informal_problem
stringlengths 47
738
| informal_answer
stringlengths 1
204
| informal_solution
stringlengths 16
303
| header
stringlengths 8
63
β | intros
listlengths 0
2
| formal_answer
stringlengths 20
24.1k
| formal_answer_type
stringlengths 1
88
| outros
listlengths 1
13
| metainfo
dict |
|---|---|---|---|---|---|---|---|---|
Find, with explanation, the maximum value of $f(x)=x^3-3x$ on the set of all real numbers $x$ satisfying $x^4+36 \leq 13x^2$. | 18 | Show that the maximum value is $18$. | null | [] | @Eq Real answer
(@OfNat.ofNat Real (nat_lit 18)
(@instOfNatAtLeastTwo Real (nat_lit 18) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 16) (instOfNatNat (nat_lit 16)))))) | β | [
{
"t": "Set β",
"v": null,
"name": "S"
},
{
"t": "β β β",
"v": null,
"name": "f"
},
{
"t": "S = {x : β | x ^ 4 + 36 β€ 13 * x ^ 2}",
"v": null,
"name": "hS"
},
{
"t": "f = fun x β¦ x ^ 3 - 3 * x",
"v": null,
"name": "hf"
},
{
"t": "IsGreatest {f x | x β S} answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1986_a1",
"tags": [
"algebra",
"analysis"
]
} |
What is the units (i.e., rightmost) digit of
\[
\left\lfloor \frac{10^{20000}}{10^{100}+3}\right\rfloor ?
\] | 3 | Show that the answer is $3$. | null | [] | @Eq Nat answer (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3))) | β | [
{
"t": "Nat.floor ((10 ^ 20000 : β) / (10 ^ 100 + 3)) % 10 = answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1986_a2",
"tags": [
"algebra"
]
} |
Evaluate $\sum_{n=0}^\infty \mathrm{Arccot}(n^2+n+1)$, where $\mathrm{Arccot}\,t$ for $t \geq 0$ denotes the number $\theta$ in the interval $0 < \theta \leq \pi/2$ with $\cot \theta = t$. | Ο / 2 | Show that the sum equals $\pi/2$. | open Real | [] | @Eq Real answer
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) Real.pi
(@OfNat.ofNat Real (nat_lit 2)
(@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))))))) | β | [
{
"t": "β β β",
"v": null,
"name": "cot"
},
{
"t": "cot = fun ΞΈ β¦ Real.cos ΞΈ / Real.sin ΞΈ",
"v": null,
"name": "fcot"
},
{
"t": "β β β",
"v": null,
"name": "arccot"
},
{
"t": "β t : β, t β₯ 0 β arccot t β Set.Ioc 0 (Real.pi / 2) β§ cot (arccot t) = t",
"v": null,
"name": "harccot"
},
{
"t": "(β' n : β, arccot (n ^ 2 + n + 1)) = answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1986_a3",
"tags": [
"analysis"
]
} |
A \emph{transversal} of an $n\times n$ matrix $A$ consists of $n$ entries of $A$, no two in the same row or column. Let $f(n)$ be the number of $n \times n$ matrices $A$ satisfying the following two conditions:
\begin{enumerate}
\item[(a)] Each entry $\alpha_{i,j}$ of $A$ is in the set
$\{-1,0,1\}$.
\item[(b)] The sum of the $n$ entries of a transversal is the same for all transversals of $A$.
\end{enumerate}
An example of such a matrix $A$ is
\[
A = \left( \begin{array}{ccc} -1 & 0 & -1 \\ 0 & 1 & 0 \\ 0 & 1 & 0
\end{array}
\right).
\]
Determine with proof a formula for $f(n)$ of the form
\[
f(n) = a_1 b_1^n + a_2 b_2^n + a_3 b_3^n + a_4,
\]
where the $a_i$'s and $b_i$'s are rational numbers. | (1, 4, 2, 3, -4, 2, 1) | Prove that $f(n) = 4^n + 2 \cdot 3^n - 4 \cdot 2^n + 1$. | open Real Equiv | [] | @Eq (Prod Rat (Prod Rat (Prod Rat (Prod Rat (Prod Rat (Prod Rat Rat)))))) answer
(@Prod.mk Rat (Prod Rat (Prod Rat (Prod Rat (Prod Rat (Prod Rat Rat)))))
(@OfNat.ofNat Rat (nat_lit 1) (@Rat.instOfNat (nat_lit 1)))
(@Prod.mk Rat (Prod Rat (Prod Rat (Prod Rat (Prod Rat Rat))))
(@OfNat.ofNat Rat (nat_lit 4) (@Rat.instOfNat (nat_lit 4)))
(@Prod.mk Rat (Prod Rat (Prod Rat (Prod Rat Rat))) (@OfNat.ofNat Rat (nat_lit 2) (@Rat.instOfNat (nat_lit 2)))
(@Prod.mk Rat (Prod Rat (Prod Rat Rat)) (@OfNat.ofNat Rat (nat_lit 3) (@Rat.instOfNat (nat_lit 3)))
(@Prod.mk Rat (Prod Rat Rat)
(@Neg.neg Rat Rat.instNeg (@OfNat.ofNat Rat (nat_lit 4) (@Rat.instOfNat (nat_lit 4))))
(@Prod.mk Rat Rat (@OfNat.ofNat Rat (nat_lit 2) (@Rat.instOfNat (nat_lit 2)))
(@OfNat.ofNat Rat (nat_lit 1) (@Rat.instOfNat (nat_lit 1))))))))) | β Γ β Γ β Γ β Γ β Γ β Γ β | [
{
"t": "β β β",
"v": null,
"name": "f"
},
{
"t": "f = fun n β¦\n Set.ncard {A : Matrix (Fin n) (Fin n) β€ |\n (β i j : Fin n, A i j β ({-1, 0, 1} : Set β€)) β§\n β S : β€, β Ο : Perm (Fin n), β i : Fin n, A i (Ο i) = S}",
"v": null,
"name": "hf"
},
{
"t": "let (a1, b1, a2, b2, a3, b3, a4) := answer;\n (β n > 0, f n = a1 * b1 ^ n + a2 * b2 ^ n + a3 * b3 ^ n + a4)",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1986_a4",
"tags": [
"linear_algebra"
]
} |
Let $a_1, a_2, \dots, a_n$ be real numbers, and let $b_1, b_2, \dots, b_n$ be distinct positive integers. Suppose that there is a polynomial $f(x)$ satisfying the identity
\[
(1-x)^n f(x) = 1 + \sum_{i=1}^n a_i x^{b_i}.
