
open Set MvPolynomial
/--
Let $\delta(x) be the greatest odd divisor of the positive integer $x$. Show that $|\sum_{n = 1}^x \delta(n)/n - 2x/3| < 1$ for all positive integers $x$.
-/
theorem putnam_1971_b6
(Ξ΄ : β€ β β€)
(hΞ΄ : Ξ΄ = fun n => sSup {t | Odd t β§ t β£ n})
: β x : β€, x > 0 β |β i in Finset.Icc 1 x, (Ξ΄ i)/(i : β) - 2*x/3| < 1 := by
| import Mathlib
open Set MvPolynomial
/--
Let $\delta(x) be the greatest odd divisor of the positive integer $x$. Show that $|\sum_{n = 1}^x \delta(n)/n - 2x/3| < 1$ for all positive integers $x$.
-/
theorem putnam_1971_b6
(Ξ΄ : β€ β β€)
(hΞ΄ : Ξ΄ = fun n => sSup {t | Odd t β§ t β£ n})
: β x : β€, x > 0 β |β i in Finset.Icc 1 x, (Ξ΄ i)/(i : β) - 2*x/3| < 1 :=
sorry
| Let $\delta(x) be the greatest odd divisor of the positive integer $x$. Show that $|\sum_{n = 1}^x \delta(n)/n - 2x/3| < 1$ for all positive integers $x$. | null | [
"number_theory"
] | null | null |
|
putnam_2002_a2 | 33ff015a-9a99-5267-a913-fc05c0233593 | train | theorem putnam_2002_a2
(unit_sphere : Set (EuclideanSpace β (Fin 3)))
(hsphere : unit_sphere = sphere 0 1)
(hemi : EuclideanSpace β (Fin 3) β Set (EuclideanSpace β (Fin 3)))
(hhemi : hemi = fun V β¦ {P : EuclideanSpace β (Fin 3) | βͺP, Vβ«_β β₯ 0})
: (β (S : Set (EuclideanSpace β (Fin 3))), S β unit_sphere β§ S.encard = 5 β β V : EuclideanSpace β (Fin 3), V β 0 β§ (S β© hemi V).encard β₯ 4) :=
sorry | import Mathlib
open Nat Metric
open scoped InnerProductSpace
/--
Given any five points on a sphere, show that some four of them must lie on a closed hemisphere.
-/
theorem putnam_2002_a2
(unit_sphere : Set (EuclideanSpace β (Fin 3)))
(hsphere : unit_sphere = sphere 0 1)
(hemi : EuclideanSpace β (Fin 3) β Set (EuclideanSpace β (Fin 3)))
(hhemi : hemi = fun V β¦ {P : EuclideanSpace β (Fin 3) | βͺP, Vβ«_β β₯ 0})
: (β (S : Set (EuclideanSpace β (Fin 3))), S β unit_sphere β§ S.encard = 5 β β V : EuclideanSpace β (Fin 3), V β 0 β§ (S β© hemi V).encard β₯ 4) := by
| import Mathlib
open Nat Metric
open scoped InnerProductSpace
/--
Given any five points on a sphere, show that some four of them must lie on a closed hemisphere.
-/
theorem putnam_2002_a2
(unit_sphere : Set (EuclideanSpace β (Fin 3)))
(hsphere : unit_sphere = sphere 0 1)
(hemi : EuclideanSpace β (Fin 3) β Set (EuclideanSpace β (Fin 3)))
(hhemi : hemi = fun V β¦ {P : EuclideanSpace β (Fin 3) | βͺP, Vβ«_β β₯ 0})
: (β (S : Set (EuclideanSpace β (Fin 3))), S β unit_sphere β§ S.encard = 5 β β V : EuclideanSpace β (Fin 3), V β 0 β§ (S β© hemi V).encard β₯ 4) :=
sorry
| Given any five points on a sphere, show that some four of them must lie on a closed hemisphere. | null | [
"geometry"
] | null | null |
|
putnam_1993_b3 | e0a38cb0-28ef-5577-8d94-9cddbadbaf90 | train | abbrev putnam_1993_b3_solution : β Γ β := sorry
-- (5 / 4, -1 / 4)
/--
Two real numbers $x$ and $y$ are chosen at random in the interval $(0,1)$ with respect to the uniform distribution. What is the probability that the closest integer to $x/y$ is even? Express the answer in the form $r+s\pi$, where $r$ and $s$ are rational numbers.
-/
theorem putnam_1993_b3 :
let (r, s) := putnam_1993_b3_solution;
(MeasureTheory.volume
{p : Fin 2 β β | 0 < p β§ p < 1 β§ Even (round (p 0 / p 1))}
).toReal
= r + s * Real.pi :=
sorry | import Mathlib
-- (5 / 4, -1 / 4)
/--
Two real numbers $x$ and $y$ are chosen at random in the interval $(0,1)$ with respect to the uniform distribution. What is the probability that the closest integer to $x/y$ is even? Express the answer in the form $r+s\pi$, where $r$ and $s$ are rational numbers.
-/
theorem putnam_1993_b3 :
let (r, s) := putnam_1993_b3_solution;
(MeasureTheory.volume
{p : Fin 2 β β | 0 < p β§ p < 1 β§ Even (round (p 0 / p 1))}
).toReal
= r + s * Real.pi := by
| import Mathlib
abbrev putnam_1993_b3_solution : β Γ β := sorry
-- (5 / 4, -1 / 4)
/--
Two real numbers $x$ and $y$ are chosen at random in the interval $(0,1)$ with respect to the uniform distribution. What is the probability that the closest integer to $x/y$ is even? Express the answer in the form $r+s\pi$, where $r$ and $s$ are rational numbers.
-/
theorem putnam_1993_b3 :
let (r, s) := putnam_1993_b3_solution;
(MeasureTheory.volume
{p : Fin 2 β β | 0 < p β§ p < 1 β§ Even (round (p 0 / p 1))}
).toReal
= r + s * Real.pi :=
sorry
| Two real numbers $x$ and $y$ are chosen at random in the interval $(0,1)$ with respect to the uniform distribution. What is the probability that the closest integer to $x/y$ is even? Express the answer in the form $r+s\pi$, where $r$ and $s$ are rational numbers. | Show that the limit is $(5-\pi)/4$. That is, $r=5/4$ and $s=-1/4$. | [
"probability",
"number_theory",
"geometry"
] | null | null |
|
putnam_2005_b6 | 64b4bbbc-8289-5418-aedd-1b695ad3084a | train | theorem putnam_2005_b6
(n : β)
(v : Equiv.Perm (Fin n) β β)
(npos : n β₯ 1)
(hv : β p : Equiv.Perm (Fin n), v p = Set.encard {i : Fin n | p i = i})
: (β p : Equiv.Perm (Fin n), (Equiv.Perm.signAux p : β€) / (v p + 1 : β)) = (-1) ^ (n + 1) * (n / (n + 1 : β)) :=
sorry | import Mathlib
open Nat Set
/--
Let $S_n$ denote the set of all permutations of the numbers $1,2,\dots,n$. For $\pi \in S_n$, let $\sigma(\pi)=1$ if $\pi$ is an even permutation and $\sigma(\pi)=-1$ if $\pi$ is an odd permutation. Also, let $\nu(\pi)$ denote the number of fixed points of $\pi$. Show that $\sum_{\pi \in S_n} \frac{\sigma(\pi)}{\nu(\pi)+1}=(-1)^{n+1}\frac{n}{n+1}$.
-/
theorem putnam_2005_b6
(n : β)
(v : Equiv.Perm (Fin n) β β)
(npos : n β₯ 1)
(hv : β p : Equiv.Perm (Fin n), v p = Set.encard {i : Fin n | p i = i})
: (β p : Equiv.Perm (Fin n), (Equiv.Perm.signAux p : β€) / (v p + 1 : β)) = (-1) ^ (n + 1) * (n / (n + 1 : β)) := by
| import Mathlib
open Nat Set
/--
Let $S_n$ denote the set of all permutations of the numbers $1,2,\dots,n$. For $\pi \in S_n$, let $\sigma(\pi)=1$ if $\pi$ is an even permutation and $\sigma(\pi)=-1$ if $\pi$ is an odd permutation. Also, let $\nu(\pi)$ denote the number of fixed points of $\pi$. Show that $\sum_{\pi \in S_n} \frac{\sigma(\pi)}{\nu(\pi)+1}=(-1)^{n+1}\frac{n}{n+1}$.
-/
theorem putnam_2005_b6
(n : β)
(v : Equiv.Perm (Fin n) β β)
(npos : n β₯ 1)
(hv : β p : Equiv.Perm (Fin n), v p = Set.encard {i : Fin n | p i = i})
: (β p : Equiv.Perm (Fin n), (Equiv.Perm.signAux p : β€) / (v p + 1 : β)) = (-1) ^ (n + 1) * (n / (n + 1 : β)) :=
sorry
| Let $S_n$ denote the set of all permutations of the numbers $1,2,\dots,n$. For $\pi \in S_n$, let $\sigma(\pi)=1$ if $\pi$ is an even permutation and $\sigma(\pi)=-1$ if $\pi$ is an odd permutation. Also, let $\nu(\pi)$ denote the number of fixed points of $\pi$. Show that $\sum_{\pi \in S_n} \frac{\sigma(\pi)}{\nu(\pi)+1}=(-1)^{n+1}\frac{n}{n+1}$. | null | [
"linear_algebra",
"algebra"
] | null | null |
|
putnam_1980_b1 | 83e138c8-4f27-570e-a433-5ab468a30e9f | train | abbrev putnam_1980_b1_solution : Set β := sorry
-- {c : β | c β₯ 1 / 2}
/--
For which real numbers $c$ is $(e^x+e^{-x})/2 \leq e^{cx^2}$ for all real $x$?
-/
theorem putnam_1980_b1
(c : β)
: (β x : β, (exp x + exp (-x)) / 2 β€ exp (c * x ^ 2)) β c β putnam_1980_b1_solution :=
sorry | import Mathlib
open Real
-- {c : β | c β₯ 1 / 2}
/--
For which real numbers $c$ is $(e^x+e^{-x})/2 \leq e^{cx^2}$ for all real $x$?
-/
theorem putnam_1980_b1
(c : β)
: (β x : β, (exp x + exp (-x)) / 2 β€ exp (c * x ^ 2)) β c β putnam_1980_b1_solution := by
| import Mathlib
open Real
abbrev putnam_1980_b1_solution : Set β := sorry
-- {c : β | c β₯ 1 / 2}
/--
For which real numbers $c$ is $(e^x+e^{-x})/2 \leq e^{cx^2}$ for all real $x$?
-/
theorem putnam_1980_b1
(c : β)
: (β x : β, (exp x + exp (-x)) / 2 β€ exp (c * x ^ 2)) β c β putnam_1980_b1_solution :=
sorry
| For which real numbers $c$ is $(e^x+e^{-x})/2 \leq e^{cx^2}$ for all real $x$? | Show that the inequality holds if and only if $c \geq 1/2$. | [
"analysis"
] | null | null |
|
putnam_2011_a2 | 0780d3f8-f6a6-5094-91db-e3a4b790d8fe | train | abbrev putnam_2011_a2_solution : β := sorry
-- 3/2
/--
Let $a_1,a_2,\dots$ and $b_1,b_2,\dots$ be sequences of positive real numbers such that $a_1 = b_1 = 1$ and $b_n = b_{n-1} a_n - 2$ for$n=2,3,\dots$. Assume that the sequence $(b_j)$ is bounded. Prove tha \[ S = \sum_{n=1}^\infty \frac{1}{a_1...a_n} \] converges, and evaluate $S$.
-/
theorem putnam_2011_a2
(a b : β β β)
(habn : β n : β, a n > 0 β§ b n > 0)
(hab1 : a 0 = 1 β§ b 0 = 1)
(hb : β n β₯ 1, b n = b (n-1) * a n - 2)
(hbnd : β B : β, β n : β, |b n| β€ B)
: Tendsto (fun n => β i : Fin n, 1/(β j : Fin (i + 1), (a j))) atTop (π putnam_2011_a2_solution) :=
sorry | import Mathlib
open Topology Filter
-- 3/2
/--
Let $a_1,a_2,\dots$ and $b_1,b_2,\dots$ be sequences of positive real numbers such that $a_1 = b_1 = 1$ and $b_n = b_{n-1} a_n - 2$ for$n=2,3,\dots$. Assume that the sequence $(b_j)$ is bounded. Prove tha \[ S = \sum_{n=1}^\infty \frac{1}{a_1...a_n} \] converges, and evaluate $S$.
-/
theorem putnam_2011_a2
(a b : β β β)
(habn : β n : β, a n > 0 β§ b n > 0)
(hab1 : a 0 = 1 β§ b 0 = 1)
(hb : β n β₯ 1, b n = b (n-1) * a n - 2)
(hbnd : β B : β, β n : β, |b n| β€ B)
: Tendsto (fun n => β i : Fin n, 1/(β j : Fin (i + 1), (a j))) atTop (π putnam_2011_a2_solution) := by
| import Mathlib
open Topology Filter
noncomputable abbrev putnam_2011_a2_solution : β := sorry
-- 3/2
/--
Let $a_1,a_2,\dots$ and $b_1,b_2,\dots$ be sequences of positive real numbers such that $a_1 = b_1 = 1$ and $b_n = b_{n-1} a_n - 2$ for$n=2,3,\dots$. Assume that the sequence $(b_j)$ is bounded. Prove tha \[ S = \sum_{n=1}^\infty \frac{1}{a_1...a_n} \] converges, and evaluate $S$.
-/
theorem putnam_2011_a2
(a b : β β β)
(habn : β n : β, a n > 0 β§ b n > 0)
(hab1 : a 0 = 1 β§ b 0 = 1)
(hb : β n β₯ 1, b n = b (n-1) * a n - 2)
(hbnd : β B : β, β n : β, |b n| β€ B)
: Tendsto (fun n => β i : Fin n, 1/(β j : Fin (i + 1), (a j))) atTop (π putnam_2011_a2_solution) :=
sorry
| Let $a_1,a_2,\dots$ and $b_1,b_2,\dots$ be sequences of positive real numbers such that $a_1 = b_1 = 1$ and $b_n = b_{n-1} a_n - 2$ for$n=2,3,\dots$. Assume that the sequence $(b_j)$ is bounded. Prove tha \[ S = \sum_{n=1}^\infty \frac{1}{a_1...a_n} \] converges, and evaluate $S$. | Show that the solution is $S = 3/2$. | [
"analysis"
] | null | null |
|
putnam_1988_b2 | 2b29edb7-5a43-59dd-8e74-92e0e440a7de | train | abbrev putnam_1988_b2_solution : Prop := sorry
-- True
/--
Prove or disprove: If $x$ and $y$ are real numbers with $y \geq 0$ and $y(y+1) \leq (x+1)^2$, then $y(y-1) \leq x^2$.
-/
theorem putnam_1988_b2
: (β x y : β, (y β₯ 0 β§ y * (y + 1) β€ (x + 1) ^ 2) β (y * (y - 1) β€ x ^ 2)) β putnam_1988_b2_solution :=
sorry | import Mathlib
open Set Filter Topology
-- True
/--
Prove or disprove: If $x$ and $y$ are real numbers with $y \geq 0$ and $y(y+1) \leq (x+1)^2$, then $y(y-1) \leq x^2$.
-/
theorem putnam_1988_b2
: (β x y : β, (y β₯ 0 β§ y * (y + 1) β€ (x + 1) ^ 2) β (y * (y - 1) β€ x ^ 2)) β putnam_1988_b2_solution := by
| import Mathlib
open Set Filter Topology
abbrev putnam_1988_b2_solution : Prop := sorry
-- True
/--
Prove or disprove: If $x$ and $y$ are real numbers with $y \geq 0$ and $y(y+1) \leq (x+1)^2$, then $y(y-1) \leq x^2$.
-/
theorem putnam_1988_b2
: (β x y : β, (y β₯ 0 β§ y * (y + 1) β€ (x + 1) ^ 2) β (y * (y - 1) β€ x ^ 2)) β putnam_1988_b2_solution :=
sorry
| Prove or disprove: If $x$ and $y$ are real numbers with $y \geq 0$ and $y(y+1) \leq (x+1)^2$, then $y(y-1) \leq x^2$. | Show that this is true. | [
"algebra"
] | null | null |
|
putnam_1998_a6 | a5c572a5-bf9e-50d4-a6e2-787b3ecb21cc | train | theorem putnam_1998_a6
(A B C : EuclideanSpace β (Fin 2))
(hint : β i : Fin 2, β a b c : β€, A i = a β§ B i = b β§ C i = c)
(htriangle : A β B β§ A β C β§ B β C)
(harea : (dist A B + dist B C) ^ 2 < 8 * (MeasureTheory.volume (convexHull β {A, B, C})).toReal + 1)
(threesquare : (EuclideanSpace β (Fin 2)) β (EuclideanSpace β (Fin 2)) β (EuclideanSpace β (Fin 2)) β Prop)
(threesquare_def : threesquare = fun P Q R β¦ dist Q P = dist Q R β§ βͺP - Q, R - Qβ«_β = 0)
: (threesquare A B C β¨ threesquare B C A β¨ threesquare C A B) :=
sorry | import Mathlib
open Set Function Metric
open scoped InnerProductSpace
/--
Let $A, B, C$ denote distinct points with integer coordinates in $\mathbb R^2$. Prove that if
\[(|AB|+|BC|)^2<8\cdot [ABC]+1\]
then $A, B, C$ are three vertices of a square. Here $|XY|$ is the length of segment $XY$ and $[ABC]$ is the area of triangle $ABC$.
-/
theorem putnam_1998_a6
(A B C : EuclideanSpace β (Fin 2))
(hint : β i : Fin 2, β a b c : β€, A i = a β§ B i = b β§ C i = c)
(htriangle : A β B β§ A β C β§ B β C)
(harea : (dist A B + dist B C) ^ 2 < 8 * (MeasureTheory.volume (convexHull β {A, B, C})).toReal + 1)
(threesquare : (EuclideanSpace β (Fin 2)) β (EuclideanSpace β (Fin 2)) β (EuclideanSpace β (Fin 2)) β Prop)
(threesquare_def : threesquare = fun P Q R β¦ dist Q P = dist Q R β§ βͺP - Q, R - Qβ«_β = 0)
: (threesquare A B C β¨ threesquare B C A β¨ threesquare C A B) := by
| import Mathlib
open Set Function Metric
open scoped InnerProductSpace
/--
Let $A, B, C$ denote distinct points with integer coordinates in $\mathbb R^2$. Prove that if
\[(|AB|+|BC|)^2<8\cdot [ABC]+1\]
then $A, B, C$ are three vertices of a square. Here $|XY|$ is the length of segment $XY$ and $[ABC]$ is the area of triangle $ABC$.
-/
theorem putnam_1998_a6
(A B C : EuclideanSpace β (Fin 2))
(hint : β i : Fin 2, β a b c : β€, A i = a β§ B i = b β§ C i = c)
(htriangle : A β B β§ A β C β§ B β C)
(harea : (dist A B + dist B C) ^ 2 < 8 * (MeasureTheory.volume (convexHull β {A, B, C})).toReal + 1)
(threesquare : (EuclideanSpace β (Fin 2)) β (EuclideanSpace β (Fin 2)) β (EuclideanSpace β (Fin 2)) β Prop)
(threesquare_def : threesquare = fun P Q R β¦ dist Q P = dist Q R β§ βͺP - Q, R - Qβ«_β = 0)
: (threesquare A B C β¨ threesquare B C A β¨ threesquare C A B) :=
sorry
| Let $A, B, C$ denote distinct points with integer coordinates in $\mathbb R^2$. Prove that if
\[(|AB|+|BC|)^2<8\cdot [ABC]+1\]
then $A, B, C$ are three vertices of a square. Here $|XY|$ is the length of segment $XY$ and $[ABC]$ is the area of triangle $ABC$. | null | [
"geometry"
] | null | null |
|
putnam_2014_a6 | b888a781-cd7c-56af-bc63-896f0383b7d4 | train | abbrev putnam_2014_a6_solution : β β β := sorry
-- (fun n : β => n ^ n)
/--
Let \( n \) be a positive integer. What is the largest \( k \) for which there exist \( n \times n \) matrices \( M_1, \ldots, M_k \) and \( N_1, \ldots, N_k \) with real entries such that for all \( i \) and \( j \), the matrix product \( M_i N_j \) has a zero entry somewhere on its diagonal if and only if \( i \neq j \)?
-/
theorem putnam_2014_a6
(n : β)
(kex : β β Prop)
(npos : n > 0)
(hkex : β k β₯ 1, kex k = β M N : Fin k β Matrix (Fin n) (Fin n) β, β i j : Fin k, ((β p : Fin n, (M i * N j) p p = 0) β i β j))
: (putnam_2014_a6_solution n β₯ 1 β§ kex (putnam_2014_a6_solution n)) β§ (β k β₯ 1, kex k β k β€ putnam_2014_a6_solution n) :=
sorry | import Mathlib
open Topology Filter Nat
-- (fun n : β => n ^ n)
/--
Let \( n \) be a positive integer. What is the largest \( k \) for which there exist \( n \times n \) matrices \( M_1, \ldots, M_k \) and \( N_1, \ldots, N_k \) with real entries such that for all \( i \) and \( j \), the matrix product \( M_i N_j \) has a zero entry somewhere on its diagonal if and only if \( i \neq j \)?
-/
theorem putnam_2014_a6
(n : β)
(kex : β β Prop)
(npos : n > 0)
(hkex : β k β₯ 1, kex k = β M N : Fin k β Matrix (Fin n) (Fin n) β, β i j : Fin k, ((β p : Fin n, (M i * N j) p p = 0) β i β j))
: (putnam_2014_a6_solution n β₯ 1 β§ kex (putnam_2014_a6_solution n)) β§ (β k β₯ 1, kex k β k β€ putnam_2014_a6_solution n) := by
| import Mathlib
open Topology Filter Nat
abbrev putnam_2014_a6_solution : β β β := sorry
-- (fun n : β => n ^ n)
/--
Let \( n \) be a positive integer. What is the largest \( k \) for which there exist \( n \times n \) matrices \( M_1, \ldots, M_k \) and \( N_1, \ldots, N_k \) with real entries such that for all \( i \) and \( j \), the matrix product \( M_i N_j \) has a zero entry somewhere on its diagonal if and only if \( i \neq j \)?
-/
theorem putnam_2014_a6
(n : β)
(kex : β β Prop)
(npos : n > 0)
(hkex : β k β₯ 1, kex k = β M N : Fin k β Matrix (Fin n) (Fin n) β, β i j : Fin k, ((β p : Fin n, (M i * N j) p p = 0) β i β j))
: (putnam_2014_a6_solution n β₯ 1 β§ kex (putnam_2014_a6_solution n)) β§ (β k β₯ 1, kex k β k β€ putnam_2014_a6_solution n) :=
sorry
| Let \( n \) be a positive integer. What is the largest \( k \) for which there exist \( n \times n \) matrices \( M_1, \ldots, M_k \) and \( N_1, \ldots, N_k \) with real entries such that for all \( i \) and \( j \), the matrix product \( M_i N_j \) has a zero entry somewhere on its diagonal if and only if \( i \neq j \)? | Show that the solution has the form k \<= n ^ n. | [
"linear_algebra"
] | null | null |
|
putnam_2012_b4 | 7cd76619-32c0-5d20-8810-f3dddfb69f0e | train | abbrev putnam_2012_b4_solution : Prop := sorry
-- True
/--
Suppose that $a_0 = 1$ and that $a_{n+1} = a_n + e^{-a_n}$ for $n=0,1,2,\dots$. Does $a_n - \log n$
have a finite limit as $n \to \infty$? (Here $\log n = \log_e n = \ln n$.)
-/
theorem putnam_2012_b4
(a : β β β)
(ha0 : a 0 = 1)
(han : β n : β, a (n + 1) = a n + exp (-a n))
: ((β L : β, Tendsto (fun n β¦ a n - Real.log n) atTop (π L)) β putnam_2012_b4_solution) :=
sorry | import Mathlib
open Matrix Function Real Topology Filter
-- True
/--
Suppose that $a_0 = 1$ and that $a_{n+1} = a_n + e^{-a_n}$ for $n=0,1,2,\dots$. Does $a_n - \log n$
have a finite limit as $n \to \infty$? (Here $\log n = \log_e n = \ln n$.)
-/
theorem putnam_2012_b4
(a : β β β)
(ha0 : a 0 = 1)
(han : β n : β, a (n + 1) = a n + exp (-a n))
: ((β L : β, Tendsto (fun n β¦ a n - Real.log n) atTop (π L)) β putnam_2012_b4_solution) := by
| import Mathlib
open Matrix Function Real Topology Filter
noncomputable abbrev putnam_2012_b4_solution : Prop := sorry
-- True
/--
Suppose that $a_0 = 1$ and that $a_{n+1} = a_n + e^{-a_n}$ for $n=0,1,2,\dots$. Does $a_n - \log n$
have a finite limit as $n \to \infty$? (Here $\log n = \log_e n = \ln n$.)
-/
theorem putnam_2012_b4
(a : β β β)
(ha0 : a 0 = 1)
(han : β n : β, a (n + 1) = a n + exp (-a n))
: ((β L : β, Tendsto (fun n β¦ a n - Real.log n) atTop (π L)) β putnam_2012_b4_solution) :=
sorry
| Suppose that $a_0 = 1$ and that $a_{n+1} = a_n + e^{-a_n}$ for $n=0,1,2,\dots$. Does $a_n - \log n$
have a finite limit as $n \to \infty$? (Here $\log n = \log_e n = \ln n$.) | Prove that the sequence has a finite limit. | [
"analysis"
] | null | null |
|
putnam_2018_a3 | 7edcecf8-44d9-5ec4-b517-79fe2288ffc9 | train | abbrev putnam_2018_a3_solution : β := sorry
-- 480/49
/--
Determine the greatest possible value of $\sum_{i=1}^{10} \cos(3x_i)$ for real numbers $x_1, x_2, \ldots, x_{10}$ satisfying $\sum_{i=1}^{10} \cos(x_i) = 0$.
-/
theorem putnam_2018_a3 :
IsGreatest
{β i, Real.cos (3 * x i) | (x : Fin 10 β β) (hx : β i, Real.cos (x i) = 0)}
putnam_2018_a3_solution :=
sorry | import Mathlib
-- 480/49
/--
Determine the greatest possible value of $\sum_{i=1}^{10} \cos(3x_i)$ for real numbers $x_1, x_2, \ldots, x_{10}$ satisfying $\sum_{i=1}^{10} \cos(x_i) = 0$.
-/
theorem putnam_2018_a3 :
IsGreatest
{β i, Real.cos (3 * x i) | (x : Fin 10 β β) (hx : β i, Real.cos (x i) = 0)}
putnam_2018_a3_solution := by
| import Mathlib
noncomputable abbrev putnam_2018_a3_solution : β := sorry
-- 480/49
/--
Determine the greatest possible value of $\sum_{i=1}^{10} \cos(3x_i)$ for real numbers $x_1, x_2, \ldots, x_{10}$ satisfying $\sum_{i=1}^{10} \cos(x_i) = 0$.
-/
theorem putnam_2018_a3 :
IsGreatest
{β i, Real.cos (3 * x i) | (x : Fin 10 β β) (hx : β i, Real.cos (x i) = 0)}
putnam_2018_a3_solution :=
sorry
| Determine the greatest possible value of $\sum_{i=1}^{10} \cos(3x_i)$ for real numbers $x_1, x_2, \ldots, x_{10}$ satisfying $\sum_{i=1}^{10} \cos(x_i) = 0$. | Show that the solution is $\frac{480}{49}$ | [
"number_theory"
] | null | null |
|
putnam_2008_a4 | be14528f-0340-54ed-996b-9a37c60a6cc6 | train | abbrev putnam_2008_a4_solution : Prop := sorry
-- False
/--
Define $f : \mathbb{R} \to \mathbb{R} by $f(x) = x$ if $x \leq e$ and $f(x) = x * f(\ln(x))$ if $x > e$. Does $\sum_{n=1}^{\infty} 1/(f(n))$ converge?
-/
theorem putnam_2008_a4
(f : β β β)
(hf : f = fun x => if x β€ Real.exp 1 then x else x * (f (Real.log x)))
: (β r : β, Tendsto (fun N : β => β n in Finset.range N, 1/(f (n + 1))) atTop (π r)) β putnam_2008_a4_solution :=
sorry | import Mathlib
open Filter Topology
-- False
/--
Define $f : \mathbb{R} \to \mathbb{R} by $f(x) = x$ if $x \leq e$ and $f(x) = x * f(\ln(x))$ if $x > e$. Does $\sum_{n=1}^{\infty} 1/(f(n))$ converge?
-/
theorem putnam_2008_a4
(f : β β β)
(hf : f = fun x => if x β€ Real.exp 1 then x else x * (f (Real.log x)))
: (β r : β, Tendsto (fun N : β => β n in Finset.range N, 1/(f (n + 1))) atTop (π r)) β putnam_2008_a4_solution := by
| import Mathlib
open Filter Topology
abbrev putnam_2008_a4_solution : Prop := sorry
-- False
/--
Define $f : \mathbb{R} \to \mathbb{R} by $f(x) = x$ if $x \leq e$ and $f(x) = x * f(\ln(x))$ if $x > e$. Does $\sum_{n=1}^{\infty} 1/(f(n))$ converge?
-/
theorem putnam_2008_a4
(f : β β β)
(hf : f = fun x => if x β€ Real.exp 1 then x else x * (f (Real.log x)))
: (β r : β, Tendsto (fun N : β => β n in Finset.range N, 1/(f (n + 1))) atTop (π r)) β putnam_2008_a4_solution :=
sorry
| Define $f : \mathbb{R} \to \mathbb{R} by $f(x) = x$ if $x \leq e$ and $f(x) = x * f(\ln(x))$ if $x > e$. Does $\sum_{n=1}^{\infty} 1/(f(n))$ converge? | Show that the sum does not converge. | [
"algebra"
] | null | null |
|
putnam_1977_b1 | eb93e639-9ac5-5029-9000-19f8ff0c0228 | train | abbrev putnam_1977_b1_solution : β := sorry
-- 2 / 3
/--
Find $\prod_{n=2}^{\infty} \frac{(n^3 - 1)}{(n^3 + 1)}$.
-/
theorem putnam_1977_b1
: Tendsto (fun N β¦ β n in Finset.Icc (2 : β€) N, ((n : β) ^ 3 - 1) / (n ^ 3 + 1)) atTop (π putnam_1977_b1_solution) :=
sorry | import Mathlib
open RingHom Set Nat Filter Topology
-- 2 / 3
/--
Find $\prod_{n=2}^{\infty} \frac{(n^3 - 1)}{(n^3 + 1)}$.
-/
theorem putnam_1977_b1
: Tendsto (fun N β¦ β n in Finset.Icc (2 : β€) N, ((n : β) ^ 3 - 1) / (n ^ 3 + 1)) atTop (π putnam_1977_b1_solution) := by
| import Mathlib
open RingHom Set Nat Filter Topology
noncomputable abbrev putnam_1977_b1_solution : β := sorry
-- 2 / 3
/--
Find $\prod_{n=2}^{\infty} \frac{(n^3 - 1)}{(n^3 + 1)}$.
-/
theorem putnam_1977_b1
: Tendsto (fun N β¦ β n in Finset.Icc (2 : β€) N, ((n : β) ^ 3 - 1) / (n ^ 3 + 1)) atTop (π putnam_1977_b1_solution) :=
sorry
| Find $\prod_{n=2}^{\infty} \frac{(n^3 - 1)}{(n^3 + 1)}$. | Prove that the product equals $\frac{2}{3}$. | [
"algebra",
"analysis"
] | null | null |
|
putnam_2001_b5 | fd950abd-7322-5bfc-a9af-00564631651c | train | theorem putnam_2001_b5
(a b : β)
(g : β β β)
(abint : 0 < a β§ a < 1 / 2 β§ 0 < b β§ b < 1 / 2)
(gcont : Continuous g)
(hg : β x : β, g (g x) = a * g x + b * x)
: β c : β, β x : β, g x = c * x :=
sorry | import Mathlib
open Topology Filter Polynomial Set
/--
Let $a$ and $b$ be real numbers in the interval $(0,1/2)$, and let $g$ be a continuous real-valued function such that $g(g(x))=ag(x)+bx$ for all real $x$. Prove that $g(x)=cx$ for some constant $c$.
-/
theorem putnam_2001_b5
(a b : β)
(g : β β β)
(abint : 0 < a β§ a < 1 / 2 β§ 0 < b β§ b < 1 / 2)
(gcont : Continuous g)
(hg : β x : β, g (g x) = a * g x + b * x)
: β c : β, β x : β, g x = c * x := by
| import Mathlib
open Topology Filter Polynomial Set
/--
Let $a$ and $b$ be real numbers in the interval $(0,1/2)$, and let $g$ be a continuous real-valued function such that $g(g(x))=ag(x)+bx$ for all real $x$. Prove that $g(x)=cx$ for some constant $c$.
-/
theorem putnam_2001_b5
(a b : β)
(g : β β β)
(abint : 0 < a β§ a < 1 / 2 β§ 0 < b β§ b < 1 / 2)
(gcont : Continuous g)
(hg : β x : β, g (g x) = a * g x + b * x)
: β c : β, β x : β, g x = c * x :=
sorry
| Let $a$ and $b$ be real numbers in the interval $(0,1/2)$, and let $g$ be a continuous real-valued function such that $g(g(x))=ag(x)+bx$ for all real $x$. Prove that $g(x)=cx$ for some constant $c$. | null | [
"analysis"
] | null | null |
|
putnam_1992_a4 | d120fea2-70e1-5cf0-a64c-0ecb4b80ff2e | train | abbrev putnam_1992_a4_solution : β β β := sorry
-- fun k β¦ ite (Even k) ((-1) ^ (k / 2) * factorial k) 0
/--
Let $f$ be an infinitely differentiable real-valued function defined on the real numbers. If
\[
f\left( \frac{1}{n} \right) = \frac{n^2}{n^2 + 1}, \qquad n = 1, 2, 3, \dots,
\]
compute the values of the derivatives $f^{(k)}(0), k = 1, 2, 3, \dots$.
-/
theorem putnam_1992_a4
(f : β β β)
(hfdiff : ContDiff β β€ f)
(hf : β n : β, n > 0 β f (1 / n) = n ^ 2 / (n ^ 2 + 1))
: (β k : β, k > 0 β iteratedDeriv k f 0 = putnam_1992_a4_solution k) :=
sorry | import Mathlib
open Topology Filter Nat Function
-- fun k β¦ ite (Even k) ((-1) ^ (k / 2) * factorial k) 0
/--
Let $f$ be an infinitely differentiable real-valued function defined on the real numbers. If
\[
f\left( \frac{1}{n} \right) = \frac{n^2}{n^2 + 1}, \qquad n = 1, 2, 3, \dots,
\]
compute the values of the derivatives $f^{(k)}(0), k = 1, 2, 3, \dots$.
-/
theorem putnam_1992_a4
(f : β β β)
(hfdiff : ContDiff β β€ f)
(hf : β n : β, n > 0 β f (1 / n) = n ^ 2 / (n ^ 2 + 1))
: (β k : β, k > 0 β iteratedDeriv k f 0 = putnam_1992_a4_solution k) := by
| import Mathlib
open Topology Filter Nat Function
abbrev putnam_1992_a4_solution : β β β := sorry
-- fun k β¦ ite (Even k) ((-1) ^ (k / 2) * factorial k) 0
/--
Let $f$ be an infinitely differentiable real-valued function defined on the real numbers. If
\[
f\left( \frac{1}{n} \right) = \frac{n^2}{n^2 + 1}, \qquad n = 1, 2, 3, \dots,
\]
compute the values of the derivatives $f^{(k)}(0), k = 1, 2, 3, \dots$.