\]
Find a simple expression (not involving any sums) for $f(1)$ in terms of $b_1, b_2, \dots, b_n$ and $n$ (but independent of $a_1, a_2, \dots, a_n$). | fun b n β¦ (β i : Finset.Icc 1 n, b i) / Nat.factorial n | Show that $f(1) = b_1 b_2 \dots b_n / n!$. | open Real Equiv | [] | @Eq ((Nat β Nat) β Nat β Real) answer fun (b_1 : Nat β Nat) (n_1 : Nat) =>
@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@Nat.cast Real Real.instNatCast
(@Finset.prod
(@Subtype Nat fun (x : Nat) =>
@Membership.mem Nat (Finset Nat) (@Finset.instMembership Nat)
(@Finset.Icc Nat Nat.instPreorder Nat.instLocallyFiniteOrder
(@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))) n_1)
x)
Nat Nat.instCommMonoid
(@Finset.univ
(@Subtype Nat fun (x : Nat) =>
@Membership.mem Nat (Finset Nat) (@Finset.instMembership Nat)
(@Finset.Icc Nat Nat.instPreorder Nat.instLocallyFiniteOrder
(@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))) n_1)
x)
(@Finset.Subtype.fintype Nat
(@Finset.Icc Nat Nat.instPreorder Nat.instLocallyFiniteOrder
(@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))) n_1)))
fun
(i :
@Subtype Nat fun (x : Nat) =>
@Membership.mem Nat (Finset Nat) (@Finset.instMembership Nat)
(@Finset.Icc Nat Nat.instPreorder Nat.instLocallyFiniteOrder
(@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))) n_1)
x) =>
b_1
(@Subtype.val Nat
(fun (x : Nat) =>
@Membership.mem Nat (Finset Nat) (@Finset.instMembership Nat)
(@Finset.Icc Nat Nat.instPreorder Nat.instLocallyFiniteOrder
(@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))) n_1)
x)
i)))
(@Nat.cast Real Real.instNatCast (Nat.factorial n_1)) | (β β β) β β β β | [
{
"t": "β",
"v": null,
"name": "n"
},
{
"t": "n > 0",
"v": null,
"name": "npos"
},
{
"t": "β β β",
"v": null,
"name": "a"
},
{
"t": "β β β",
"v": null,
"name": "b"
},
{
"t": "β i β Finset.Icc 1 n, b i > 0",
"v": null,
"name": "bpos"
},
{
"t": "β i β Finset.Icc 1 n, β j β Finset.Icc 1 n, b i = b j β i = j",
"v": null,
"name": "binj"
},
{
"t": "Polynomial β",
"v": null,
"name": "f"
},
{
"t": "β x : β, (1 - x) ^ n * f.eval x = 1 + β i : Finset.Icc 1 n, (a i) * x ^ (b i)",
"v": null,
"name": "hf"
},
{
"t": "f.eval 1 = answer b n",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1986_a6",
"tags": [
"algebra"
]
} |
Inscribe a rectangle of base $b$ and height $h$ and an isosceles triangle of base $b$ (against a corresponding side of the rectangle and pointed in the other direction) in a circle of radius one. For what value of $h$ do the rectangle and triangle have the same area? | $2/5$ | Show that the only such value of $h$ is $2/5$. | open Real Equiv | [] | @Eq Real answer
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@OfNat.ofNat Real (nat_lit 2)
(@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))))))
(@OfNat.ofNat Real (nat_lit 5)
(@instOfNatAtLeastTwo Real (nat_lit 5) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3))))))) | β | [
{
"t": "β",
"v": null,
"name": "b"
},
{
"t": "β",
"v": null,
"name": "h"
},
{
"t": "b > 0 β§ h > 0 β§ b ^ 2 + h ^ 2 = 2 ^ 2",
"v": null,
"name": "hbh"
},
{
"t": "b * h = 0.5 * b * (1 - h / 2)",
"v": null,
"name": "areaeq"
},
{
"t": "h = answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1986_b1",
"tags": [
"geometry",
"algebra"
]
} |
Prove that there are only a finite number of possibilities for the ordered triple $T=(x-y,y-z,z-x)$, where $x,y,z$ are complex numbers satisfying the simultaneous equations
\[
x(x-1)+2yz = y(y-1)+2zx = z(z-1)+2xy,
\]
and list all such triples $T$. | {(0, 0, 0), (0, -1, 1), (1, 0, -1), (-1, 1, 0)} | Show that the possibilities for $T$ are $(0, 0, 0), \, (0, -1, 1), \, (1, 0, -1), \, (-1, 1, 0)$. | open Real Equiv | [] | @Eq (Finset (Prod Complex (Prod Complex Complex))) answer
(@Insert.insert (Prod Complex (Prod Complex Complex)) (Finset (Prod Complex (Prod Complex Complex)))