-/
theorem putnam_1992_a4
(f : β β β)
(hfdiff : ContDiff β β€ f)
(hf : β n : β, n > 0 β f (1 / n) = n ^ 2 / (n ^ 2 + 1))
: (β k : β, k > 0 β iteratedDeriv k f 0 = putnam_1992_a4_solution k) :=
sorry
| Let $f$ be an infinitely differentiable real-valued function defined on the real numbers. If
\[
f\left( \frac{1}{n} \right) = \frac{n^2}{n^2 + 1}, \qquad n = 1, 2, 3, \dots,
\]
compute the values of the derivatives $f^{(k)}(0), k = 1, 2, 3, \dots$. | Prove that
\[
f^{(k)}(0) =
\begin{cases}
(-1)^{k/2}k! & \text{if $k$ is even;} \\
0 & \text{if $k$ is odd.} \\
\end{cases}
\] | [
"analysis"
] | null | null |
|
putnam_2008_a1 | 6d7df049-6437-5add-92c8-c3ce134af4fe | train | theorem putnam_2008_a1
(f : β β β β β)
(hf : β x y z : β, f x y + f y z + f z x = 0)
: β g : β β β, β x y : β, f x y = g x - g y :=
sorry | import Mathlib
/--
Let $f:\mathbb{R}^2 \to \mathbb{R}$ be a function such that $f(x,y)+f(y,z)+f(z,x)=0$ for all real numbers $x$, $y$, and $z$. Prove that there exists a function $g:\mathbb{R} \to \mathbb{R}$ such that $f(x,y)=g(x)-g(y)$ for all real numbers $x$ and $y$.
-/
theorem putnam_2008_a1
(f : β β β β β)
(hf : β x y z : β, f x y + f y z + f z x = 0)
: β g : β β β, β x y : β, f x y = g x - g y := by
| import Mathlib
/--
Let $f:\mathbb{R}^2 \to \mathbb{R}$ be a function such that $f(x,y)+f(y,z)+f(z,x)=0$ for all real numbers $x$, $y$, and $z$. Prove that there exists a function $g:\mathbb{R} \to \mathbb{R}$ such that $f(x,y)=g(x)-g(y)$ for all real numbers $x$ and $y$.
-/
theorem putnam_2008_a1
(f : β β β β β)
(hf : β x y z : β, f x y + f y z + f z x = 0)
: β g : β β β, β x y : β, f x y = g x - g y :=
sorry
| Let $f:\mathbb{R}^2 \to \mathbb{R}$ be a function such that $f(x,y)+f(y,z)+f(z,x)=0$ for all real numbers $x$, $y$, and $z$. Prove that there exists a function $g:\mathbb{R} \to \mathbb{R}$ such that $f(x,y)=g(x)-g(y)$ for all real numbers $x$ and $y$. | null | [
"algebra"
] | null | null |
|
putnam_2005_b3 | 4b5ad976-c1e1-5623-a4f4-0aee525e6514 | train | abbrev putnam_2005_b3_solution : Set (β β β) := sorry
-- {f : β β β | βα΅ (c > 0) (d > (0 : β)), (d = 1 β c = 1) β§ (Ioi 0).EqOn f (fun x β¦ c * x ^ d)}
/--
Find all differentiable functions $f:(0,\infty) \to (0,\infty)$ for which there is a positive real number $a$ such that $f'(\frac{a}{x})=\frac{x}{f(x)}$ for all $x>0$.
-/
theorem putnam_2005_b3
(f : β β β)
(hf : β x > 0, 0 < f x)
(hf' : DifferentiableOn β f (Ioi 0)) :
(β a > 0, β x > 0, deriv f (a / x) = x / f x) β f β putnam_2005_b3_solution :=
sorry | import Mathlib
open Nat Set
-- {f : β β β | βα΅ (c > 0) (d > (0 : β)), (d = 1 β c = 1) β§ (Ioi 0).EqOn f (fun x β¦ c * x ^ d)}
/--
Find all differentiable functions $f:(0,\infty) \to (0,\infty)$ for which there is a positive real number $a$ such that $f'(\frac{a}{x})=\frac{x}{f(x)}$ for all $x>0$.
-/
theorem putnam_2005_b3
(f : β β β)
(hf : β x > 0, 0 < f x)
(hf' : DifferentiableOn β f (Ioi 0)) :
(β a > 0, β x > 0, deriv f (a / x) = x / f x) β f β putnam_2005_b3_solution := by
| import Mathlib
open Nat Set
abbrev putnam_2005_b3_solution : Set (β β β) := sorry
-- {f : β β β | βα΅ (c > 0) (d > (0 : β)), (d = 1 β c = 1) β§ (Ioi 0).EqOn f (fun x β¦ c * x ^ d)}
/--
Find all differentiable functions $f:(0,\infty) \to (0,\infty)$ for which there is a positive real number $a$ such that $f'(\frac{a}{x})=\frac{x}{f(x)}$ for all $x>0$.
-/
theorem putnam_2005_b3
(f : β β β)
(hf : β x > 0, 0 < f x)
(hf' : DifferentiableOn β f (Ioi 0)) :
(β a > 0, β x > 0, deriv f (a / x) = x / f x) β f β putnam_2005_b3_solution :=
sorry
| Find all differentiable functions $f:(0,\infty) \to (0,\infty)$ for which there is a positive real number $a$ such that $f'(\frac{a}{x})=\frac{x}{f(x)}$ for all $x>0$. | Show that the functions are precisely $f(x)=cx^d$ for $c,d>0$ arbitrary except that we must take $c=1$ in case $d=1$. | [
"analysis"
] | null | null |
|
putnam_1969_a6 | 8e3b26fc-0a0f-519a-9608-8cc135ca43a3 | train | theorem putnam_1969_a6
(x : β β β)
(y : β β β)
(hy1 : β n β₯ 2, y n = x (n-1) + 2 * (x n))
(hy2 : β c : β, Tendsto y atTop (π c))
: β C : β, Tendsto x atTop (π C) :=
sorry | import Mathlib
open Matrix Filter Topology Set Nat
/--
Let $(x_n)$ be a sequence, and let $y_n = x_{n-1} + 2*x_n$ for $n \geq 2$. Suppose that $(y_n)$ converges, then prove that $(x_n)$ converges.
-/
theorem putnam_1969_a6
(x : β β β)
(y : β β β)
(hy1 : β n β₯ 2, y n = x (n-1) + 2 * (x n))
(hy2 : β c : β, Tendsto y atTop (π c))
: β C : β, Tendsto x atTop (π C) := by
| import Mathlib
open Matrix Filter Topology Set Nat
/--
Let $(x_n)$ be a sequence, and let $y_n = x_{n-1} + 2*x_n$ for $n \geq 2$. Suppose that $(y_n)$ converges, then prove that $(x_n)$ converges.
-/
theorem putnam_1969_a6
(x : β β β)
(y : β β β)
(hy1 : β n β₯ 2, y n = x (n-1) + 2 * (x n))
(hy2 : β c : β, Tendsto y atTop (π c))
: β C : β, Tendsto x atTop (π C) :=
sorry
| Let $(x_n)$ be a sequence, and let $y_n = x_{n-1} + 2*x_n$ for $n \geq 2$. Suppose that $(y_n)$ converges, then prove that $(x_n)$ converges. | null | [
"analysis"
] | null | null |
|
putnam_1982_a5 | f701eba0-626e-520f-bdcc-42a308b147a2 | train | theorem putnam_1982_a5
(a b c d : β€)
(hpos : a > 0 β§ b > 0 β§ c > 0 β§ d > 0)
(hac : a + c β€ 1982)
(hfrac : (a : β) / b + (c : β) / d < 1)
: (1 - (a : β) / b - (c : β) / d > 1 / 1983 ^ 3) :=
sorry | import Mathlib
/--
Let $a, b, c, d$ be positive integers satisfying $a + c \leq 1982$ and $\frac{a}{b} + \frac{c}{d} < 1$. Prove that $1 - \frac{a}{b} - \frac{c}{d} > \frac{1}{1983^3}$.
-/
theorem putnam_1982_a5
(a b c d : β€)
(hpos : a > 0 β§ b > 0 β§ c > 0 β§ d > 0)
(hac : a + c β€ 1982)
(hfrac : (a : β) / b + (c : β) / d < 1)
: (1 - (a : β) / b - (c : β) / d > 1 / 1983 ^ 3) := by
| import Mathlib
/--
Let $a, b, c, d$ be positive integers satisfying $a + c \leq 1982$ and $\frac{a}{b} + \frac{c}{d} < 1$. Prove that $1 - \frac{a}{b} - \frac{c}{d} > \frac{1}{1983^3}$.
-/
theorem putnam_1982_a5
(a b c d : β€)
(hpos : a > 0 β§ b > 0 β§ c > 0 β§ d > 0)
(hac : a + c β€ 1982)
(hfrac : (a : β) / b + (c : β) / d < 1)
: (1 - (a : β) / b - (c : β) / d > 1 / 1983 ^ 3) :=
sorry
| Let $a, b, c, d$ be positive integers satisfying $a + c \leq 1982$ and $\frac{a}{b} + \frac{c}{d} < 1$. Prove that $1 - \frac{a}{b} - \frac{c}{d} > \frac{1}{1983^3}$. | null | [
"algebra"
] | null | null |
|
putnam_1975_b6 | 4fb9d4d4-02de-51c4-b5ce-7eb3d501289f | train | theorem putnam_1975_b6
(s : β β β)
(hs : s = fun (n : β) => β i in Finset.Icc 1 n, 1/(i : β))
: (β n : β, n > 1 β n * (n+1 : β)^(1/(n : β)) < n + s n) β§ (β n : β, n > 2 β ((n : β) - 1)*((n : β)^(-1/(n-1 : β))) < n - s n) :=
sorry | import Mathlib
open Polynomial Real Complex Matrix Filter Topology Multiset
/--
Show that if $s_n = 1 + \frac{1}{2} + \frac{1}{3} + \dots + 1/n, then $n(n+1)^{1/n} < n + s_n$ whenever $n > 1$ and $(n-1)n^{-1/(n-1)} < n - s_n$ whenever $n > 2$.
-/
theorem putnam_1975_b6
(s : β β β)
(hs : s = fun (n : β) => β i in Finset.Icc 1 n, 1/(i : β))
: (β n : β, n > 1 β n * (n+1 : β)^(1/(n : β)) < n + s n) β§ (β n : β, n > 2 β ((n : β) - 1)*((n : β)^(-1/(n-1 : β))) < n - s n) := by
| import Mathlib
open Polynomial Real Complex Matrix Filter Topology Multiset
/--
Show that if $s_n = 1 + \frac{1}{2} + \frac{1}{3} + \dots + 1/n, then $n(n+1)^{1/n} < n + s_n$ whenever $n > 1$ and $(n-1)n^{-1/(n-1)} < n - s_n$ whenever $n > 2$.
-/
theorem putnam_1975_b6
(s : β β β)
(hs : s = fun (n : β) => β i in Finset.Icc 1 n, 1/(i : β))
: (β n : β, n > 1 β n * (n+1 : β)^(1/(n : β)) < n + s n) β§ (β n : β, n > 2 β ((n : β) - 1)*((n : β)^(-1/(n-1 : β))) < n - s n) :=
sorry
| Show that if $s_n = 1 + \frac{1}{2} + \frac{1}{3} + \dots + 1/n, then $n(n+1)^{1/n} < n + s_n$ whenever $n > 1$ and $(n-1)n^{-1/(n-1)} < n - s_n$ whenever $n > 2$. | null | [
"analysis"
] | null | null |
|
putnam_1974_a1 | ee7be205-62d1-56aa-b1b9-4e43faf1f31e | train | abbrev putnam_1974_a1_solution : β := sorry
-- 11
/--
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?
-/
theorem putnam_1974_a1
(conspiratorial : Set β€ β Prop)
(hconspiratorial : β S, conspiratorial S β β a β S, β b β S, β c β S, (a > 0 β§ b > 0 β§ c > 0) β§ ((a β b β§ b β c β§ a β c) β (Int.gcd a b > 1 β¨ Int.gcd b c > 1 β¨ Int.gcd a c > 1))) :
IsGreatest {k | β S, S β Icc 1 16 β§ conspiratorial S β§ S.encard = k} putnam_1974_a1_solution :=
sorry | import Mathlib
open Set
-- 11
/--
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?
-/
theorem putnam_1974_a1
(conspiratorial : Set β€ β Prop)
(hconspiratorial : β S, conspiratorial S β β a β S, β b β S, β c β S, (a > 0 β§ b > 0 β§ c > 0) β§ ((a β b β§ b β c β§ a β c) β (Int.gcd a b > 1 β¨ Int.gcd b c > 1 β¨ Int.gcd a c > 1))) :
IsGreatest {k | β S, S β Icc 1 16 β§ conspiratorial S β§ S.encard = k} putnam_1974_a1_solution := by
| import Mathlib
open Set
abbrev putnam_1974_a1_solution : β := sorry
-- 11
/--
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?
-/
theorem putnam_1974_a1
(conspiratorial : Set β€ β Prop)
(hconspiratorial : β S, conspiratorial S β β a β S, β b β S, β c β S, (a > 0 β§ b > 0 β§ c > 0) β§ ((a β b β§ b β c β§ a β c) β (Int.gcd a b > 1 β¨ Int.gcd b c > 1 β¨ Int.gcd a c > 1))) :
IsGreatest {k | β S, S β Icc 1 16 β§ conspiratorial S β§ S.encard = k} putnam_1974_a1_solution :=
sorry
| 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? | Show that the answer is 11. | [
"number_theory"
] | null | null |
|
putnam_1988_a6 | 191347d6-3e44-5e26-8377-1ef46aae7197 | train | abbrev putnam_1988_a6_solution : Prop := sorry
-- True
/--
If a linear transformation $A$ on an $n$-dimensional vector space has $n+1$ eigenvectors such that any $n$ of them are linearly independent, does it follow that $A$ is a scalar multiple of the identity? Prove your answer.
-/
theorem putnam_1988_a6
: (β (F V : Type*) (_ : Field F) (_ : AddCommGroup V) (_ : Module F V) (_ : FiniteDimensional F V) (n : β) (A : Module.End F V) (evecs : Set V), (n = FiniteDimensional.finrank F V β§ evecs β {v : V | β f : F, A.HasEigenvector f v} β§ evecs.encard = n + 1 β§ (β sevecs : Fin n β V, (Set.range sevecs β evecs β§ (Set.range sevecs).encard = n) β LinearIndependent F sevecs)) β (β c : F, A = c β’ LinearMap.id)) β putnam_1988_a6_solution :=
sorry | import Mathlib
open Set Filter Topology
-- True
/--
If a linear transformation $A$ on an $n$-dimensional vector space has $n+1$ eigenvectors such that any $n$ of them are linearly independent, does it follow that $A$ is a scalar multiple of the identity? Prove your answer.
-/
theorem putnam_1988_a6
: (β (F V : Type*) (_ : Field F) (_ : AddCommGroup V) (_ : Module F V) (_ : FiniteDimensional F V) (n : β) (A : Module.End F V) (evecs : Set V), (n = FiniteDimensional.finrank F V β§ evecs β {v : V | β f : F, A.HasEigenvector f v} β§ evecs.encard = n + 1 β§ (β sevecs : Fin n β V, (Set.range sevecs β evecs β§ (Set.range sevecs).encard = n) β LinearIndependent F sevecs)) β (β c : F, A = c β’ LinearMap.id)) β putnam_1988_a6_solution := by
| import Mathlib
open Set Filter Topology
abbrev putnam_1988_a6_solution : Prop := sorry
-- True
/--
If a linear transformation $A$ on an $n$-dimensional vector space has $n+1$ eigenvectors such that any $n$ of them are linearly independent, does it follow that $A$ is a scalar multiple of the identity? Prove your answer.
-/
theorem putnam_1988_a6
: (β (F V : Type*) (_ : Field F) (_ : AddCommGroup V) (_ : Module F V) (_ : FiniteDimensional F V) (n : β) (A : Module.End F V) (evecs : Set V), (n = FiniteDimensional.finrank F V β§ evecs β {v : V | β f : F, A.HasEigenvector f v} β§ evecs.encard = n + 1 β§ (β sevecs : Fin n β V, (Set.range sevecs β evecs β§ (Set.range sevecs).encard = n) β LinearIndependent F sevecs)) β (β c : F, A = c β’ LinearMap.id)) β putnam_1988_a6_solution :=
sorry
| If a linear transformation $A$ on an $n$-dimensional vector space has $n+1$ eigenvectors such that any $n$ of them are linearly independent, does it follow that $A$ is a scalar multiple of the identity? Prove your answer. | Show that the answer is yes, $A$ must be a scalar multiple of the identity. | [
"linear_algebra"
] | null | null |
|
putnam_1966_a1 | ff995b33-22fc-5efb-9320-b20dd5cc09ce | train | theorem putnam_1966_a1
(f : β€ β β€)
(hf : f = fun n : β€ => β m in Finset.Icc 0 n, (if Even m then m / 2 else (m - 1)/2))
: β x y : β€, x > 0 β§ y > 0 β§ x > y β x * y = f (x + y) - f (x - y) :=
sorry | import Mathlib
/--
Let $a_n$ denote the sequence $0, 1, 1, 2, 2, 3, \dots$, where $a_n = \frac{n}{2}$ if $n$ is even and $\frac{n - 1}{2}$ if n is odd. Furthermore, let $f(n)$ denote the sum of the first $n$ terms of $a_n$. Prove that all positive integers $x$ and $y$ with $x > y$ satisfy $xy = f(x + y) - f(x - y)$.
-/
theorem putnam_1966_a1
(f : β€ β β€)
(hf : f = fun n : β€ => β m in Finset.Icc 0 n, (if Even m then m / 2 else (m - 1)/2))
: β x y : β€, x > 0 β§ y > 0 β§ x > y β x * y = f (x + y) - f (x - y) := by
| import Mathlib
/--
Let $a_n$ denote the sequence $0, 1, 1, 2, 2, 3, \dots$, where $a_n = \frac{n}{2}$ if $n$ is even and $\frac{n - 1}{2}$ if n is odd. Furthermore, let $f(n)$ denote the sum of the first $n$ terms of $a_n$. Prove that all positive integers $x$ and $y$ with $x > y$ satisfy $xy = f(x + y) - f(x - y)$.