(@Finset.instInsert (Prod Complex (Prod Complex Complex)) fun (a b : Prod Complex (Prod Complex Complex)) =>
@instDecidableEqProd Complex (Prod Complex Complex) Complex.instDecidableEq
(fun (a_1 b_1 : Prod Complex Complex) =>
@instDecidableEqProd Complex Complex Complex.instDecidableEq Complex.instDecidableEq a_1 b_1)
a b)
(@Prod.mk Complex (Prod Complex Complex)
(@OfNat.ofNat Complex (nat_lit 0) (@Zero.toOfNat0 Complex Complex.instZero))
(@Prod.mk Complex Complex (@OfNat.ofNat Complex (nat_lit 0) (@Zero.toOfNat0 Complex Complex.instZero))
(@OfNat.ofNat Complex (nat_lit 0) (@Zero.toOfNat0 Complex Complex.instZero))))
(@Insert.insert (Prod Complex (Prod Complex Complex)) (Finset (Prod Complex (Prod Complex Complex)))
(@Finset.instInsert (Prod Complex (Prod Complex Complex)) fun (a b : Prod Complex (Prod Complex Complex)) =>
@instDecidableEqProd Complex (Prod Complex Complex) Complex.instDecidableEq
(fun (a_1 b_1 : Prod Complex Complex) =>
@instDecidableEqProd Complex Complex Complex.instDecidableEq Complex.instDecidableEq a_1 b_1)
a b)
(@Prod.mk Complex (Prod Complex Complex)
(@OfNat.ofNat Complex (nat_lit 0) (@Zero.toOfNat0 Complex Complex.instZero))
(@Prod.mk Complex Complex
(@Neg.neg Complex Complex.instNeg (@OfNat.ofNat Complex (nat_lit 1) (@One.toOfNat1 Complex Complex.instOne)))
(@OfNat.ofNat Complex (nat_lit 1) (@One.toOfNat1 Complex Complex.instOne))))
(@Insert.insert (Prod Complex (Prod Complex Complex)) (Finset (Prod Complex (Prod Complex Complex)))
(@Finset.instInsert (Prod Complex (Prod Complex Complex)) fun (a b : Prod Complex (Prod Complex Complex)) =>
@instDecidableEqProd Complex (Prod Complex Complex) Complex.instDecidableEq
(fun (a_1 b_1 : Prod Complex Complex) =>
@instDecidableEqProd Complex Complex Complex.instDecidableEq Complex.instDecidableEq a_1 b_1)
a b)
(@Prod.mk Complex (Prod Complex Complex)
(@OfNat.ofNat Complex (nat_lit 1) (@One.toOfNat1 Complex Complex.instOne))
(@Prod.mk Complex Complex (@OfNat.ofNat Complex (nat_lit 0) (@Zero.toOfNat0 Complex Complex.instZero))
(@Neg.neg Complex Complex.instNeg
(@OfNat.ofNat Complex (nat_lit 1) (@One.toOfNat1 Complex Complex.instOne)))))
(@Singleton.singleton (Prod Complex (Prod Complex Complex)) (Finset (Prod Complex (Prod Complex Complex)))
(@Finset.instSingleton (Prod Complex (Prod Complex Complex)))
(@Prod.mk Complex (Prod Complex Complex)
(@Neg.neg Complex Complex.instNeg
(@OfNat.ofNat Complex (nat_lit 1) (@One.toOfNat1 Complex Complex.instOne)))
(@Prod.mk Complex Complex (@OfNat.ofNat Complex (nat_lit 1) (@One.toOfNat1 Complex Complex.instOne))
(@OfNat.ofNat Complex (nat_lit 0) (@Zero.toOfNat0 Complex Complex.instZero)))))))) | Finset (β Γ β Γ β) | [
{
"t": "{T : β Γ β Γ β | β x y z : β, T = (x - y, y - z, z - x) β§ x * (x - 1) + 2 * y * z = y * (y - 1) + 2 * z * x β§ y * (y - 1) + 2 * z * x = z * (z - 1) + 2 * x * y} = answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1986_b2",
"tags": [
"algebra"
]
} |
For a positive real number $r$, let $G(r)$ be the minimum value of $|r - \sqrt{m^2+2n^2}|$ for all integers $m$ and $n$. Prove or disprove the assertion that $\lim_{r\to \infty}G(r)$ exists and equals $0$. | True | Show that the limit exists and equals $0$. | open Real Equiv Polynomial Filter Topology | [] | @Eq Prop answer True | Prop | [
{
"t": "β β β",
"v": null,
"name": "G"
},
{
"t": "β r : β, β m n : β€, G r = |r - Real.sqrt (m ^ 2 + 2 * n ^ 2)|",
"v": null,
"name": "hGeq"
},
{
"t": "β r : β, β m n : β€, G r β€ |r - Real.sqrt (m ^ 2 + 2 * n ^ 2)|",
"v": null,
"name": "hGlb"
},
{
"t": "answer β Tendsto G atTop (π 0)",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1986_b4",
"tags": [
"analysis"
]
} |
Let $f(x,y,z) = x^2+y^2+z^2+xyz$. Let $p(x,y,z), q(x,y,z)$, $r(x,y,z)$ be polynomials with real coefficients satisfying
\[
f(p(x,y,z), q(x,y,z), r(x,y,z)) = f(x,y,z).
\]
Prove or disprove the assertion that the sequence $p,q,r$ consists of some permutation of $\pm x, \pm y, \pm z$, where the number of minus signs is $0$ or $2$. | False | Prove that the assertion is false. | open Real Equiv Polynomial Filter Topology MvPolynomial | [] | @Eq Prop answer False | Prop | [
{
"t": "MvPolynomial (Fin 3) β",
"v": null,
"name": "f"
},
{
"t": "Set (Set (MvPolynomial (Fin 3) β))",
"v": null,
"name": "perms"
},
{
"t": "f = (X 0) ^ 2 + (X 1) ^ 2 + (X 2) ^ 2 + (X 0) * (X 1) * (X 2)",
"v": null,
"name": "hf"
},
{
"t": "perms = {{X 0, X 1, X 2}, {X 0, -X 1, -X 2}, {-X 0, X 1, -X 2}, {-X 0, -X 1, X 2}}",
"v": null,
"name": "hperms"
},
{
"t": "answer β (β pqr : Fin 3 β MvPolynomial (Fin 3) β,\n (β xyz : Fin 3 β β, MvPolynomial.eval (fun i β¦ MvPolynomial.eval xyz (pqr i)) f = MvPolynomial.eval xyz f) β\n ({pqr 0, pqr 1, pqr 2} β perms))",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1986_b5",
"tags": [
"algebra"
]
} |
The sequence of digits $123456789101112131415161718192021 \dots$ is obtained by writing the positive integers in order. If the $10^n$-th digit in this sequence occurs in the part of the sequence in which the $m$-digit numbers are placed, define $f(n)$ to be $m$. For example, $f(2)=2$ because the $100$th digit enters the sequence in the placement of the two-digit integer $55$. Find, with proof, $f(1987)$. | 1984 | Show that the value of $f(1987)$ is $1984$. | null | [] | @Eq Nat answer (@OfNat.ofNat Nat (nat_lit 1984) (instOfNatNat (nat_lit 1984))) | β | [
{
"t": "β β β",
"v": null,
"name": "seqind"
},
{
"t": "β β β",
"v": null,
"name": "seqsize"
},
{
"t": "β β β",
"v": null,
"name": "f"
},
{
"t": "seqind 1 = 1 β§ β i β₯ 2, seqind i = seqind (i - 1) + (Nat.digits 10 (i - 1)).length",
"v": null,
"name": "hseqind"
},
{
"t": "β i β₯ 1, β j : Fin ((Nat.digits 10 i).length), seqsize (seqind i + j) = (Nat.digits 10 i).length",
"v": null,
"name": "hseqsize"
},
{
"t": "β n : β, f n = seqsize (10 ^ n)",
"v": null,
"name": "hf"
},
{
"t": "f 1987 = answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1987_a2",
"tags": [
"algebra"
]
} |
Let $P$ be a polynomial, with real coefficients, in three variables and $F$ be a function of two variables such that
\[
P(ux, uy, uz) = u^2 F(y-x,z-x) \quad \mbox{for all real $x,y,z,u$},
\]
and such that $P(1,0,0)=4$, $P(0,1,0)=5$, and $P(0,0,1)=6$. Also let $A,B,C$ be complex numbers with $P(A,B,C)=0$ and $|B-A|=10$. Find $|C-A|$. | $\frac{5}{3}\sqrt{30}$ | Prove that $|C - A| = \frac{5}{3}\sqrt{30}$. | open MvPolynomial Real | [] | @Eq Complex answer
(@HMul.hMul Complex Complex Complex (@instHMul Complex Complex.instMul)
(@HDiv.hDiv Complex Complex Complex (@instHDiv Complex (@DivInvMonoid.toDiv Complex Complex.instDivInvMonoid))
(@OfNat.ofNat Complex (nat_lit 5)
(@instOfNatAtLeastTwo Complex (nat_lit 5)
(@AddMonoidWithOne.toNatCast Complex (@AddGroupWithOne.toAddMonoidWithOne Complex Complex.addGroupWithOne))
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3))))))
(@OfNat.ofNat Complex (nat_lit 3)
(@instOfNatAtLeastTwo Complex (nat_lit 3)
(@AddMonoidWithOne.toNatCast Complex (@AddGroupWithOne.toAddMonoidWithOne Complex Complex.addGroupWithOne))
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))))
β(Real.sqrt
(@OfNat.ofNat Real (nat_lit 30)
(@instOfNatAtLeastTwo Real (nat_lit 30) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 28) (instOfNatNat (nat_lit 28)))))))) | β | [
{
"t": "MvPolynomial (Fin 3) β",
"v": null,
"name": "P"
},
{
"t": "β i : Fin 3 ββ β, (coeff i P).im = 0",
"v": null,
"name": "hPreal"
},
{
"t": "β β β β β",
"v": null,
"name": "F"
},
{
"t": "β β β β β β (Fin 3 β β)",
"v": null,
"name": "vars"
},
{
"t": "vars = fun a b c β¦ fun i β¦ ite (i = 0) a (ite (i = 1) b c)",
"v": null,
"name": "hvars"
},
{
"t": "β x y z u : β, eval (vars (u * x) (u * y) (u * z)) P = u ^ 2 * F (y - x) (z - x)",
"v": null,
"name": "h"
},
{
"t": "eval (vars 1 0 0) P = 4 β§ eval (vars 0 1 0) P = 5 β§ eval (vars 0 0 1) P = 6",
"v": null,
"name": "hPval"
},
{
"t": "β",
"v": null,
"name": "A"
},
{
"t": "β",
"v": null,
"name": "B"
},
{
"t": "β",
"v": null,
"name": "C"
},
{
"t": "eval (vars A B C) P = 0",
"v": null,
"name": "hPABC"
},
{
"t": "βB - Aβ = 10",
"v": null,
"name": "habs"
},
{
"t": "βC - Aβ = answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1987_a4",
"tags": [
"algebra"
]
} |
Let $\vec{G}(x,y)=\left(\frac{-y}{x^2+4y^2},\frac{x}{x^2+4y^2},0\right)$. Prove or disprove that there is a vector-valued function $\vec{F}(x,y,z)=(M(x,y,z),N(x,y,z),P(x,y,z))$ with the following properties:
\begin{enumerate}
\item[(i)] $M$, $N$, $P$ have continuous partial derivatives for all $(x,y,z) \neq (0,0,0)$;
\item[(ii)] $\text{Curl}\,\vec{F}=\vec{0}$ for all $(x,y,z) \neq (0,0,0)$;
\item[(iii)] $\vec{F}(x,y,0)=\vec{G}(x,y)$.