-/
theorem putnam_1966_a1
(f : β€ β β€)
(hf : f = fun n : β€ => β m in Finset.Icc 0 n, (if Even m then m / 2 else (m - 1)/2))
: β x y : β€, x > 0 β§ y > 0 β§ x > y β x * y = f (x + y) - f (x - y) :=
sorry
| Let $a_n$ denote the sequence $0, 1, 1, 2, 2, 3, \dots$, where $a_n = \frac{n}{2}$ if $n$ is even and $\frac{n - 1}{2}$ if n is odd. Furthermore, let $f(n)$ denote the sum of the first $n$ terms of $a_n$. Prove that all positive integers $x$ and $y$ with $x > y$ satisfy $xy = f(x + y) - f(x - y)$. | null | [
"algebra"
] | null | null |
|
putnam_1965_a1 | 2f004992-d4e7-5932-96fa-2bdb5fe7855d | train | abbrev putnam_1965_a1_solution : β := sorry
-- Real.pi / 15
/--
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 $\angle CAB$.
-/
theorem putnam_1965_a1
(A B C X Y : EuclideanSpace β (Fin 2))
(hABC : Β¬Collinear β {A, B, C})
(hangles : β C A B < β B C A β§ β B C A < Ο/2 β§ Ο/2 < β A B C)
(hX : Collinear β {X, B, C} β§ β X A B = (Ο - β C A B)/2 β§ dist A X = dist A B)
(hY : Collinear β {Y, C, A} β§ β Y B C = (Ο - β A B C)/2 β§ dist B Y = dist A B)
: β C A B = putnam_1965_a1_solution :=
sorry | import Mathlib
open EuclideanGeometry Real
-- Real.pi / 15
/--
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 $\angle CAB$.
-/
theorem putnam_1965_a1
(A B C X Y : EuclideanSpace β (Fin 2))
(hABC : Β¬Collinear β {A, B, C})
(hangles : β C A B < β B C A β§ β B C A < Ο/2 β§ Ο/2 < β A B C)
(hX : Collinear β {X, B, C} β§ β X A B = (Ο - β C A B)/2 β§ dist A X = dist A B)
(hY : Collinear β {Y, C, A} β§ β Y B C = (Ο - β A B C)/2 β§ dist B Y = dist A B)
: β C A B = putnam_1965_a1_solution := by
| import Mathlib
open EuclideanGeometry Real
noncomputable abbrev putnam_1965_a1_solution : β := sorry
-- Real.pi / 15
/--
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 $\angle CAB$.
-/
theorem putnam_1965_a1
(A B C X Y : EuclideanSpace β (Fin 2))
(hABC : Β¬Collinear β {A, B, C})
(hangles : β C A B < β B C A β§ β B C A < Ο/2 β§ Ο/2 < β A B C)
(hX : Collinear β {X, B, C} β§ β X A B = (Ο - β C A B)/2 β§ dist A X = dist A B)
(hY : Collinear β {Y, C, A} β§ β Y B C = (Ο - β A B C)/2 β§ dist B Y = dist A B)
: β C A B = putnam_1965_a1_solution :=
sorry
| 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 $\angle CAB$. | Show that the solution is $\angle CAB = \frac{\pi}{15}$. | [
"geometry"
] | null | null |
|
putnam_2019_b1 | d395469b-9a3a-5dcf-a1c6-c2e88094b924 | train | abbrev putnam_2019_b1_solution : β β β := sorry
-- (fun n : β => 5 * n + 1)
/--
Denote by $\mathbb{Z}^2$ the set of all points $(x,y)$ in the plane with integer coordinates. For each integer $n \geq 0$, let $P_n$ be the subset of $\mathbb{Z}^2$ consisting of the point $(0,0)$ together with all points $(x,y)$ such that $x^2+y^2=2^k$ for some integer $k \leq n$. Determine, as a function of $n$, the number of four-point subsets of $P_n$ whose elements are the vertices of a square.
-/
theorem putnam_2019_b1
(n : β)
(Pn : Set (Fin 2 β β€))
(pZtoR : (Fin 2 β β€) β EuclideanSpace β (Fin 2))
(sPnsquare : Finset (Fin 2 β β€) β Prop)
(hPn : Pn = {p | (p 0 = 0 β§ p 1 = 0) β¨ (β k β€ n, (p 0) ^ 2 + (p 1) ^ 2 = 2 ^ k)})
(hpZtoR : β p i, (pZtoR p) i = p i)
(sPnsquare_def : β sPn : Finset (Fin 2 β β€), sPnsquare sPn β (sPn.card = 4 β§ β p4 : Fin 4 β (Fin 2 β β€), Set.range p4 = sPn β§ (β s > 0, β i : Fin 4, dist (pZtoR (p4 i) : EuclideanSpace β (Fin 2)) (pZtoR (p4 (i + 1))) = s) β§ (dist (pZtoR (p4 0)) (pZtoR (p4 2)) = dist (pZtoR (p4 1)) (pZtoR (p4 3)))))
: {sPn : Finset (Fin 2 β β€) | (sPn : Set (Fin 2 β β€)) β Pn β§ sPnsquare sPn}.encard = putnam_2019_b1_solution n :=
sorry | import Mathlib
open Topology Filter
-- (fun n : β => 5 * n + 1)
/--
Denote by $\mathbb{Z}^2$ the set of all points $(x,y)$ in the plane with integer coordinates. For each integer $n \geq 0$, let $P_n$ be the subset of $\mathbb{Z}^2$ consisting of the point $(0,0)$ together with all points $(x,y)$ such that $x^2+y^2=2^k$ for some integer $k \leq n$. Determine, as a function of $n$, the number of four-point subsets of $P_n$ whose elements are the vertices of a square.
-/
theorem putnam_2019_b1
(n : β)
(Pn : Set (Fin 2 β β€))
(pZtoR : (Fin 2 β β€) β EuclideanSpace β (Fin 2))
(sPnsquare : Finset (Fin 2 β β€) β Prop)
(hPn : Pn = {p | (p 0 = 0 β§ p 1 = 0) β¨ (β k β€ n, (p 0) ^ 2 + (p 1) ^ 2 = 2 ^ k)})
(hpZtoR : β p i, (pZtoR p) i = p i)
(sPnsquare_def : β sPn : Finset (Fin 2 β β€), sPnsquare sPn β (sPn.card = 4 β§ β p4 : Fin 4 β (Fin 2 β β€), Set.range p4 = sPn β§ (β s > 0, β i : Fin 4, dist (pZtoR (p4 i) : EuclideanSpace β (Fin 2)) (pZtoR (p4 (i + 1))) = s) β§ (dist (pZtoR (p4 0)) (pZtoR (p4 2)) = dist (pZtoR (p4 1)) (pZtoR (p4 3)))))
: {sPn : Finset (Fin 2 β β€) | (sPn : Set (Fin 2 β β€)) β Pn β§ sPnsquare sPn}.encard = putnam_2019_b1_solution n := by
| import Mathlib
open Topology Filter
abbrev putnam_2019_b1_solution : β β β := sorry
-- (fun n : β => 5 * n + 1)
/--
Denote by $\mathbb{Z}^2$ the set of all points $(x,y)$ in the plane with integer coordinates. For each integer $n \geq 0$, let $P_n$ be the subset of $\mathbb{Z}^2$ consisting of the point $(0,0)$ together with all points $(x,y)$ such that $x^2+y^2=2^k$ for some integer $k \leq n$. Determine, as a function of $n$, the number of four-point subsets of $P_n$ whose elements are the vertices of a square.
-/
theorem putnam_2019_b1
(n : β)
(Pn : Set (Fin 2 β β€))
(pZtoR : (Fin 2 β β€) β EuclideanSpace β (Fin 2))
(sPnsquare : Finset (Fin 2 β β€) β Prop)
(hPn : Pn = {p | (p 0 = 0 β§ p 1 = 0) β¨ (β k β€ n, (p 0) ^ 2 + (p 1) ^ 2 = 2 ^ k)})
(hpZtoR : β p i, (pZtoR p) i = p i)
(sPnsquare_def : β sPn : Finset (Fin 2 β β€), sPnsquare sPn β (sPn.card = 4 β§ β p4 : Fin 4 β (Fin 2 β β€), Set.range p4 = sPn β§ (β s > 0, β i : Fin 4, dist (pZtoR (p4 i) : EuclideanSpace β (Fin 2)) (pZtoR (p4 (i + 1))) = s) β§ (dist (pZtoR (p4 0)) (pZtoR (p4 2)) = dist (pZtoR (p4 1)) (pZtoR (p4 3)))))
: {sPn : Finset (Fin 2 β β€) | (sPn : Set (Fin 2 β β€)) β Pn β§ sPnsquare sPn}.encard = putnam_2019_b1_solution n :=
sorry
| Denote by $\mathbb{Z}^2$ the set of all points $(x,y)$ in the plane with integer coordinates. For each integer $n \geq 0$, let $P_n$ be the subset of $\mathbb{Z}^2$ consisting of the point $(0,0)$ together with all points $(x,y)$ such that $x^2+y^2=2^k$ for some integer $k \leq n$. Determine, as a function of $n$, the number of four-point subsets of $P_n$ whose elements are the vertices of a square. | Show that the answer is $5n+1$. | [
"geometry"
] | null | null |
|
putnam_1966_b4 | fd2dacf0-d81a-5fd5-8b3c-3e603392301e | train | theorem putnam_1966_b4
(m n : β)
(S : Finset β)
(hS : (β i β S, i > 0) β§ S.card = m * n + 1)
: β T β S, (T.card = m + 1 β§ β j β T, β i β T, i β j β Β¬(j β£ i)) β¨ (T.card = n + 1 β§ β i β T, β j β T, j < i β j β£ i) :=
sorry | import Mathlib
open Topology Filter
/--
Let $a_1, a_2, ...$ be an increasing sequence of $mn + 1$ positive integers. Prove that there exists either a subset of $m + 1$ $a_i$ such that no element of the subset divides any other, or a subset of $n + 1$ $a_i$ such that each element of the subset (except the greatest) divides the next greatest element.
-/
theorem putnam_1966_b4
(m n : β)
(S : Finset β)
(hS : (β i β S, i > 0) β§ S.card = m * n + 1)
: β T β S, (T.card = m + 1 β§ β j β T, β i β T, i β j β Β¬(j β£ i)) β¨ (T.card = n + 1 β§ β i β T, β j β T, j < i β j β£ i) := by
| import Mathlib
open Topology Filter
/--
Let $a_1, a_2, ...$ be an increasing sequence of $mn + 1$ positive integers. Prove that there exists either a subset of $m + 1$ $a_i$ such that no element of the subset divides any other, or a subset of $n + 1$ $a_i$ such that each element of the subset (except the greatest) divides the next greatest element.
-/
theorem putnam_1966_b4
(m n : β)
(S : Finset β)
(hS : (β i β S, i > 0) β§ S.card = m * n + 1)
: β T β S, (T.card = m + 1 β§ β j β T, β i β T, i β j β Β¬(j β£ i)) β¨ (T.card = n + 1 β§ β i β T, β j β T, j < i β j β£ i) :=
sorry
| Let $a_1, a_2, ...$ be an increasing sequence of $mn + 1$ positive integers. Prove that there exists either a subset of $m + 1$ $a_i$ such that no element of the subset divides any other, or a subset of $n + 1$ $a_i$ such that each element of the subset (except the greatest) divides the next greatest element. | null | [
"number_theory",
"combinatorics"
] | null | null |
|
putnam_1964_b1 | 2f557f70-dfbb-50a9-a6cd-09d0a245ec49 | train | theorem putnam_1964_b1
(a b : β β β)
(h : β n, 0 < a n)
(h' : Summable fun n β¦ (1 : β) / a n)
(h'' : β n, b n = {k | a k β€ n}.ncard) :
Tendsto (fun n β¦ (b n : β) / n) atTop (π 0) :=
sorry | import Mathlib
open Set Function Filter Topology
/--
Let $a_n$ be a sequence of positive integers such that $\sum_{n=1}^{\infty} 1/a_n$ converges. For all $n$, let $b_n$ be the number of $a_n$ which are at most $n$. Prove that $\lim_{n \to \infty} b_n/n = 0$.
-/
theorem putnam_1964_b1
(a b : β β β)
(h : β n, 0 < a n)
(h' : Summable fun n β¦ (1 : β) / a n)
(h'' : β n, b n = {k | a k β€ n}.ncard) :
Tendsto (fun n β¦ (b n : β) / n) atTop (π 0) := by
| import Mathlib
open Set Function Filter Topology
/--
Let $a_n$ be a sequence of positive integers such that $\sum_{n=1}^{\infty} 1/a_n$ converges. For all $n$, let $b_n$ be the number of $a_n$ which are at most $n$. Prove that $\lim_{n \to \infty} b_n/n = 0$.
-/
theorem putnam_1964_b1
(a b : β β β)
(h : β n, 0 < a n)
(h' : Summable fun n β¦ (1 : β) / a n)
(h'' : β n, b n = {k | a k β€ n}.ncard) :
Tendsto (fun n β¦ (b n : β) / n) atTop (π 0) :=
sorry
| Let $a_n$ be a sequence of positive integers such that $\sum_{n=1}^{\infty} 1/a_n$ converges. For all $n$, let $b_n$ be the number of $a_n$ which are at most $n$. Prove that $\lim_{n \to \infty} b_n/n = 0$. | null | [
"analysis"
] | null | null |
|
putnam_2014_b3 | c8b4e812-cf58-5a95-8466-a23392f5b365 | train | theorem putnam_2014_b3
(m n : β)
(A : Matrix (Fin m) (Fin n) β)
(mnpos : 0 < m β§ 0 < n)
(Aprime : {p : β | p.Prime β§ β (i : Fin m) (j : Fin n), |A i j| = p}.encard β₯ m + n)
: A.rank β₯ 2 :=
sorry | import Mathlib
open Topology Filter Nat
/--
Let $A$ be an $m \times n$ matrix with rational entries. Suppose that there are at least $m+n$ distinct prime numbers among the absolute values of the entries of $A$. Show that the rank of $A$ is at least 2.
-/
theorem putnam_2014_b3
(m n : β)
(A : Matrix (Fin m) (Fin n) β)
(mnpos : 0 < m β§ 0 < n)
(Aprime : {p : β | p.Prime β§ β (i : Fin m) (j : Fin n), |A i j| = p}.encard β₯ m + n)
: A.rank β₯ 2 := by
| import Mathlib
open Topology Filter Nat
/--
Let $A$ be an $m \times n$ matrix with rational entries. Suppose that there are at least $m+n$ distinct prime numbers among the absolute values of the entries of $A$. Show that the rank of $A$ is at least 2.
-/
theorem putnam_2014_b3
(m n : β)
(A : Matrix (Fin m) (Fin n) β)
(mnpos : 0 < m β§ 0 < n)
(Aprime : {p : β | p.Prime β§ β (i : Fin m) (j : Fin n), |A i j| = p}.encard β₯ m + n)
: A.rank β₯ 2 :=
sorry
| Let $A$ be an $m \times n$ matrix with rational entries. Suppose that there are at least $m+n$ distinct prime numbers among the absolute values of the entries of $A$. Show that the rank of $A$ is at least 2. | null | [
"linear_algebra",
"number_theory"
] | null | null |
|
putnam_1990_b5 | 014c9d3b-2f71-5b66-afd8-0077233c0f07 | train | abbrev putnam_1990_b5_solution : Prop := sorry
-- True
/--
Is there an infinite sequence $a_0,a_1,a_2,\dots$ of nonzero real numbers such that for $n=1,2,3,\dots$ the polynomial $p_n(x)=a_0+a_1x+a_2x^2+\cdots+a_nx^n$ has exactly $n$ distinct real roots?
-/
theorem putnam_1990_b5 :
(β a : β β β, (β i, a i β 0) β§
(β n β₯ 1, (β i in Finset.Iic n, a i β’ X ^ i : Polynomial β).roots.toFinset.card = n)) β
putnam_1990_b5_solution :=
sorry | import Mathlib
open Filter Polynomial Topology Nat
-- True
/--
Is there an infinite sequence $a_0,a_1,a_2,\dots$ of nonzero real numbers such that for $n=1,2,3,\dots$ the polynomial $p_n(x)=a_0+a_1x+a_2x^2+\cdots+a_nx^n$ has exactly $n$ distinct real roots?
-/
theorem putnam_1990_b5 :
(β a : β β β, (β i, a i β 0) β§
(β n β₯ 1, (β i in Finset.Iic n, a i β’ X ^ i : Polynomial β).roots.toFinset.card = n)) β
putnam_1990_b5_solution := by
| import Mathlib
open Filter Polynomial Topology Nat
abbrev putnam_1990_b5_solution : Prop := sorry
-- True
/--
Is there an infinite sequence $a_0,a_1,a_2,\dots$ of nonzero real numbers such that for $n=1,2,3,\dots$ the polynomial $p_n(x)=a_0+a_1x+a_2x^2+\cdots+a_nx^n$ has exactly $n$ distinct real roots?
-/
theorem putnam_1990_b5 :
(β a : β β β, (β i, a i β 0) β§
(β n β₯ 1, (β i in Finset.Iic n, a i β’ X ^ i : Polynomial β).roots.toFinset.card = n)) β
putnam_1990_b5_solution :=
sorry
| Is there an infinite sequence $a_0,a_1,a_2,\dots$ of nonzero real numbers such that for $n=1,2,3,\dots$ the polynomial $p_n(x)=a_0+a_1x+a_2x^2+\cdots+a_nx^n$ has exactly $n$ distinct real roots? | Show that the answer is yes, such an infinite sequence exists. | [
"algebra",
"analysis"
] | null | null |
|
putnam_1999_b4 | 7042e16d-925b-53a2-b877-1020e61be3ea | train | theorem putnam_1999_b4
(f : β β β)
(hf : ContDiff β 3 f)
(hpos: β n β€ 3, β x : β, iteratedDeriv n f x > 0)
(hle : β x : β, iteratedDeriv 3 f x β€ f x)
: β x : β, deriv f x < 2 * (f x) :=
sorry | import Mathlib
open Filter Topology Metric
/--
Let $f$ be a real function with a continuous third derivative such that $f(x), f'(x), f''(x), f'''(x)$ are positive for all $x$. Suppose that $f'''(x)\leq f(x)$ for all $x$. Show that $f'(x)<2f(x)$ for all $x$.
-/
theorem putnam_1999_b4
(f : β β β)
(hf : ContDiff β 3 f)
(hpos: β n β€ 3, β x : β, iteratedDeriv n f x > 0)
(hle : β x : β, iteratedDeriv 3 f x β€ f x)
: β x : β, deriv f x < 2 * (f x) := by
| import Mathlib
open Filter Topology Metric
/--
Let $f$ be a real function with a continuous third derivative such that $f(x), f'(x), f''(x), f'''(x)$ are positive for all $x$. Suppose that $f'''(x)\leq f(x)$ for all $x$. Show that $f'(x)<2f(x)$ for all $x$.
-/
theorem putnam_1999_b4
(f : β β β)
(hf : ContDiff β 3 f)
(hpos: β n β€ 3, β x : β, iteratedDeriv n f x > 0)
(hle : β x : β, iteratedDeriv 3 f x β€ f x)
: β x : β, deriv f x < 2 * (f x) :=
sorry
| Let $f$ be a real function with a continuous third derivative such that $f(x), f'(x), f''(x), f'''(x)$ are positive for all $x$. Suppose that $f'''(x)\leq f(x)$ for all $x$. Show that $f'(x)<2f(x)$ for all $x$. | null | [
"analysis"
] | null | null |
|
putnam_1999_a3 | c14b2304-e1c8-5e6c-8cc8-3b77cb0e221d | train | theorem putnam_1999_a3
(f : β β β)
(hf : f = fun x β¦ 1 / (1 - 2 * x - x ^ 2))
(a : β β β)
(hf' : βαΆ x in π 0, Tendsto (fun N : β β¦ β n in Finset.range N, (a n) * x ^ n) atTop (π (f x)))
(n : β) :
β m : β, (a n) ^ 2 + (a (n + 1)) ^ 2 = a m :=
sorry | import Mathlib
open Filter Topology Metric
/--
Consider the power series expansion \[\frac{1}{1-2x-x^2} = \sum_{n=0}^\infty a_n x^n.\] Prove that, for each integer $n\geq 0$, there is an integer $m$ such that \[a_n^2 + a_{n+1}^2 = a_m .\]
-/
theorem putnam_1999_a3
(f : β β β)
(hf : f = fun x β¦ 1 / (1 - 2 * x - x ^ 2))
(a : β β β)
(hf' : βαΆ x in π 0, Tendsto (fun N : β β¦ β n in Finset.range N, (a n) * x ^ n) atTop (π (f x)))
(n : β) :
β m : β, (a n) ^ 2 + (a (n + 1)) ^ 2 = a m := by
| import Mathlib
open Filter Topology Metric
/--
Consider the power series expansion \[\frac{1}{1-2x-x^2} = \sum_{n=0}^\infty a_n x^n.\] Prove that, for each integer $n\geq 0$, there is an integer $m$ such that \[a_n^2 + a_{n+1}^2 = a_m .\]
-/
theorem putnam_1999_a3
(f : β β β)
(hf : f = fun x β¦ 1 / (1 - 2 * x - x ^ 2))
(a : β β β)
(hf' : βαΆ x in π 0, Tendsto (fun N : β β¦ β n in Finset.range N, (a n) * x ^ n) atTop (π (f x)))
(n : β) :
β m : β, (a n) ^ 2 + (a (n + 1)) ^ 2 = a m :=
sorry
| Consider the power series expansion \[\frac{1}{1-2x-x^2} = \sum_{n=0}^\infty a_n x^n.\] Prove that, for each integer $n\geq 0$, there is an integer $m$ such that \[a_n^2 + a_{n+1}^2 = a_m .\] | null | [
"algebra"
] | null | null |
|
putnam_2012_a3 | b9a62d17-524f-5c75-9bbd-3c7dc1723b2a | train | abbrev putnam_2012_a3_solution : β β β := sorry
-- fun x : β => Real.sqrt (1 - x^2)
/--
Let $f: [-1, 1] \to \mathbb{R}$ be a continuous function such that
\begin{itemize}
\item[(i)]
$f(x) = \frac{2-x^2}{2} f \left( \frac{x^2}{2-x^2} \right)$ for every $x$ in $[-1, 1]$,
\item[(ii)]
$f(0) = 1$, and
\item[(iii)]
$\lim_{x \to 1^-} \frac{f(x)}{\sqrt{1-x}}$ exists and is finite.
\end{itemize}
Prove that $f$ is unique, and express $f(x)$ in closed form.
-/
theorem putnam_2012_a3
(S : Set β)
(hS : S = Set.Icc (-1 : β) 1)
(fsat : (β β β) β Prop)
(hfsat : fsat = fun f : β β β => ContinuousOn f S β§
(β x β S, f x = ((2 - x^2)/2)*f (x^2/(2 - x^2))) β§ f 0 = 1 β§
(β y : β, leftLim (fun x : β => (f x)/Real.sqrt (1 - x)) 1 = y))
: fsat putnam_2012_a3_solution β§ β f : β β β, fsat f β β x β S, f x = putnam_2012_a3_solution x :=
sorry | import Mathlib
open Matrix Function
-- Note: uses (β β β) instead of (Set.Icc (-1 : β) 1 β β)
-- fun x : β => Real.sqrt (1 - x^2)
/--
Let $f: [-1, 1] \to \mathbb{R}$ be a continuous function such that
\begin{itemize}
\item[(i)]
$f(x) = \frac{2-x^2}{2} f \left( \frac{x^2}{2-x^2} \right)$ for every $x$ in $[-1, 1]$,
\item[(ii)]
$f(0) = 1$, and
\item[(iii)]
$\lim_{x \to 1^-} \frac{f(x)}{\sqrt{1-x}}$ exists and is finite.
\end{itemize}
Prove that $f$ is unique, and express $f(x)$ in closed form.
-/
theorem putnam_2012_a3
(S : Set β)
(hS : S = Set.Icc (-1 : β) 1)
(fsat : (β β β) β Prop)
(hfsat : fsat = fun f : β β β => ContinuousOn f S β§
(β x β S, f x = ((2 - x^2)/2)*f (x^2/(2 - x^2))) β§ f 0 = 1 β§
(β y : β, leftLim (fun x : β => (f x)/Real.sqrt (1 - x)) 1 = y))
: fsat putnam_2012_a3_solution β§ β f : β β β, fsat f β β x β S, f x = putnam_2012_a3_solution x := by
| import Mathlib
open Matrix Function
-- Note: uses (β β β) instead of (Set.Icc (-1 : β) 1 β β)
noncomputable abbrev putnam_2012_a3_solution : β β β := sorry
-- fun x : β => Real.sqrt (1 - x^2)
/--
Let $f: [-1, 1] \to \mathbb{R}$ be a continuous function such that
\begin{itemize}
\item[(i)]
$f(x) = \frac{2-x^2}{2} f \left( \frac{x^2}{2-x^2} \right)$ for every $x$ in $[-1, 1]$,
\item[(ii)]
$f(0) = 1$, and
\item[(iii)]
$\lim_{x \to 1^-} \frac{f(x)}{\sqrt{1-x}}$ exists and is finite.
\end{itemize}
Prove that $f$ is unique, and express $f(x)$ in closed form.
-/
theorem putnam_2012_a3
(S : Set β)
(hS : S = Set.Icc (-1 : β) 1)
(fsat : (β β β) β Prop)
(hfsat : fsat = fun f : β β β => ContinuousOn f S β§
(β x β S, f x = ((2 - x^2)/2)*f (x^2/(2 - x^2))) β§ f 0 = 1 β§
(β y : β, leftLim (fun x : β => (f x)/Real.sqrt (1 - x)) 1 = y))
: fsat putnam_2012_a3_solution β§ β f : β β β, fsat f β β x β S, f x = putnam_2012_a3_solution x :=
sorry
| Let $f: [-1, 1] \to \mathbb{R}$ be a continuous function such that
\begin{itemize}
\item[(i)]
$f(x) = \frac{2-x^2}{2} f \left( \frac{x^2}{2-x^2} \right)$ for every $x$ in $[-1, 1]$,
\item[(ii)]
$f(0) = 1$, and
\item[(iii)]
$\lim_{x \to 1^-} \frac{f(x)}{\sqrt{1-x}}$ exists and is finite.
\end{itemize}
Prove that $f$ is unique, and express $f(x)$ in closed form. | $f(x) = \sqrt{1-x^2}$ for all $x \in [-1,1]$. | [
"analysis",
"algebra"
] | null | null |
|
putnam_1989_a2 | 54c7515d-f54c-5087-8261-31a97bac6c9b | train | abbrev putnam_1989_a2_solution : β β β β β := sorry
-- (fun a b : β => (Real.exp (a ^ 2 * b ^ 2) - 1) / (a * b))
/--
Evaluate $\int_0^a \int_0^b e^{\max\{b^2x^2,a^2y^2\}}\,dy\,dx$ where $a$ and $b$ are positive.
-/
theorem putnam_1989_a2
(a b : β)
(abpos : a > 0 β§ b > 0)
: β« x in Set.Ioo 0 a, β« y in Set.Ioo 0 b, Real.exp (max (b ^ 2 * x ^ 2) (a ^ 2 * y ^ 2)) = putnam_1989_a2_solution a b :=
sorry | import Mathlib
-- (fun a b : β => (Real.exp (a ^ 2 * b ^ 2) - 1) / (a * b))
/--
Evaluate $\int_0^a \int_0^b e^{\max\{b^2x^2,a^2y^2\}}\,dy\,dx$ where $a$ and $b$ are positive.
-/
theorem putnam_1989_a2
(a b : β)
(abpos : a > 0 β§ b > 0)
: β« x in Set.Ioo 0 a, β« y in Set.Ioo 0 b, Real.exp (max (b ^ 2 * x ^ 2) (a ^ 2 * y ^ 2)) = putnam_1989_a2_solution a b := by
| import Mathlib
noncomputable abbrev putnam_1989_a2_solution : β β β β β := sorry
-- (fun a b : β => (Real.exp (a ^ 2 * b ^ 2) - 1) / (a * b))
/--
Evaluate $\int_0^a \int_0^b e^{\max\{b^2x^2,a^2y^2\}}\,dy\,dx$ where $a$ and $b$ are positive.
-/
theorem putnam_1989_a2
(a b : β)
(abpos : a > 0 β§ b > 0)
: β« x in Set.Ioo 0 a, β« y in Set.Ioo 0 b, Real.exp (max (b ^ 2 * x ^ 2) (a ^ 2 * y ^ 2)) = putnam_1989_a2_solution a b :=
sorry
| Evaluate $\int_0^a \int_0^b e^{\max\{b^2x^2,a^2y^2\}}\,dy\,dx$ where $a$ and $b$ are positive. | Show that the value of the integral is $(e^{a^2b^2}-1)/(ab)$. | [
"analysis"
] | null | null |
|
putnam_1978_a1 | 277e9810-c62b-5c72-b632-95c8753b05b9 | train | theorem putnam_1978_a1
(S T : Set β€)
(hS : S = {k | β j : β€, 0 β€ j β§ j β€ 33 β§ k = 3 * j + 1})
(hT : T β S β§ T.ncard = 20)
: (β m β T, β n β T, m β n β§ m + n = 104) :=
sorry | import Mathlib
/--
Let $S = \{1, 4, 7, 10, 13, 16, \dots , 100\}$. Let $T$ be a subset of $20$ elements of $S$. Show that we can find two distinct elements of $T$ with sum $104$.
-/
theorem putnam_1978_a1
(S T : Set β€)
(hS : S = {k | β j : β€, 0 β€ j β§ j β€ 33 β§ k = 3 * j + 1})
(hT : T β S β§ T.ncard = 20)
: (β m β T, β n β T, m β n β§ m + n = 104) := by
| import Mathlib
/--
Let $S = \{1, 4, 7, 10, 13, 16, \dots , 100\}$. Let $T$ be a subset of $20$ elements of $S$. Show that we can find two distinct elements of $T$ with sum $104$.
-/
theorem putnam_1978_a1
(S T : Set β€)
(hS : S = {k | β j : β€, 0 β€ j β§ j β€ 33 β§ k = 3 * j + 1})
(hT : T β S β§ T.ncard = 20)
: (β m β T, β n β T, m β n β§ m + n = 104) :=
sorry
| Let $S = \{1, 4, 7, 10, 13, 16, \dots , 100\}$. Let $T$ be a subset of $20$ elements of $S$. Show that we can find two distinct elements of $T$ with sum $104$. | null | [
"algebra"
] | null | null |
|
putnam_2003_b5 | da006359-d77d-561d-a75b-0aee46035c4a | train | theorem putnam_2003_b5
(A B C : EuclideanSpace β (Fin 2))
(hABC : dist 0 A = 1 β§ dist 0 B = 1 β§ dist 0 C = 1 β§ dist A B = dist A C β§ dist A B = dist B C)
: (β f : β β β, β P : EuclideanSpace β (Fin 2), dist 0 P < 1 β β X Y Z : EuclideanSpace β (Fin 2),
dist X Y = dist P A β§ dist Y Z = dist P B β§ dist X Z = dist P C β§
(MeasureTheory.volume (convexHull β {X, Y, Z})).toReal = f (dist 0 P)) :=
sorry | import Mathlib
open MvPolynomial Set Nat
/--
Let $A,B$, and $C$ be equidistant points on the circumference of a circle of unit radius centered at $O$, and let $P$ be any point in the circle's interior. Let $a, b, c$ be the distance from $P$ to $A, B, C$, respectively. Show that there is a triangle with side lengths $a, b, c$, and that the area of this triangle depends only on the distance from $P$ to $O$.
-/
theorem putnam_2003_b5
(A B C : EuclideanSpace β (Fin 2))
(hABC : dist 0 A = 1 β§ dist 0 B = 1 β§ dist 0 C = 1 β§ dist A B = dist A C β§ dist A B = dist B C)
: (β f : β β β, β P : EuclideanSpace β (Fin 2), dist 0 P < 1 β β X Y Z : EuclideanSpace β (Fin 2),
dist X Y = dist P A β§ dist Y Z = dist P B β§ dist X Z = dist P C β§
(MeasureTheory.volume (convexHull β {X, Y, Z})).toReal = f (dist 0 P)) := by
| import Mathlib
open MvPolynomial Set Nat
/--
Let $A,B$, and $C$ be equidistant points on the circumference of a circle of unit radius centered at $O$, and let $P$ be any point in the circle's interior. Let $a, b, c$ be the distance from $P$ to $A, B, C$, respectively. Show that there is a triangle with side lengths $a, b, c$, and that the area of this triangle depends only on the distance from $P$ to $O$.
-/
theorem putnam_2003_b5
(A B C : EuclideanSpace β (Fin 2))
(hABC : dist 0 A = 1 β§ dist 0 B = 1 β§ dist 0 C = 1 β§ dist A B = dist A C β§ dist A B = dist B C)
: (β f : β β β, β P : EuclideanSpace β (Fin 2), dist 0 P < 1 β β X Y Z : EuclideanSpace β (Fin 2),
dist X Y = dist P A β§ dist Y Z = dist P B β§ dist X Z = dist P C β§
(MeasureTheory.volume (convexHull β {X, Y, Z})).toReal = f (dist 0 P)) :=
sorry
| Let $A,B$, and $C$ be equidistant points on the circumference of a circle of unit radius centered at $O$, and let $P$ be any point in the circle's interior. Let $a, b, c$ be the distance from $P$ to $A, B, C$, respectively. Show that there is a triangle with side lengths $a, b, c$, and that the area of this triangle depends only on the distance from $P$ to $O$. | null | [
"geometry"
] | null | null |
|
putnam_1963_a3 | fa9b5963-bd9b-56f4-bd81-fa989896fc68 | train | abbrev putnam_1963_a3_solution : (β β β) β β β β β β β β := sorry
-- fun (f : β β β) (n : β) (x : β) (t : β) β¦ (x - t) ^ (n - 1) * (f t) / ((n - 1)! * t ^ n)
/--
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 $x\frac{d}{dx}$.
-/
theorem putnam_1963_a3
(P : β β (β β β) β (β β β))
(hP : P 0 = id β§ β i y, P (i + 1) y = P i (fun x β¦ x * deriv y x - i * y x))
(n : β)
(hn : 0 < n)
(f y : β β β)
(hf : ContinuousOn f (Ici 1))
(hy : ContDiffOn β n y (Ici 1)) :
(β i < n, deriv^[i] y 1 = 0) β§ (Ici 1).EqOn (P n y) f β
β x β₯ 1, y x = β« t in (1 : β)..x, putnam_1963_a3_solution f n x t :=
sorry | import Mathlib
open Nat Set Topology Filter
-- fun (f : β β β) (n : β) (x : β) (t : β) β¦ (x - t) ^ (n - 1) * (f t) / ((n - 1)! * t ^ n)
/--
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 $x\frac{d}{dx}$.
-/
theorem putnam_1963_a3
(P : β β (β β β) β (β β β))
(hP : P 0 = id β§ β i y, P (i + 1) y = P i (fun x β¦ x * deriv y x - i * y x))
(n : β)
(hn : 0 < n)
(f y : β β β)
(hf : ContinuousOn f (Ici 1))
(hy : ContDiffOn β n y (Ici 1)) :
(β i < n, deriv^[i] y 1 = 0) β§ (Ici 1).EqOn (P n y) f β
β x β₯ 1, y x = β« t in (1 : β)..x, putnam_1963_a3_solution f n x t := by
| import Mathlib
open Nat Set Topology Filter
noncomputable abbrev putnam_1963_a3_solution : (β β β) β β β β β β β β := sorry
-- fun (f : β β β) (n : β) (x : β) (t : β) β¦ (x - t) ^ (n - 1) * (f t) / ((n - 1)! * t ^ n)
/--
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 $x\frac{d}{dx}$.
-/
theorem putnam_1963_a3
(P : β β (β β β) β (β β β))
(hP : P 0 = id β§ β i y, P (i + 1) y = P i (fun x β¦ x * deriv y x - i * y x))
(n : β)
(hn : 0 < n)
(f y : β β β)
(hf : ContinuousOn f (Ici 1))
(hy : ContDiffOn β n y (Ici 1)) :
(β i < n, deriv^[i] y 1 = 0) β§ (Ici 1).EqOn (P n y) f β
β x β₯ 1, y x = β« t in (1 : β)..x, putnam_1963_a3_solution f n x t :=
sorry
| 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 $x\frac{d}{dx}$. | Show that the solution is $$y(x) = \int_{1}^{x} \frac{(x - t)^{n - 1} f(t)}{(n - 1)!t^n} dt$$. | [
"analysis"
] | null | null |
|
putnam_1973_a2 | 8e962ce2-3591-5987-857b-36090ee5c1eb | train | abbrev putnam_1973_a2_solution : Prop := sorry
-- True
/--
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?