\end{enumerate} | False | Show that there is no such $\vec{F}$. | null | [] | @Eq Prop answer False | Prop | [
{
"t": "((Fin 3 β β) β (Fin 3 β β)) β ((Fin 3 β β) β (Fin 3 β β))",
"v": null,
"name": "curl"
},
{
"t": "β f x, curl f x = ![\n fderiv β f x (Pi.single 1 1) 2 - fderiv β f x (Pi.single 2 1) 1,\n fderiv β f x (Pi.single 2 1) 0 - fderiv β f x (Pi.single 0 1) 2,\n fderiv β f x (Pi.single 0 1) 1 - fderiv β f x (Pi.single 1 1) 0]",
"v": null,
"name": "curl_def"
},
{
"t": "(Fin 2 β β) β (Fin 3 β β)",
"v": null,
"name": "G"
},
{
"t": "β x y, G ![x, y] = ![(-y / (x ^ 2 + 4 * y ^ 2)), (x / (x ^ 2 + 4 * y ^ 2)), 0]",
"v": null,
"name": "G_def"
},
{
"t": "(β F : (Fin 3 β β) β (Fin 3 β β),\n ContDiffOn β 1 F {v | v β ![0,0,0]} β§\n (β x, x β 0 β curl F x = 0) β§\n β x y, F ![x, y, 0] = G ![x, y]) β answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1987_a5",
"tags": [
"analysis"
]
} |
For each positive integer $n$, let $a(n)$ be the number of zeroes in the base $3$ representation of $n$. For which positive real numbers $x$ does the series
\[
\sum_{n=1}^\infty \frac{x^{a(n)}}{n^3}
\]
converge? | the set of positive real numbers $x$ such that $0 < x < 25$ | Show that for positive $x$, the series converges if and only if $x < 25$. | open MvPolynomial Real Nat | [] | @Eq (Set Real) answer
(@setOf Real fun (x : Real) =>
And (@GT.gt Real Real.instLT x (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)))
(@LT.lt Real Real.instLT x
(@OfNat.ofNat Real (nat_lit 25)
(@instOfNatAtLeastTwo Real (nat_lit 25) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 23) (instOfNatNat (nat_lit 23)))))))) | Set β | [
{
"t": "β β β",
"v": null,
"name": "a"
},
{
"t": "a = fun n β¦ {i | (digits 3 n).get i = 0}.ncard",
"v": null,
"name": "ha"
},
{
"t": "{x : β | x > 0 β§ Summable (fun n β¦ x ^ (a n) / (n ^ 3))} = answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1987_a6",
"tags": [
"algebra",
"analysis"
]
} |
Evaluate
\[
\int_2^4 \frac{\sqrt{\ln(9-x)}\,dx}{\sqrt{\ln(9-x)}+\sqrt{\ln(x+3)}}.
\] | 1 | Prove that the integral evaluates to $1$. | open MvPolynomial Real Nat | [] | @Eq Real answer (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)) | β | [
{
"t": "answer = β« x in (2)..4, sqrt (log (9 - x)) / (sqrt (log (9 - x)) + sqrt (log (x + 3)))",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1987_b1",
"tags": [
"analysis"
]
} |
Let $(x_1,y_1) = (0.8, 0.6)$ and let $x_{n+1} = x_n \cos y_n - y_n \sin y_n$ and $y_{n+1}= x_n \sin y_n + y_n \cos y_n$ for $n=1,2,3,\dots$. For each of $\lim_{n\to \infty} x_n$ and $\lim_{n \to \infty} y_n$, prove that the limit exists and find it or prove that the limit does not exist. | (True, -1, True, 0) | Show that $\lim_{n \to \infty} x_n = -1$ and $\lim_{n \to \infty} y_n = 0$. | open MvPolynomial Real Nat Filter Topology | [] | @Eq (Prod Prop (Prod Real (Prod Prop Real))) answer
(@Prod.mk Prop (Prod Real (Prod Prop Real)) True
(@Prod.mk Real (Prod Prop Real)
(@Neg.neg Real Real.instNeg (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)))
(@Prod.mk Prop Real True (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero))))) | Prop Γ β Γ Prop Γ β | [
{
"t": "β β β",
"v": null,
"name": "x"
},
{
"t": "β β β",
"v": null,
"name": "y"
},
{
"t": "(x 1, y 1) = (0.8, 0.6)",
"v": null,
"name": "hxy1"
},
{
"t": "β n β₯ 1, x (n + 1) = (x n) * cos (y n) - (y n) * sin (y n)",
"v": null,
"name": "hx"
},
{
"t": "β n β₯ 1, y (n + 1) = (x n) * sin (y n) + (y n) * cos (y n)",
"v": null,
"name": "hy"
},
{
"t": "let (existsx, limx, existsy, limy) := answer\n((β c : β, Tendsto x atTop (π c)) β existsx) β§\n(existsx β Tendsto x atTop (π limx)) β§\n((β c : β, Tendsto y atTop (π c)) β existsy) β§\n(existsy β Tendsto y atTop (π limy))",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1987_b4",
"tags": [
"analysis"
]
} |
Let $R$ be the region consisting of the points $(x,y)$ of the cartesian plane satisfying both $|x|-|y| \leq 1$ and $|y| \leq 1$. Find the area of $R$. | 6 | Show that the area of $R$ is $6$. | open MeasureTheory | [] | @Eq Real answer
(@OfNat.ofNat Real (nat_lit 6)
(@instOfNatAtLeastTwo Real (nat_lit 6) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 4) (instOfNatNat (nat_lit 4)))))) | β | [
{
"t": "Set (Fin 2 β β)",
"v": null,
"name": "R"
},
{
"t": "R = {p | |p 0| - |p 1| β€ 1 β§ |p 1| β€ 1}",
"v": null,
"name": "hR"
},
{
"t": "(volume R).toReal = answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1988_a1",
"tags": [
"geometry"
]
} |
A not uncommon calculus mistake is to believe that the product rule for derivatives says that $(fg)' = f'g'$. If $f(x)=e^{x^2}$, determine, with proof, whether there exists an open interval $(a,b)$ and a nonzero function $g$ defined on $(a,b)$ such that this wrong product rule is true for $x$ in $(a,b)$. | True | Show that such $(a,b)$ and $g$ exist. | open Set | [] | @Eq Prop answer True | Prop | [
{
"t": "β β β",
"v": null,
"name": "f"
},
{
"t": "f = fun x β¦ Real.exp (x ^ 2)",
"v": null,
"name": "hf"
},
{
"t": "answer β\n (β a b : β,\n a < b β§\n β g : β β β,\n (β x β Ioo a b, g x β 0) β§ \n DifferentiableOn β g (Ioo a b) β§ \n β x β Ioo a b, deriv (fun y β¦ f y * g y) x = (deriv f x) * (deriv g x))",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1988_a2",
"tags": [
"analysis"
]
} |
Determine, with proof, the set of real numbers $x$ for which
\[
\sum_{n=1}^\infty \left( \frac{1}{n} \csc \frac{1}{n} - 1 \right)^x
\]
converges. | {x | x > 1 / 2} | Show that the series converges if and only if $x > \frac{1}{2}$. | open Set Filter Topology | [] | @Eq (Set Real) answer
(@setOf Real fun (x : Real) =>
@GT.gt Real Real.instLT x
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))
(@OfNat.ofNat Real (nat_lit 2)
(@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0)))))))) | Set β | [
{
"t": "answer = {x : β | β L : β, Tendsto (fun t β¦ β n in Finset.Icc (1 : β) t, (((1 / n) / Real.sin (1 / n) - 1) ^ x)) atTop (π L)}",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1988_a3",
"tags": [
"analysis"
]
} |
\begin{enumerate}
\item[(a)] If every point of the plane is painted one of three colors, do there necessarily exist two points of the same color exactly one inch apart?