-/
theorem putnam_1973_a2
(L : List β)
(hL : L.length = 8 β§ β i : Fin L.length, L[i] = 1 β¨ L[i] = -1)
(pluses : β)
(hpluses : pluses = {i : Fin L.length | L[i] = 1}.ncard)
(S : β β β)
(hS : S = fun n : β β¦ β i in Finset.Icc 1 n, L[i % 8]/i)
: (pluses = 4 β β l : β, Tendsto S atTop (π l)) β§ (putnam_1973_a2_solution β ((β l : β, Tendsto S atTop (π l)) β pluses = 4)) :=
sorry | import Mathlib
open Nat Set MeasureTheory Topology Filter
-- True
/--
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?
-/
theorem putnam_1973_a2
(L : List β)
(hL : L.length = 8 β§ β i : Fin L.length, L[i] = 1 β¨ L[i] = -1)
(pluses : β)
(hpluses : pluses = {i : Fin L.length | L[i] = 1}.ncard)
(S : β β β)
(hS : S = fun n : β β¦ β i in Finset.Icc 1 n, L[i % 8]/i)
: (pluses = 4 β β l : β, Tendsto S atTop (π l)) β§ (putnam_1973_a2_solution β ((β l : β, Tendsto S atTop (π l)) β pluses = 4)) := by
| import Mathlib
open Nat Set MeasureTheory Topology Filter
abbrev putnam_1973_a2_solution : Prop := sorry
-- True
/--
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?
-/
theorem putnam_1973_a2
(L : List β)
(hL : L.length = 8 β§ β i : Fin L.length, L[i] = 1 β¨ L[i] = -1)
(pluses : β)
(hpluses : pluses = {i : Fin L.length | L[i] = 1}.ncard)
(S : β β β)
(hS : S = fun n : β β¦ β i in Finset.Icc 1 n, L[i % 8]/i)
: (pluses = 4 β β l : β, Tendsto S atTop (π l)) β§ (putnam_1973_a2_solution β ((β l : β, Tendsto S atTop (π l)) β pluses = 4)) :=
sorry
| 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? | Show that the condition is necessary. | [
"analysis"
] | null | null |
|
putnam_2018_b2 | ee3302b3-48c7-506a-bfa2-e42b7023af87 | train | theorem putnam_2018_b2
(n : β)
(hn : n > 0)
(f : β β β β β)
(hf : β z : β, f n z = β i in Finset.range n, (n - i) * z^i)
: β z : β, βzβ β€ 1 β f n z β 0 :=
sorry | import Mathlib
/--
Let $n$ be a positive integer, and let $f_n(z) = n + (n-1)z + (n-2)z^2 + \cdots + z^{n-1}$. Prove that $f_n$ has no roots in the closed unit disk $\{z \in \mathbb{C}: |z| \leq 1\}$.
-/
theorem putnam_2018_b2
(n : β)
(hn : n > 0)
(f : β β β β β)
(hf : β z : β, f n z = β i in Finset.range n, (n - i) * z^i)
: β z : β, βzβ β€ 1 β f n z β 0 := by
| import Mathlib
/--
Let $n$ be a positive integer, and let $f_n(z) = n + (n-1)z + (n-2)z^2 + \cdots + z^{n-1}$. Prove that $f_n$ has no roots in the closed unit disk $\{z \in \mathbb{C}: |z| \leq 1\}$.
-/
theorem putnam_2018_b2
(n : β)
(hn : n > 0)
(f : β β β β β)
(hf : β z : β, f n z = β i in Finset.range n, (n - i) * z^i)
: β z : β, βzβ β€ 1 β f n z β 0 :=
sorry
| Let $n$ be a positive integer, and let $f_n(z) = n + (n-1)z + (n-2)z^2 + \cdots + z^{n-1}$. Prove that $f_n$ has no roots in the closed unit disk $\{z \in \mathbb{C}: |z| \leq 1\}$. | null | [
"analysis"
] | null | null |
|
putnam_1997_a4 | be1f113c-b3e6-5cc2-ac5a-10f3aadb79cd | train | theorem putnam_1997_a4
(G : Type*)
[Group G]
(Ο : G β G)
(hΟ : β g1 g2 g3 h1 h2 h3 : G, (g1 * g2 * g3 = 1 β§ h1 * h2 * h3 = 1) β Ο g1 * Ο g2 * Ο g3 = Ο h1 * Ο h2 * Ο h3)
: β a : G, let Ο := fun g => a * Ο g; β x y : G, Ο (x * y) = Ο x * Ο y :=
sorry | import Mathlib
open Filter Topology
/--
Let $G$ be a group with identity $e$ and $\phi:G\rightarrow G$ a function such that \[\phi(g_1)\phi(g_2)\phi(g_3)=\phi(h_1)\phi(h_2)\phi(h_3)\] whenever $g_1g_2g_3=e=h_1h_2h_3$. Prove that there exists an element $a\in G$ such that $\psi(x)=a\phi(x)$ is a homomorphism (i.e. $\psi(xy)=\psi(x)\psi(y)$ for all $x,y\in G$).
-/
theorem putnam_1997_a4
(G : Type*)
[Group G]
(Ο : G β G)
(hΟ : β g1 g2 g3 h1 h2 h3 : G, (g1 * g2 * g3 = 1 β§ h1 * h2 * h3 = 1) β Ο g1 * Ο g2 * Ο g3 = Ο h1 * Ο h2 * Ο h3)
: β a : G, let Ο := fun g => a * Ο g; β x y : G, Ο (x * y) = Ο x * Ο y := by
| import Mathlib
open Filter Topology
/--
Let $G$ be a group with identity $e$ and $\phi:G\rightarrow G$ a function such that \[\phi(g_1)\phi(g_2)\phi(g_3)=\phi(h_1)\phi(h_2)\phi(h_3)\] whenever $g_1g_2g_3=e=h_1h_2h_3$. Prove that there exists an element $a\in G$ such that $\psi(x)=a\phi(x)$ is a homomorphism (i.e. $\psi(xy)=\psi(x)\psi(y)$ for all $x,y\in G$).
-/
theorem putnam_1997_a4
(G : Type*)
[Group G]
(Ο : G β G)
(hΟ : β g1 g2 g3 h1 h2 h3 : G, (g1 * g2 * g3 = 1 β§ h1 * h2 * h3 = 1) β Ο g1 * Ο g2 * Ο g3 = Ο h1 * Ο h2 * Ο h3)
: β a : G, let Ο := fun g => a * Ο g; β x y : G, Ο (x * y) = Ο x * Ο y :=
sorry
| Let $G$ be a group with identity $e$ and $\phi:G\rightarrow G$ a function such that \[\phi(g_1)\phi(g_2)\phi(g_3)=\phi(h_1)\phi(h_2)\phi(h_3)\] whenever $g_1g_2g_3=e=h_1h_2h_3$. Prove that there exists an element $a\in G$ such that $\psi(x)=a\phi(x)$ is a homomorphism (i.e. $\psi(xy)=\psi(x)\psi(y)$ for all $x,y\in G$). | null | [
"abstract_algebra"
] | null | null |
|
putnam_1964_a6 | 1bbef2cc-00a3-51e1-b699-d6d765ae91e4 | train | theorem putnam_1964_a6
(S : Finset β)
(pairs : Set (β Γ β))
(hpairs : pairs = {(a, b) | (a β S) β§ (b β S) β§ (a < b)})
(distance : β Γ β β β)
(hdistance : distance = fun (a, b) β¦ b - a)
(hrepdist : β p β pairs, (β m β pairs, distance m > distance p) β β q β pairs, q β p β§ distance p = distance q)
: (β p q : pairs, q β p β β r : β, distance p / distance q = r) :=
sorry | import Mathlib
open Set Function Filter Topology
/--
Let $S$ be a finite set of collinear points. Let $k$ be the maximum distance between any two points of $S$. Given a pair of points of $S$ a distance $d < k$ apart, we can find another pair of points of $S$ also a distance $d$ apart. Prove that if two pairs of points of $S$ are distances $a$ and $b$ apart, then $\frac{a}{b}$ is rational.
-/
theorem putnam_1964_a6
(S : Finset β)
(pairs : Set (β Γ β))
(hpairs : pairs = {(a, b) | (a β S) β§ (b β S) β§ (a < b)})
(distance : β Γ β β β)
(hdistance : distance = fun (a, b) β¦ b - a)
(hrepdist : β p β pairs, (β m β pairs, distance m > distance p) β β q β pairs, q β p β§ distance p = distance q)
: (β p q : pairs, q β p β β r : β, distance p / distance q = r) := by
| import Mathlib
open Set Function Filter Topology
/--
Let $S$ be a finite set of collinear points. Let $k$ be the maximum distance between any two points of $S$. Given a pair of points of $S$ a distance $d < k$ apart, we can find another pair of points of $S$ also a distance $d$ apart. Prove that if two pairs of points of $S$ are distances $a$ and $b$ apart, then $\frac{a}{b}$ is rational.
-/
theorem putnam_1964_a6
(S : Finset β)
(pairs : Set (β Γ β))
(hpairs : pairs = {(a, b) | (a β S) β§ (b β S) β§ (a < b)})
(distance : β Γ β β β)
(hdistance : distance = fun (a, b) β¦ b - a)
(hrepdist : β p β pairs, (β m β pairs, distance m > distance p) β β q β pairs, q β p β§ distance p = distance q)
: (β p q : pairs, q β p β β r : β, distance p / distance q = r) :=
sorry
| Let $S$ be a finite set of collinear points. Let $k$ be the maximum distance between any two points of $S$. Given a pair of points of $S$ a distance $d < k$ apart, we can find another pair of points of $S$ also a distance $d$ apart. Prove that if two pairs of points of $S$ are distances $a$ and $b$ apart, then $\frac{a}{b}$ is rational. | null | [
"geometry"
] | null | null |
|
putnam_2012_a5 | 3cefb131-7239-5240-8ac7-f19da68252ec | train | abbrev putnam_2012_a5_solution : Set (β Γ β) := sorry
-- {q | let β¨n, _β© := q; n = 1} βͺ {(2,2)}
/--
Let $\FF_p$ denote the field of integers modulo a prime $p$, and let $n$ be a positive integer. Let $v$ be a fixed vector in $\FF_p^n$, let $M$ be an $n \times n$ matrix with entries of $\FF_p$, and define $G: \FF_p^n \to \FF_p^n$ by $G(x) = v + Mx$. Let $G^{(k)}$ denote the $k$-fold composition of $G$ with itself, that is, $G^{(1)}(x) = G(x)$ and $G^{(k+1)}(x) = G(G^{(k)}(x))$. Determine all pairs $p, n$ for which there exist $v$ and $M$ such that the $p^n$ vectors $G^{(k)}(0)$, $k=1,2,\dots,p^n$ are distinct.
-/
theorem putnam_2012_a5
(n p : β)
(hn : n > 0)
(hp : Nat.Prime p)
{F : Type*} [Field F] [Fintype F]
(hK : Fintype.card F = p)
(G : Matrix (Fin n) (Fin n) F β (Fin n β F) β (Fin n β F) β (Fin n β F))
(hG : β M v x, G M v x = v + mulVec M x) :
(n, p) β putnam_2012_a5_solution β
βα΅ (M : Matrix (Fin n) (Fin n) F) (v : (Fin n β F)),
Β¬(β i j : Finset.range (p^n), i β j β§ (G M v)^[i + 1] 0 = (G M v)^[j + 1] 0) :=
sorry | import Mathlib
open Matrix Function
-- {q | let β¨n, _β© := q; n = 1} βͺ {(2,2)}
/--
Let $\FF_p$ denote the field of integers modulo a prime $p$, and let $n$ be a positive integer. Let $v$ be a fixed vector in $\FF_p^n$, let $M$ be an $n \times n$ matrix with entries of $\FF_p$, and define $G: \FF_p^n \to \FF_p^n$ by $G(x) = v + Mx$. Let $G^{(k)}$ denote the $k$-fold composition of $G$ with itself, that is, $G^{(1)}(x) = G(x)$ and $G^{(k+1)}(x) = G(G^{(k)}(x))$. Determine all pairs $p, n$ for which there exist $v$ and $M$ such that the $p^n$ vectors $G^{(k)}(0)$, $k=1,2,\dots,p^n$ are distinct.
-/
theorem putnam_2012_a5
(n p : β)
(hn : n > 0)
(hp : Nat.Prime p)
{F : Type*} [Field F] [Fintype F]
(hK : Fintype.card F = p)
(G : Matrix (Fin n) (Fin n) F β (Fin n β F) β (Fin n β F) β (Fin n β F))
(hG : β M v x, G M v x = v + mulVec M x) :
(n, p) β putnam_2012_a5_solution β
βα΅ (M : Matrix (Fin n) (Fin n) F) (v : (Fin n β F)),
Β¬(β i j : Finset.range (p^n), i β j β§ (G M v)^[i + 1] 0 = (G M v)^[j + 1] 0) := by
| import Mathlib
open Matrix Function
abbrev putnam_2012_a5_solution : Set (β Γ β) := sorry
-- {q | let β¨n, _β© := q; n = 1} βͺ {(2,2)}
/--
Let $\FF_p$ denote the field of integers modulo a prime $p$, and let $n$ be a positive integer. Let $v$ be a fixed vector in $\FF_p^n$, let $M$ be an $n \times n$ matrix with entries of $\FF_p$, and define $G: \FF_p^n \to \FF_p^n$ by $G(x) = v + Mx$. Let $G^{(k)}$ denote the $k$-fold composition of $G$ with itself, that is, $G^{(1)}(x) = G(x)$ and $G^{(k+1)}(x) = G(G^{(k)}(x))$. Determine all pairs $p, n$ for which there exist $v$ and $M$ such that the $p^n$ vectors $G^{(k)}(0)$, $k=1,2,\dots,p^n$ are distinct.
-/
theorem putnam_2012_a5
(n p : β)
(hn : n > 0)
(hp : Nat.Prime p)
{F : Type*} [Field F] [Fintype F]
(hK : Fintype.card F = p)
(G : Matrix (Fin n) (Fin n) F β (Fin n β F) β (Fin n β F) β (Fin n β F))
(hG : β M v x, G M v x = v + mulVec M x) :
(n, p) β putnam_2012_a5_solution β
βα΅ (M : Matrix (Fin n) (Fin n) F) (v : (Fin n β F)),
Β¬(β i j : Finset.range (p^n), i β j β§ (G M v)^[i + 1] 0 = (G M v)^[j + 1] 0) :=
sorry
| Let $\FF_p$ denote the field of integers modulo a prime $p$, and let $n$ be a positive integer. Let $v$ be a fixed vector in $\FF_p^n$, let $M$ be an $n \times n$ matrix with entries of $\FF_p$, and define $G: \FF_p^n \to \FF_p^n$ by $G(x) = v + Mx$. Let $G^{(k)}$ denote the $k$-fold composition of $G$ with itself, that is, $G^{(1)}(x) = G(x)$ and $G^{(k+1)}(x) = G(G^{(k)}(x))$. Determine all pairs $p, n$ for which there exist $v$ and $M$ such that the $p^n$ vectors $G^{(k)}(0)$, $k=1,2,\dots,p^n$ are distinct. | Show that the solution is the pairs $(p,n)$ with $n = 1$ as well as the single pair $(2,2)$. | [
"linear_algebra"
] | null | null |
|
putnam_2005_a3 | f5dc6208-b40f-56d7-a34d-94bfc18c7894 | train | theorem putnam_2005_a3
(p : Polynomial β)
(n : β)
(hn : 0 < n)
(g : β β β)
(pdeg : p.degree = n)
(pzeros : β z : β, p.eval z = 0 β Complex.abs z = 1)
(hg : β z : β, g z = (p.eval z) / z ^ ((n : β) / 2))
(z : β)
(hz : z β 0 β§ deriv g z = 0) :
Complex.abs z = 1 :=
sorry | import Mathlib
open Nat Set
/--
Let $p(z)$ be a polynomial of degree $n$ all of whose zeros have absolute value $1$ in the complex plane. Put $g(z)=p(z)/z^{n/2}$. Show that all zeros of $g'(z)=0$ have absolute value $1$.
-/
theorem putnam_2005_a3
(p : Polynomial β)
(n : β)
(hn : 0 < n)
(g : β β β)
(pdeg : p.degree = n)
(pzeros : β z : β, p.eval z = 0 β Complex.abs z = 1)
(hg : β z : β, g z = (p.eval z) / z ^ ((n : β) / 2))
(z : β)
(hz : z β 0 β§ deriv g z = 0) :
Complex.abs z = 1 := by
| import Mathlib
open Nat Set
/--
Let $p(z)$ be a polynomial of degree $n$ all of whose zeros have absolute value $1$ in the complex plane. Put $g(z)=p(z)/z^{n/2}$. Show that all zeros of $g'(z)=0$ have absolute value $1$.
-/
theorem putnam_2005_a3
(p : Polynomial β)
(n : β)
(hn : 0 < n)
(g : β β β)
(pdeg : p.degree = n)
(pzeros : β z : β, p.eval z = 0 β Complex.abs z = 1)
(hg : β z : β, g z = (p.eval z) / z ^ ((n : β) / 2))
(z : β)
(hz : z β 0 β§ deriv g z = 0) :
Complex.abs z = 1 :=
sorry
| Let $p(z)$ be a polynomial of degree $n$ all of whose zeros have absolute value $1$ in the complex plane. Put $g(z)=p(z)/z^{n/2}$. Show that all zeros of $g'(z)=0$ have absolute value $1$. | null | [
"analysis",
"algebra"
] | null | null |
|
putnam_1987_b3 | 20fe6b6c-b847-5fa3-b3f7-325e89e44c3d | train | theorem putnam_1987_b3
(F : Type*) [Field F]
(hF : (1 : F) + 1 β 0)
: {(x, y) : F Γ F | x ^ 2 + y ^ 2 = 1} = {(1, 0)} βͺ {((r ^ 2 - 1) / (r ^ 2 + 1), (2 * r) / (r ^ 2 + 1)) | r β {r' : F | r' ^ 2 β -1}} :=
sorry | import Mathlib
open MvPolynomial Real Nat
/--
Let $F$ be a field in which $1+1 \neq 0$. Show that the set of solutions to the equation $x^2+y^2=1$ with $x$ and $y$ in $F$ is given by $(x,y)=(1,0)$ and $(x,y)=\left(\frac{r^2-1}{r^2+1},\frac{2r}{r^2+1}\right)$, where $r$ runs through the elements of $F$ such that $r^2 \neq -1$.
-/
theorem putnam_1987_b3
(F : Type*) [Field F]
(hF : (1 : F) + 1 β 0)
: {(x, y) : F Γ F | x ^ 2 + y ^ 2 = 1} = {(1, 0)} βͺ {((r ^ 2 - 1) / (r ^ 2 + 1), (2 * r) / (r ^ 2 + 1)) | r β {r' : F | r' ^ 2 β -1}} := by
| import Mathlib
open MvPolynomial Real Nat
/--
Let $F$ be a field in which $1+1 \neq 0$. Show that the set of solutions to the equation $x^2+y^2=1$ with $x$ and $y$ in $F$ is given by $(x,y)=(1,0)$ and $(x,y)=\left(\frac{r^2-1}{r^2+1},\frac{2r}{r^2+1}\right)$, where $r$ runs through the elements of $F$ such that $r^2 \neq -1$.
-/
theorem putnam_1987_b3
(F : Type*) [Field F]
(hF : (1 : F) + 1 β 0)
: {(x, y) : F Γ F | x ^ 2 + y ^ 2 = 1} = {(1, 0)} βͺ {((r ^ 2 - 1) / (r ^ 2 + 1), (2 * r) / (r ^ 2 + 1)) | r β {r' : F | r' ^ 2 β -1}} :=
sorry
| Let $F$ be a field in which $1+1 \neq 0$. Show that the set of solutions to the equation $x^2+y^2=1$ with $x$ and $y$ in $F$ is given by $(x,y)=(1,0)$ and $(x,y)=\left(\frac{r^2-1}{r^2+1},\frac{2r}{r^2+1}\right)$, where $r$ runs through the elements of $F$ such that $r^2 \neq -1$. | null | [
"abstract_algebra"
] | null | null |
|
putnam_2017_b1 | 7cffdc75-5fa5-53b4-8c4b-92df7a5579c5 | train | theorem putnam_2017_b1
(lines : Set (Set (Fin 2 β β)))
(L1 L2 : Set (Fin 2 β β))
(L1L2lines : L1 β lines β§ L2 β lines)
(L1L2distinct : L1 β L2)
(hlines : lines = {L | β v w : Fin 2 β β, w β 0 β§ L = {p | β t : β, p = v + t β’ w}}) :
L1 β© L2 β β
β (β lambda : β, lambda β 0 β
β P, (P β L1 β§ P β L2) β β A1 A2, A1 β L1 β§ A2 β L2 β§ (A2 - P = lambda β’ (A1 - P))) :=
sorry | import Mathlib
open Topology Filter
/--
Let $L_1$ and $L_2$ be distinct lines in the plane. Prove that $L_1$ and $L_2$ intersect if and only if, for every real number $\lambda \neq 0$ and every point $P$ not on $L_1$ or $L_2$, there exist points $A_1$ on $L_1$ and $A_2$ on $L_2$ such that $\overrightarrow{PA_2}=\lambda \overrightarrow{PA_1}$.
-/
theorem putnam_2017_b1
(lines : Set (Set (Fin 2 β β)))
(L1 L2 : Set (Fin 2 β β))
(L1L2lines : L1 β lines β§ L2 β lines)
(L1L2distinct : L1 β L2)
(hlines : lines = {L | β v w : Fin 2 β β, w β 0 β§ L = {p | β t : β, p = v + t β’ w}}) :
L1 β© L2 β β
β (β lambda : β, lambda β 0 β
β P, (P β L1 β§ P β L2) β β A1 A2, A1 β L1 β§ A2 β L2 β§ (A2 - P = lambda β’ (A1 - P))) := by
| import Mathlib
open Topology Filter
/--
Let $L_1$ and $L_2$ be distinct lines in the plane. Prove that $L_1$ and $L_2$ intersect if and only if, for every real number $\lambda \neq 0$ and every point $P$ not on $L_1$ or $L_2$, there exist points $A_1$ on $L_1$ and $A_2$ on $L_2$ such that $\overrightarrow{PA_2}=\lambda \overrightarrow{PA_1}$.
-/
theorem putnam_2017_b1
(lines : Set (Set (Fin 2 β β)))
(L1 L2 : Set (Fin 2 β β))
(L1L2lines : L1 β lines β§ L2 β lines)
(L1L2distinct : L1 β L2)
(hlines : lines = {L | β v w : Fin 2 β β, w β 0 β§ L = {p | β t : β, p = v + t β’ w}}) :
L1 β© L2 β β
β (β lambda : β, lambda β 0 β
β P, (P β L1 β§ P β L2) β β A1 A2, A1 β L1 β§ A2 β L2 β§ (A2 - P = lambda β’ (A1 - P))) :=
sorry
| Let $L_1$ and $L_2$ be distinct lines in the plane. Prove that $L_1$ and $L_2$ intersect if and only if, for every real number $\lambda \neq 0$ and every point $P$ not on $L_1$ or $L_2$, there exist points $A_1$ on $L_1$ and $A_2$ on $L_2$ such that $\overrightarrow{PA_2}=\lambda \overrightarrow{PA_1}$. | null | [
"geometry"
] | null | null |
|
putnam_2003_b3 | f3f0f6af-3bc2-5343-8ec2-307178319241 | train | theorem putnam_2003_b3 (n : β) :
n ! = β i in Finset.Icc 1 n, ((List.range βn / iββ).map succ).foldl Nat.lcm 1 :=
sorry | import Mathlib
open MvPolynomial Set Nat
/--
Show that for each positive integer $n$, $n!=\prod_{i=1}^n \text{lcm}\{1,2,\dots,\lfloor n/i \rfloor\}$. (Here lcm denotes the least common multiple, and $\lfloor x \rfloor$ denotes the greatest integer $\leq x$.)
-/
theorem putnam_2003_b3 (n : β) :
n ! = β i in Finset.Icc 1 n, ((List.range βn / iββ).map succ).foldl Nat.lcm 1 := by
| import Mathlib
open MvPolynomial Set Nat
/--
Show that for each positive integer $n$, $n!=\prod_{i=1}^n \text{lcm}\{1,2,\dots,\lfloor n/i \rfloor\}$. (Here lcm denotes the least common multiple, and $\lfloor x \rfloor$ denotes the greatest integer $\leq x$.)
-/
theorem putnam_2003_b3 (n : β) :
n ! = β i in Finset.Icc 1 n, ((List.range βn / iββ).map succ).foldl Nat.lcm 1 :=
sorry
| Show that for each positive integer $n$, $n!=\prod_{i=1}^n \text{lcm}\{1,2,\dots,\lfloor n/i \rfloor\}$. (Here lcm denotes the least common multiple, and $\lfloor x \rfloor$ denotes the greatest integer $\leq x$.) | null | [
"number_theory"
] | null | null |
|
putnam_2023_a4 | 4f23c0ee-5835-5c4f-8638-60cf144e89d9 | train | theorem putnam_2023_a4
(v : Fin 12 β EuclideanSpace β (Fin 3))
(hv :
letI Ο : β := (1 + β5) / 2
letI e : (Fin 3 β β) β EuclideanSpace β (Fin 3) := (WithLp.equiv _ _).symm
letI s := β(1 + Ο ^ 2)
β g : EuclideanSpace β (Fin 3) ββα΅’[β] EuclideanSpace β (Fin 3),
g β v = sβ»ΒΉ β’ e β ![![1, Ο, 0], ![-1, Ο, 0], ![ 1, -Ο, 0], ![-1, -Ο, 0],
![Ο, 0, 1], ![ Ο, 0, -1], ![-Ο, 0, 1], ![-Ο, 0, -1],
![0, 1, Ο], ![ 0, -1, Ο], ![ 0, 1, -Ο], ![ 0, -1, -Ο]])
(w : EuclideanSpace β (Fin 3))
(Ξ΅ : β) (hΞ΅ : Ξ΅ > 0) :
β a : Fin 12 β β€, ββ i, a i β’ v i - wβ < Ξ΅ :=
sorry | import Mathlib
/--
Let $v_1, \ldots, v_{12}$ be unit vectors in $\mathbb{R}^3$ from the origin to the vertices of a regular icosahedron. Show that for every vector $v \in \mathbb{R}^3$ and every $\epsilon > 0$, there exist integers $a_1, \ldots, a_{12}$ such that $\|a_1v_1 + \cdots + a_{12}v_{12} - v\| < Ξ΅$.
-/
theorem putnam_2023_a4
(v : Fin 12 β EuclideanSpace β (Fin 3))
(hv :
letI Ο : β := (1 + β5) / 2
letI e : (Fin 3 β β) β EuclideanSpace β (Fin 3) := (WithLp.equiv _ _).symm
letI s := β(1 + Ο ^ 2)
β g : EuclideanSpace β (Fin 3) ββα΅’[β] EuclideanSpace β (Fin 3),
g β v = sβ»ΒΉ β’ e β ![![1, Ο, 0], ![-1, Ο, 0], ![ 1, -Ο, 0], ![-1, -Ο, 0],
![Ο, 0, 1], ![ Ο, 0, -1], ![-Ο, 0, 1], ![-Ο, 0, -1],
![0, 1, Ο], ![ 0, -1, Ο], ![ 0, 1, -Ο], ![ 0, -1, -Ο]])
(w : EuclideanSpace β (Fin 3))
(Ξ΅ : β) (hΞ΅ : Ξ΅ > 0) :
β a : Fin 12 β β€, ββ i, a i β’ v i - wβ < Ξ΅ := by
| import Mathlib
/--
Let $v_1, \ldots, v_{12}$ be unit vectors in $\mathbb{R}^3$ from the origin to the vertices of a regular icosahedron. Show that for every vector $v \in \mathbb{R}^3$ and every $\epsilon > 0$, there exist integers $a_1, \ldots, a_{12}$ such that $\|a_1v_1 + \cdots + a_{12}v_{12} - v\| < Ξ΅$.
-/
theorem putnam_2023_a4
(v : Fin 12 β EuclideanSpace β (Fin 3))
(hv :
letI Ο : β := (1 + β5) / 2
letI e : (Fin 3 β β) β EuclideanSpace β (Fin 3) := (WithLp.equiv _ _).symm
letI s := β(1 + Ο ^ 2)
β g : EuclideanSpace β (Fin 3) ββα΅’[β] EuclideanSpace β (Fin 3),
g β v = sβ»ΒΉ β’ e β ![![1, Ο, 0], ![-1, Ο, 0], ![ 1, -Ο, 0], ![-1, -Ο, 0],
![Ο, 0, 1], ![ Ο, 0, -1], ![-Ο, 0, 1], ![-Ο, 0, -1],
![0, 1, Ο], ![ 0, -1, Ο], ![ 0, 1, -Ο], ![ 0, -1, -Ο]])
(w : EuclideanSpace β (Fin 3))
(Ξ΅ : β) (hΞ΅ : Ξ΅ > 0) :
β a : Fin 12 β β€, ββ i, a i β’ v i - wβ < Ξ΅ :=
sorry
| Let $v_1, \ldots, v_{12}$ be unit vectors in $\mathbb{R}^3$ from the origin to the vertices of a regular icosahedron. Show that for every vector $v \in \mathbb{R}^3$ and every $\epsilon > 0$, there exist integers $a_1, \ldots, a_{12}$ such that $\|a_1v_1 + \cdots + a_{12}v_{12} - v\| < Ξ΅$. | null | [
"geometry",
"number_theory"
] | null | null |
|
putnam_2010_b2 | a5c706cf-fb40-5017-8867-a89191b90482 | train | abbrev putnam_2010_b2_solution : β := sorry
-- 3
/--
Given that $A$, $B$, and $C$ are noncollinear points in the plane with integer coordinates such that the distances $AB$, $AC$, and $BC$ are integers, what is the smallest possible value of $AB$?
-/
theorem putnam_2010_b2
(ABCintcoords ABCintdists ABCall: EuclideanSpace β (Fin 2) β EuclideanSpace β (Fin 2) β EuclideanSpace β (Fin 2) β Prop)
(hABCintcoords : β A B C, ABCintcoords A B C β (β i : Fin 2, A i = round (A i) β§ B i = round (B i) β§ C i = round (C i)))
(hABCintdists : β A B C, ABCintdists A B C β (dist A B = round (dist A B) β§ dist A C = round (dist A C) β§ dist B C = round (dist B C)))
(hABCall : β A B C, ABCall A B C β (Β¬Collinear β {A, B, C} β§ ABCintcoords A B C β§ ABCintdists A B C)) :
IsLeast {y | β A B C, ABCall A B C β§ y = dist A B} putnam_2010_b2_solution :=
sorry | import Mathlib
open Filter Topology Set
-- 3
/--
Given that $A$, $B$, and $C$ are noncollinear points in the plane with integer coordinates such that the distances $AB$, $AC$, and $BC$ are integers, what is the smallest possible value of $AB$?
-/
theorem putnam_2010_b2
(ABCintcoords ABCintdists ABCall: EuclideanSpace β (Fin 2) β EuclideanSpace β (Fin 2) β EuclideanSpace β (Fin 2) β Prop)
(hABCintcoords : β A B C, ABCintcoords A B C β (β i : Fin 2, A i = round (A i) β§ B i = round (B i) β§ C i = round (C i)))
(hABCintdists : β A B C, ABCintdists A B C β (dist A B = round (dist A B) β§ dist A C = round (dist A C) β§ dist B C = round (dist B C)))
(hABCall : β A B C, ABCall A B C β (Β¬Collinear β {A, B, C} β§ ABCintcoords A B C β§ ABCintdists A B C)) :
IsLeast {y | β A B C, ABCall A B C β§ y = dist A B} putnam_2010_b2_solution := by
| import Mathlib
open Filter Topology Set
abbrev putnam_2010_b2_solution : β := sorry
-- 3
/--
Given that $A$, $B$, and $C$ are noncollinear points in the plane with integer coordinates such that the distances $AB$, $AC$, and $BC$ are integers, what is the smallest possible value of $AB$?
-/
theorem putnam_2010_b2
(ABCintcoords ABCintdists ABCall: EuclideanSpace β (Fin 2) β EuclideanSpace β (Fin 2) β EuclideanSpace β (Fin 2) β Prop)
(hABCintcoords : β A B C, ABCintcoords A B C β (β i : Fin 2, A i = round (A i) β§ B i = round (B i) β§ C i = round (C i)))
(hABCintdists : β A B C, ABCintdists A B C β (dist A B = round (dist A B) β§ dist A C = round (dist A C) β§ dist B C = round (dist B C)))
(hABCall : β A B C, ABCall A B C β (Β¬Collinear β {A, B, C} β§ ABCintcoords A B C β§ ABCintdists A B C)) :
IsLeast {y | β A B C, ABCall A B C β§ y = dist A B} putnam_2010_b2_solution :=
sorry
| Given that $A$, $B$, and $C$ are noncollinear points in the plane with integer coordinates such that the distances $AB$, $AC$, and $BC$ are integers, what is the smallest possible value of $AB$? | Show that the smallest distance is $3$. | [
"geometry"
] | null | null |
|
putnam_1980_a3 | b64554da-c53c-53c8-b28d-06f34d1b9ffe | train | abbrev putnam_1980_a3_solution : β := sorry
-- Real.pi / 4
/--
Evaluate $\int_0^{\pi/2}\frac{dx}{1+(\tan x)^{\sqrt{2}}}$.
-/
theorem putnam_1980_a3
: β« x in Set.Ioo 0 (Real.pi / 2), 1 / (1 + (Real.tan x) ^ (Real.sqrt 2)) = putnam_1980_a3_solution :=
sorry | import Mathlib
-- Real.pi / 4
/--
Evaluate $\int_0^{\pi/2}\frac{dx}{1+(\tan x)^{\sqrt{2}}}$.
-/
theorem putnam_1980_a3
: β« x in Set.Ioo 0 (Real.pi / 2), 1 / (1 + (Real.tan x) ^ (Real.sqrt 2)) = putnam_1980_a3_solution := by
| import Mathlib
noncomputable abbrev putnam_1980_a3_solution : β := sorry
-- Real.pi / 4
/--
Evaluate $\int_0^{\pi/2}\frac{dx}{1+(\tan x)^{\sqrt{2}}}$.
-/
theorem putnam_1980_a3
: β« x in Set.Ioo 0 (Real.pi / 2), 1 / (1 + (Real.tan x) ^ (Real.sqrt 2)) = putnam_1980_a3_solution :=
sorry
| Evaluate $\int_0^{\pi/2}\frac{dx}{1+(\tan x)^{\sqrt{2}}}$. | Show that the integral is $\pi/4$. | [
"analysis"
] | null | null |
|
putnam_1993_b5: | b330b48b-becb-5787-810d-f27c374e1398 | train | theorem putnam_1993_b5:
Β¬β p : Fin 4 β (EuclideanSpace β (Fin 2)),
β i j, i β j β
(β n : β€, dist (p i) (p j) = n β§ Odd n) :=
sorry | import Mathlib
/--
Show there do not exist four points in the Euclidean plane such that the pairwise distances between the points are all odd integers.
-/
theorem putnam_1993_b5:
Β¬β p : Fin 4 β (EuclideanSpace β (Fin 2)),
β i j, i β j β
(β n : β€, dist (p i) (p j) = n β§ Odd n) := by
| import Mathlib
/--
Show there do not exist four points in the Euclidean plane such that the pairwise distances between the points are all odd integers.