\item[(b)] What if ``three'' is replaced by ``nine''?
\end{enumerate} | (True, False) | Prove that the points must exist with three colors, but not necessarily with nine. | open Set Filter Topology | [] | @Eq (Prod Prop Prop) answer (@Prod.mk Prop Prop True False) | Prop Γ Prop | [
{
"t": "β β Prop",
"v": null,
"name": "p"
},
{
"t": "β n, p n β\n β color : (EuclideanSpace β (Fin 2)) β Fin n,\n β p q : EuclideanSpace β (Fin 2),\n color p = color q β§ dist p q = 1",
"v": null,
"name": "hp"
},
{
"t": "let (a, b) := answer; (p 3 β a) β§ (p 9 β b)",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1988_a4",
"tags": [
"geometry",
"combinatorics"
]
} |
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. | True | Show that the answer is yes, $A$ must be a scalar multiple of the identity. | open Set Filter Topology | [] | @Eq Prop answer True | Prop | [
{
"t": "(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 = Module.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 answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1988_a6",
"tags": [
"linear_algebra"
]
} |
Prove or disprove: If $x$ and $y$ are real numbers with $y \geq 0$ and $y(y+1) \leq (x+1)^2$, then $y(y-1) \leq x^2$. | True | Show that this is true. | open Set Filter Topology | [] | @Eq Prop answer True | Prop | [
{
"t": "β",
"v": null,
"name": "x"
},
{
"t": "β",
"v": null,
"name": "y"
},
{
"t": "y β₯ 0",
"v": null,
"name": "h_y_nonneg"
},
{
"t": "y * (y + 1) β€ (x + 1) ^ 2",
"v": null,
"name": "h_ineq"
},
{
"t": "answer = (y * (y - 1) β€ x ^ 2)",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1988_b2",
"tags": [
"algebra"
]
} |
For every $n$ in the set $N=\{1,2,\dots\}$ of positive integers, let $r_n$ be the minimum value of $|c-d \sqrt{3}|$ for all nonnegative integers $c$ and $d$ with $c+d=n$. Find, with proof, the smallest positive real number $g$ with $r_n \leq g$ for all $n \in N$. | (1 + β3) / 2 | Show that the smallest such $g$ is $(1+\sqrt{3})/2$. | open Set Filter Topology | [] | @Eq Real answer
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd)
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))
(Real.sqrt
(@OfNat.ofNat Real (nat_lit 3)
(@instOfNatAtLeastTwo Real (nat_lit 3) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))))))
(@OfNat.ofNat Real (nat_lit 2)
(@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))))))) | β | [
{
"t": "β€ β β",
"v": null,
"name": "r"
},
{
"t": "β n β₯ 1,\n (β c d : β€,\n (c β₯ 0 β§ d β₯ 0) β§\n c + d = n β§ r n = |c - d * Real.sqrt 3|) β§\n (β c d : β€, (c β₯ 0 β§ d β₯ 0 β§ c + d = n) β |c - d * Real.sqrt 3| β₯ r n)",
"v": null,
"name": "hr"
},
{
"t": "IsLeast {g : β | g > 0 β§ (β n : β€, n β₯ 1 β r n β€ g)} answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1988_b3",
"tags": [
"algebra"
]
} |
For positive integers $n$, let $M_n$ be the $2n+1$ by $2n+1$ skew-symmetric matrix for which each entry in the first $n$ subdiagonals below the main diagonal is $1$ and each of the remaining entries below the main diagonal is $-1$. Find, with proof, the rank of $M_n$. (According to one definition, the rank of a matrix is the largest $k$ such that there is a $k \times k$ submatrix with nonzero determinant.) One may note that
\begin{align*}
M_1&=\begin{pmatrix} 0 & -1 & 1 \\ 1 & 0 & -1 \\ -1 & 1 & 0 \end{pmatrix} \\
M_2&=\begin{pmatrix} 0 & -1 & -1 & 1 & 1 \\ 1 & 0 & -1 & -1 & 1 \\ 1 & 1 & 0 & -1 & -1 \\ -1 & 1 & 1 & 0 & -1 \\ -1 & -1 & 1 & 1 & 0 \end{pmatrix}.
\end{align*} | 2n | Show that the rank of $M_n$ equals $2n$. | open Set Filter Topology | [] | @Eq (Nat β Nat) answer fun (n_1 : Nat) =>
@HMul.hMul Nat Nat Nat (@instHMul Nat instMulNat) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) n_1 | β β β | [
{
"t": "β",
"v": null,
"name": "n"
},
{
"t": "n > 0",
"v": null,
"name": "hn"
},
{
"t": "Matrix (Fin (2 * n + 1)) (Fin (2 * n + 1)) β",
"v": null,
"name": "Mn"
},
{
"t": "β i j, Mn i j = -(Mn j i)",
"v": null,
"name": "Mnskewsymm"
},
{
"t": "β i j, (1 β€ (i.1 : β€) - j.1 β§ (i.1 : β€) - j.1 β€ n) β Mn i j = 1",
"v": null,
"name": "hMn1"
},
{
"t": "β i j, (i.1 : β€) - j.1 > n β Mn i j = -1",
"v": null,
"name": "hMnn1"
},
{
"t": "Mn.rank = answer n",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1988_b5",
"tags": [
"linear_algebra"
]
} |
How many primes among the positive integers, written as usual in base $10$, are alternating $1$'s and $0$'s, beginning and ending with $1$? | 1 | Show that there is only one such prime. | null | [] | @Eq ENat answer
(@OfNat.ofNat ENat (nat_lit 1)
(@One.toOfNat1 ENat
(@AddMonoidWithOne.toOne ENat
(@AddCommMonoidWithOne.toAddMonoidWithOne ENat
(@NonAssocSemiring.toAddCommMonoidWithOne ENat
(@Semiring.toNonAssocSemiring ENat
(@OrderedSemiring.toSemiring ENat
(@OrderedCommSemiring.toOrderedSemiring ENat
(@CanonicallyOrderedCommSemiring.toOrderedCommSemiring ENat
instENatCanonicallyOrderedCommSemiring))))))))) | ββ | [
{
"t": "List β β Prop",
"v": null,
"name": "pdigalt"
},
{
"t": "β l, pdigalt l β Odd l.length β§ (β i, l.get i = if Even (i : β) then 1 else 0)",
"v": null,
"name": "hpdigalt"
},
{
"t": "{p : β | p.Prime β§ pdigalt (Nat.digits 10 p)}.encard = answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1989_a1",
"tags": [
"algebra",
"number_theory"
]
} |
Evaluate $\int_0^a \int_0^b e^{\max\{b^2x^2,a^2y^2\}}\,dy\,dx$ where $a$ and $b$ are positive. | $\frac{e^{a^2 b^2} - 1}{a b}$ | Show that the value of the integral is $(e^{a^2b^2}-1)/(ab)$. | null | [] | @Eq (Real β Real β Real) answer fun (a_1 b_1 : Real) =>
@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@HSub.hSub Real Real Real (@instHSub Real Real.instSub)
(Real.exp
(@HMul.hMul Real Real Real (@instHMul Real Real.instMul)
(@HPow.hPow Real Nat Real (@instHPow Real Nat (@Monoid.toNatPow Real Real.instMonoid)) a_1
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))
(@HPow.hPow Real Nat Real (@instHPow Real Nat (@Monoid.toNatPow Real Real.instMonoid)) b_1
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))))
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)))
(@HMul.hMul Real Real Real (@instHMul Real Real.instMul) a_1 b_1) | β β β β β | [
{
"t": "β",
"v": null,
"name": "a"
},
{
"t": "β",
"v": null,
"name": "b"
},
{
"t": "a > 0 β§ b > 0",
"v": null,
"name": "abpos"
},
{
"t": "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)) = answer a b",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1989_a2",
"tags": [
"analysis"
]
} |
A dart, thrown at random, hits a square target. Assuming that any two parts of the target of equal area are equally likely to be hit, find the probability that the point hit is nearer to the center than to any edge. Express your answer in the form $(a\sqrt{b}+c)/d$, where $a$, $b$, $c$, $d$ are integers and $b$, $d$ are positive. | (4, 2, -5, 3) | Show that the probability is $(4\sqrt{2}-5)/3$. | open Nat MeasureTheory | [] | @Eq (Prod Int (Prod Int (Prod Int Int))) answer
(@Prod.mk Int (Prod Int (Prod Int Int)) (@OfNat.ofNat Int (nat_lit 4) (@instOfNat (nat_lit 4)))
(@Prod.mk Int (Prod Int Int) (@OfNat.ofNat Int (nat_lit 2) (@instOfNat (nat_lit 2)))
(@Prod.mk Int Int (@Neg.neg Int Int.instNegInt (@OfNat.ofNat Int (nat_lit 5) (@instOfNat (nat_lit 5))))
(@OfNat.ofNat Int (nat_lit 3) (@instOfNat (nat_lit 3)))))) | β€ Γ β€ Γ β€ Γ β€ | [
{
"t": "Set (EuclideanSpace β (Fin 2))",
"v": null,
"name": "square"
},
{
"t": "Set (EuclideanSpace β (Fin 2))",
"v": null,
"name": "Scloser"
},
{
"t": "Set (EuclideanSpace β (Fin 2))",
"v": null,
"name": "perimeter"
},
{
"t": "EuclideanSpace β (Fin 2)",
"v": null,
"name": "center"
},
{
"t": "square = {p | β i : Fin 2, p i β Set.Icc 0 1}",
"v": null,
"name": "square_def"
},
{
"t": "perimeter = {p β square | p 0 = 0 β¨ p 0 = 1 β¨ p 1 = 0 β¨ p 1 = 1}",
"v": null,
"name": "perimeter_def"
},
{
"t": "center = (fun i : Fin 2 => 1 / 2)",
"v": null,
"name": "center_def"
},
{
"t": "Scloser = {p β square | β q β perimeter, dist p center < dist p q}",
"v": null,
"name": "hScloser"
},
{
"t": "let (a, b, c, d) := answer;\n b > 0 β§ d > 0 β§ (Β¬β n : β€, n^2 = b) β§\n (volume Scloser).toReal / (volume square).toReal = (a * Real.sqrt b + c) / d",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1989_b1",
"tags": [
"probability",
"geometry"
]
} |
Let $S$ be a non-empty set with an associative operation that is left and right cancellative ($xy=xz$ implies $y=z$, and $yx=zx$ implies $y=z$). Assume that for every $a$ in $S$ the set $\{a^n:\,n=1, 2, 3, \ldots\}$ is finite. Must $S$ be a group? | True | Prove that $S$ must be a group. | open Nat | [] | @Eq Prop answer True | Prop | [
{
"t": "(β (S : Type) [Nonempty S] [Semigroup S] [IsCancelMul S]\n (h_fin : β a : S, {(a * Β·)^[n] a | n : β}.Finite),\n β e : S, β x, e * x = x β§ x * e = x β§ β y, x * y = e β§ y * x = e) β\n answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1989_b2",
"tags": [
"abstract_algebra"
]
} |
Let $f$ be a function on $[0,\infty)$, differentiable and satisfying
\[
f'(x)=-3f(x)+6f(2x)
\]
for $x>0$. Assume that $|f(x)|\le e^{-\sqrt{x}}$ for $x\ge 0$ (so that $f(x)$ tends rapidly to $0$ as $x$ increases). For $n$ a non-negative integer, define
\[
\mu_n=\int_0^\infty x^n f(x)\,dx
\]
(sometimes called the $n$th moment of $f$).
\begin{enumerate}
\item[a)] Express $\mu_n$ in terms of $\mu_0$.