-/
theorem putnam_1993_b5:
Β¬β p : Fin 4 β (EuclideanSpace β (Fin 2)),
β i j, i β j β
(β n : β€, dist (p i) (p j) = n β§ Odd n) :=
sorry
| null | null | [] | null | null |
|
putnam_2015_a4 | 2479b41e-543b-537e-b3c2-adf085f47650 | train | abbrev putnam_2015_a4_solution : β := sorry
-- 4 / 7
/--
For each real number $x$, let
\[
f(x) = \sum_{n\in S_x} \frac{1}{2^n},
\]
where $S_x$ is the set of positive integers $n$ for which $\lfloor nx \rfloor$ is even. What is the largest real number $L$ such that $f(x) \geq L$ for all $x \in [0,1)$? (As usual, $\lfloor z \rfloor$ denotes the greatest integer less than or equal to $z$.)
-/
theorem putnam_2015_a4
(S : β β Set β€)
(f : β β β)
(p : β β Prop)
(hS : S = fun (x : β) β¦ {n : β€ | n > 0 β§ Even βn * xβ})
(hf : f = fun (x : β) β¦ β' n : S x, 1 / 2 ^ (n : β€))
(hp : β l, p l β β x β Set.Ico 0 1, f x β₯ l)
: IsGreatest p putnam_2015_a4_solution :=
sorry | import Mathlib
-- 4 / 7
/--
For each real number $x$, let
\[
f(x) = \sum_{n\in S_x} \frac{1}{2^n},
\]
where $S_x$ is the set of positive integers $n$ for which $\lfloor nx \rfloor$ is even. What is the largest real number $L$ such that $f(x) \geq L$ for all $x \in [0,1)$? (As usual, $\lfloor z \rfloor$ denotes the greatest integer less than or equal to $z$.)
-/
theorem putnam_2015_a4
(S : β β Set β€)
(f : β β β)
(p : β β Prop)
(hS : S = fun (x : β) β¦ {n : β€ | n > 0 β§ Even βn * xβ})
(hf : f = fun (x : β) β¦ β' n : S x, 1 / 2 ^ (n : β€))
(hp : β l, p l β β x β Set.Ico 0 1, f x β₯ l)
: IsGreatest p putnam_2015_a4_solution := by
| import Mathlib
noncomputable abbrev putnam_2015_a4_solution : β := sorry
-- 4 / 7
/--
For each real number $x$, let
\[
f(x) = \sum_{n\in S_x} \frac{1}{2^n},
\]
where $S_x$ is the set of positive integers $n$ for which $\lfloor nx \rfloor$ is even. What is the largest real number $L$ such that $f(x) \geq L$ for all $x \in [0,1)$? (As usual, $\lfloor z \rfloor$ denotes the greatest integer less than or equal to $z$.)
-/
theorem putnam_2015_a4
(S : β β Set β€)
(f : β β β)
(p : β β Prop)
(hS : S = fun (x : β) β¦ {n : β€ | n > 0 β§ Even βn * xβ})
(hf : f = fun (x : β) β¦ β' n : S x, 1 / 2 ^ (n : β€))
(hp : β l, p l β β x β Set.Ico 0 1, f x β₯ l)
: IsGreatest p putnam_2015_a4_solution :=
sorry
| For each real number $x$, let
\[
f(x) = \sum_{n\in S_x} \frac{1}{2^n},
\]
where $S_x$ is the set of positive integers $n$ for which $\lfloor nx \rfloor$ is even. What is the largest real number $L$ such that $f(x) \geq L$ for all $x \in [0,1)$? (As usual, $\lfloor z \rfloor$ denotes the greatest integer less than or equal to $z$.) | Prove that $L = \frac{4}{7}$. | [
"analysis"
] | null | null |
|
putnam_1975_b2 | 514faec7-30c2-5434-875f-be27db5b3db0 | train | theorem putnam_1975_b2
(slab : (Fin 3 β β) β β β β β Set (Fin 3 β β))
(hslab : slab = fun normal offset thickness => {x : Fin 3 β β | offset < normal β¬α΅₯ x β§ normal β¬α΅₯ x < offset + thickness})
(normals : β β (Fin 3 β β))
(offsets : β β β)
(thicknesses : β β β)
(hnormalsunit : β i : β, βnormals iβ = 1)
(hthicknessespos : β i : β, thicknesses i > 0)
(hthicknessesconv : β C : β, Tendsto (fun i : β => β j in Finset.range i, thicknesses j) atTop (π C))
: Set.univ β β i : β, slab (normals i) (offsets i) (thicknesses i) :=
sorry | import Mathlib
open Polynomial Real Complex Matrix Filter Topology
/--
In three-dimensional Euclidean space, define a \emph{slab} to be the open set of points lying between two parallel planes. The distance between the planes is called the \emph{thickness} of the slab. Given an infinite sequence $S_1, S_2, \dots$ of slabs of thicknesses $d_1, d_2, \dots,$ respectively, such that $\Sigma_{i=1}^{\infty} d_i$ converges, prove that there is some point in the space which is not contained in any of the slabs.
-/
theorem putnam_1975_b2
(slab : (Fin 3 β β) β β β β β Set (Fin 3 β β))
(hslab : slab = fun normal offset thickness => {x : Fin 3 β β | offset < normal β¬α΅₯ x β§ normal β¬α΅₯ x < offset + thickness})
(normals : β β (Fin 3 β β))
(offsets : β β β)
(thicknesses : β β β)
(hnormalsunit : β i : β, βnormals iβ = 1)
(hthicknessespos : β i : β, thicknesses i > 0)
(hthicknessesconv : β C : β, Tendsto (fun i : β => β j in Finset.range i, thicknesses j) atTop (π C))
: Set.univ β β i : β, slab (normals i) (offsets i) (thicknesses i) := by
| import Mathlib
open Polynomial Real Complex Matrix Filter Topology
/--
In three-dimensional Euclidean space, define a \emph{slab} to be the open set of points lying between two parallel planes. The distance between the planes is called the \emph{thickness} of the slab. Given an infinite sequence $S_1, S_2, \dots$ of slabs of thicknesses $d_1, d_2, \dots,$ respectively, such that $\Sigma_{i=1}^{\infty} d_i$ converges, prove that there is some point in the space which is not contained in any of the slabs.
-/
theorem putnam_1975_b2
(slab : (Fin 3 β β) β β β β β Set (Fin 3 β β))
(hslab : slab = fun normal offset thickness => {x : Fin 3 β β | offset < normal β¬α΅₯ x β§ normal β¬α΅₯ x < offset + thickness})
(normals : β β (Fin 3 β β))
(offsets : β β β)
(thicknesses : β β β)
(hnormalsunit : β i : β, βnormals iβ = 1)
(hthicknessespos : β i : β, thicknesses i > 0)
(hthicknessesconv : β C : β, Tendsto (fun i : β => β j in Finset.range i, thicknesses j) atTop (π C))
: Set.univ β β i : β, slab (normals i) (offsets i) (thicknesses i) :=
sorry
| In three-dimensional Euclidean space, define a \emph{slab} to be the open set of points lying between two parallel planes. The distance between the planes is called the \emph{thickness} of the slab. Given an infinite sequence $S_1, S_2, \dots$ of slabs of thicknesses $d_1, d_2, \dots,$ respectively, such that $\Sigma_{i=1}^{\infty} d_i$ converges, prove that there is some point in the space which is not contained in any of the slabs. | null | [
"analysis",
"geometry"
] | null | null |
|
putnam_2021_a2 | a98b9c7c-d02f-5a66-b9b3-82e295fb9669 | train | abbrev putnam_2021_a2_solution : β := sorry
-- Real.exp 1
/--
For every positive real number $x$, let $g(x)=\lim_{r \to 0}((x+1)^{r+1}-x^{r+1})^\frac{1}{r}$. Find $\lim_{x \to \infty}\frac{g(x)}{x}$.
-/
theorem putnam_2021_a2
(g : β β β)
(hg : β x > 0, Tendsto (fun r : β => ((x + 1) ^ (r + 1) - x ^ (r + 1)) ^ (1 / r)) (π[>] 0) (π (g x)))
: Tendsto (fun x : β => g x / x) atTop (π putnam_2021_a2_solution) :=
sorry | import Mathlib
open Filter Topology
-- Real.exp 1
/--
For every positive real number $x$, let $g(x)=\lim_{r \to 0}((x+1)^{r+1}-x^{r+1})^\frac{1}{r}$. Find $\lim_{x \to \infty}\frac{g(x)}{x}$.
-/
theorem putnam_2021_a2
(g : β β β)
(hg : β x > 0, Tendsto (fun r : β => ((x + 1) ^ (r + 1) - x ^ (r + 1)) ^ (1 / r)) (π[>] 0) (π (g x)))
: Tendsto (fun x : β => g x / x) atTop (π putnam_2021_a2_solution) := by
| import Mathlib
open Filter Topology
abbrev putnam_2021_a2_solution : β := sorry
-- Real.exp 1
/--
For every positive real number $x$, let $g(x)=\lim_{r \to 0}((x+1)^{r+1}-x^{r+1})^\frac{1}{r}$. Find $\lim_{x \to \infty}\frac{g(x)}{x}$.
-/
theorem putnam_2021_a2
(g : β β β)
(hg : β x > 0, Tendsto (fun r : β => ((x + 1) ^ (r + 1) - x ^ (r + 1)) ^ (1 / r)) (π[>] 0) (π (g x)))
: Tendsto (fun x : β => g x / x) atTop (π putnam_2021_a2_solution) :=
sorry
| For every positive real number $x$, let $g(x)=\lim_{r \to 0}((x+1)^{r+1}-x^{r+1})^\frac{1}{r}$. Find $\lim_{x \to \infty}\frac{g(x)}{x}$. | Show that the limit is $e$. | [
"analysis"
] | null | null |
|
putnam_2007_b3 | b84c58c1-8982-55f0-93fd-d66429a20d70 | train | abbrev putnam_2007_b3_solution : β := sorry
-- (2 ^ 2006 / Real.sqrt 5) * (((1 + Real.sqrt 5) / 2) ^ 3997 - ((1 + Real.sqrt 5) / 2) ^ (-3997 : β€))
/--
Let $x_0 = 1$ and for $n \geq 0$, let $x_{n+1} = 3x_n + \lfloor x_n \sqrt{5} \rfloor$. In particular, $x_1 = 5$, $x_2 = 26$, $x_3 = 136$, $x_4 = 712$. Find a closed-form expression for $x_{2007}$. ($\lfloor a \rfloor$ means the largest integer $\leq a$.)
-/
theorem putnam_2007_b3
(x : β β β)
(hx0 : x 0 = 1)
(hx : β n : β, x (n + 1) = 3 * (x n) + β(x n) * Real.sqrt 5β)
: (x 2007 = putnam_2007_b3_solution) :=
sorry | import Mathlib
open Set Nat Function
-- (2 ^ 2006 / Real.sqrt 5) * (((1 + Real.sqrt 5) / 2) ^ 3997 - ((1 + Real.sqrt 5) / 2) ^ (-3997 : β€))
/--
Let $x_0 = 1$ and for $n \geq 0$, let $x_{n+1} = 3x_n + \lfloor x_n \sqrt{5} \rfloor$. In particular, $x_1 = 5$, $x_2 = 26$, $x_3 = 136$, $x_4 = 712$. Find a closed-form expression for $x_{2007}$. ($\lfloor a \rfloor$ means the largest integer $\leq a$.)
-/
theorem putnam_2007_b3
(x : β β β)
(hx0 : x 0 = 1)
(hx : β n : β, x (n + 1) = 3 * (x n) + β(x n) * Real.sqrt 5β)
: (x 2007 = putnam_2007_b3_solution) := by
| import Mathlib
open Set Nat Function
noncomputable abbrev putnam_2007_b3_solution : β := sorry
-- (2 ^ 2006 / Real.sqrt 5) * (((1 + Real.sqrt 5) / 2) ^ 3997 - ((1 + Real.sqrt 5) / 2) ^ (-3997 : β€))
/--
Let $x_0 = 1$ and for $n \geq 0$, let $x_{n+1} = 3x_n + \lfloor x_n \sqrt{5} \rfloor$. In particular, $x_1 = 5$, $x_2 = 26$, $x_3 = 136$, $x_4 = 712$. Find a closed-form expression for $x_{2007}$. ($\lfloor a \rfloor$ means the largest integer $\leq a$.)
-/
theorem putnam_2007_b3
(x : β β β)
(hx0 : x 0 = 1)
(hx : β n : β, x (n + 1) = 3 * (x n) + β(x n) * Real.sqrt 5β)
: (x 2007 = putnam_2007_b3_solution) :=
sorry
| Let $x_0 = 1$ and for $n \geq 0$, let $x_{n+1} = 3x_n + \lfloor x_n \sqrt{5} \rfloor$. In particular, $x_1 = 5$, $x_2 = 26$, $x_3 = 136$, $x_4 = 712$. Find a closed-form expression for $x_{2007}$. ($\lfloor a \rfloor$ means the largest integer $\leq a$.) | Prove that $x_{2007} = \frac{2^{2006}}{\sqrt{5}}(\alpha^{3997}-\alpha^{-3997})$, where $\alpha = \frac{1+\sqrt{5}}{2}$. | [
"analysis"
] | null | null |
|
putnam_2015_a2 | 456f40dc-4d77-59d9-a484-5ddce1c391c3 | train | abbrev putnam_2015_a2_solution : β := sorry
-- 181
/--
Let $a_0=1$, $a_1=2$, and $a_n=4a_{n-1}-a_{n-2}$ for $n \geq 2$. Find an odd prime factor of $a_{2015}$.
-/
theorem putnam_2015_a2
(a : β β β€)
(abase : a 0 = 1 β§ a 1 = 2)
(arec : β n β₯ 2, a n = 4 * a (n - 1) - a (n - 2))
: Odd putnam_2015_a2_solution β§ putnam_2015_a2_solution.Prime β§ ((putnam_2015_a2_solution : β€) β£ a 2015) :=
sorry | import Mathlib
-- Note: this problem admits several possible correct solutions; this is the one shown on the solutions document
-- 181
/--
Let $a_0=1$, $a_1=2$, and $a_n=4a_{n-1}-a_{n-2}$ for $n \geq 2$. Find an odd prime factor of $a_{2015}$.
-/
theorem putnam_2015_a2
(a : β β β€)
(abase : a 0 = 1 β§ a 1 = 2)
(arec : β n β₯ 2, a n = 4 * a (n - 1) - a (n - 2))
: Odd putnam_2015_a2_solution β§ putnam_2015_a2_solution.Prime β§ ((putnam_2015_a2_solution : β€) β£ a 2015) := by
| import Mathlib
-- Note: this problem admits several possible correct solutions; this is the one shown on the solutions document
abbrev putnam_2015_a2_solution : β := sorry
-- 181
/--
Let $a_0=1$, $a_1=2$, and $a_n=4a_{n-1}-a_{n-2}$ for $n \geq 2$. Find an odd prime factor of $a_{2015}$.
-/
theorem putnam_2015_a2
(a : β β β€)
(abase : a 0 = 1 β§ a 1 = 2)
(arec : β n β₯ 2, a n = 4 * a (n - 1) - a (n - 2))
: Odd putnam_2015_a2_solution β§ putnam_2015_a2_solution.Prime β§ ((putnam_2015_a2_solution : β€) β£ a 2015) :=
sorry
| Let $a_0=1$, $a_1=2$, and $a_n=4a_{n-1}-a_{n-2}$ for $n \geq 2$. Find an odd prime factor of $a_{2015}$. | Show that one possible answer is $181$. | [
"number_theory"
] | null | null |
|
putnam_2020_a5 | 242c410b-bcb9-5f8c-9489-a901a613893e | train | abbrev putnam_2020_a5_solution : β€ := sorry
-- (Nat.fib 4040) - 1
/--
Let $a_n$ be the number of sets $S$ of positive integers for which
\[
\sum_{k \in S} F_k = n,
\]
where the Fibonacci sequence $(F_k)_{k \geq 1}$ satisfies $F_{k+2} = F_{k+1} + F_k$ and begins $F_1 = 1, F_2 = 1, F_3 = 2, F_4 = 3$. Find the largest integer $n$ such that $a_n = 2020$.
-/
theorem putnam_2020_a5
(a : β€ β β)
(ha : a = fun n : β€ => {S : Finset β | (β k β S, k > 0) β§ β k : S, Nat.fib k = n}.ncard) :
IsGreatest {n | a n = 2020} putnam_2020_a5_solution :=
sorry | import Mathlib
open Filter Topology Set
-- (Nat.fib 4040) - 1
/--
Let $a_n$ be the number of sets $S$ of positive integers for which
\[
\sum_{k \in S} F_k = n,
\]
where the Fibonacci sequence $(F_k)_{k \geq 1}$ satisfies $F_{k+2} = F_{k+1} + F_k$ and begins $F_1 = 1, F_2 = 1, F_3 = 2, F_4 = 3$. Find the largest integer $n$ such that $a_n = 2020$.
-/
theorem putnam_2020_a5
(a : β€ β β)
(ha : a = fun n : β€ => {S : Finset β | (β k β S, k > 0) β§ β k : S, Nat.fib k = n}.ncard) :
IsGreatest {n | a n = 2020} putnam_2020_a5_solution := by
| import Mathlib
open Filter Topology Set
abbrev putnam_2020_a5_solution : β€ := sorry
-- (Nat.fib 4040) - 1
/--
Let $a_n$ be the number of sets $S$ of positive integers for which
\[
\sum_{k \in S} F_k = n,
\]
where the Fibonacci sequence $(F_k)_{k \geq 1}$ satisfies $F_{k+2} = F_{k+1} + F_k$ and begins $F_1 = 1, F_2 = 1, F_3 = 2, F_4 = 3$. Find the largest integer $n$ such that $a_n = 2020$.
-/
theorem putnam_2020_a5
(a : β€ β β)
(ha : a = fun n : β€ => {S : Finset β | (β k β S, k > 0) β§ β k : S, Nat.fib k = n}.ncard) :
IsGreatest {n | a n = 2020} putnam_2020_a5_solution :=
sorry
| Let $a_n$ be the number of sets $S$ of positive integers for which
\[
\sum_{k \in S} F_k = n,
\]
where the Fibonacci sequence $(F_k)_{k \geq 1}$ satisfies $F_{k+2} = F_{k+1} + F_k$ and begins $F_1 = 1, F_2 = 1, F_3 = 2, F_4 = 3$. Find the largest integer $n$ such that $a_n = 2020$. | The answer is $n=F_{4040}-1$. | [
"number_theory",
"combinatorics"
] | null | null |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.