\item[b)] Prove that the sequence $\{\mu_n \frac{3^n}{n!}\}$ always converges, and that the limit is $0$ only if $\mu_0=0$.
\end{enumerate} | fun n c β¦ c * n ! / (3 ^ n * β m in Finset.Icc (1 : β€) n, (1 - 2 ^ (-m))) | Show that for each $n \geq 0$, $\mu_n = \frac{n!}{3^n} \left( \prod_{m=1}^{n}(1 - 2^{-m}) \right)^{-1} \mu_0$. | open Nat Filter Topology | [] | @Eq (Nat β Real β Real) answer fun (n : Nat) (c : Real) =>
@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@HMul.hMul Real Real Real (@instHMul Real Real.instMul) c (@Nat.cast Real Real.instNatCast (Nat.factorial n)))
(@HMul.hMul Real Real Real (@instHMul Real Real.instMul)
(@HPow.hPow Real Nat Real (@instHPow Real Nat (@Monoid.toNatPow Real Real.instMonoid))
(@OfNat.ofNat Real (nat_lit 3)
(@instOfNatAtLeastTwo Real (nat_lit 3) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))))
n)
(@Finset.prod Int Real Real.instCommMonoid
(@Finset.Icc Int
(@PartialOrder.toPreorder Int
(@OrderedAddCommGroup.toPartialOrder Int
(@StrictOrderedRing.toOrderedAddCommGroup Int
(@LinearOrderedRing.toStrictOrderedRing Int
(@LinearOrderedCommRing.toLinearOrderedRing Int Int.instLinearOrderedCommRing)))))
Int.instLocallyFiniteOrder (@OfNat.ofNat Int (nat_lit 1) (@instOfNat (nat_lit 1)))
(@Nat.cast Int instNatCastInt n))
fun (m : Int) =>
@HSub.hSub Real Real Real (@instHSub Real Real.instSub)
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))
(@HPow.hPow Real Int Real (@instHPow Real Int (@DivInvMonoid.toZPow Real Real.instDivInvMonoid))
(@OfNat.ofNat Real (nat_lit 2)
(@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))))))
(@Neg.neg Int Int.instNegInt m)))) | β β β β β | [
{
"t": "β β β",
"v": null,
"name": "f"
},
{
"t": "Differentiable β f",
"v": null,
"name": "hfdiff"
},
{
"t": "β x > 0, deriv f x = -3 * f x + 6 * f (2 * x)",
"v": null,
"name": "hfderiv"
},
{
"t": "β x β₯ 0, |f x| β€ Real.exp (- βx)",
"v": null,
"name": "hdecay"
},
{
"t": "β β β",
"v": null,
"name": "ΞΌ"
},
{
"t": "β n, ΞΌ n = β« x in Set.Ioi 0, x ^ n * f x",
"v": null,
"name": "ΞΌ_def"
},
{
"t": "(β n, ΞΌ n = answer n (ΞΌ 0)) β§\n (β L, Tendsto (fun n β¦ (ΞΌ n) * 3 ^ n / n !) atTop (π L)) β§\n (Tendsto (fun n β¦ (ΞΌ n) * 3 ^ n / n !) atTop (π 0) β ΞΌ 0 = 0)",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1989_b3",
"tags": [
"analysis"
]
} |
Can a countably infinite set have an uncountable collection of non-empty subsets such that the intersection of any two of them is finite? | True | Prove that such a collection exists. | open Nat Filter Topology Set | [] | @Eq Prop answer True | Prop | [
{
"t": "answer β\n (β S : Type,\n Countable S β§ Infinite S β§\n β C : Set (Set S),\n Β¬Countable C β§\n (β R β C, R β β
) β§\n (β A β C, β B β C, A β B β (A β© B).Finite)\n )",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1989_b4",
"tags": [
"set_theory"
]
} |
Let $T_0=2,T_1=3,T_2=6$, and for $n \geq 3$, $T_n=(n+4)T_{n-1}-4nT_{n-2}+(4n-8)T_{n-3}$. The first few terms are $2,3,6,14,40,152,784,5168,40576$. Find, with proof, a formula for $T_n$ of the form $T_n=A_n+B_n$, where $\{A_n\}$ and $\{B_n\}$ are well-known sequences. | fun n : β => (n)! + 2 ^ n | Show that we have $T_n=n!+2^n$. | open Filter Topology Nat | [] | @Eq (Nat β Nat) answer fun (n : Nat) =>
@HAdd.hAdd Nat Nat Nat (@instHAdd Nat instAddNat) (Nat.factorial n)
(@HPow.hPow Nat Nat Nat (@instHPow Nat Nat (@Monoid.toNatPow Nat Nat.instMonoid))
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) n) | (β β β) | [
{
"t": "β β β",
"v": null,
"name": "T"
},
{
"t": "T 0 = 2 β§ T 1 = 3 β§ T 2 = 6 β§ T 3 = 14",
"v": null,
"name": "hT012"
},
{
"t": "β n : β, T (n + 3) = (n + 7) * T (n + 2) - 4 * (n + 3) * T (n + 1) + (4 * n + 4) * T n",
"v": null,
"name": "hTn"
},
{
"t": "T = answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1990_a1",
"tags": [
"algebra"
]
} |
Is $\sqrt{2}$ the limit of a sequence of numbers of the form $\sqrt[3]{n}-\sqrt[3]{m}$ ($n,m=0,1,2,\dots$)? | True | Show that the answer is yes. | open Filter Topology Nat | [] | @Eq Prop answer True | Prop | [
{
"t": "β β Prop",
"v": null,
"name": "numform"
},
{
"t": "β x : β, numform x β β n m : β, x = n ^ ((1 : β) / 3) - m ^ ((1 : β) / 3)",
"v": null,
"name": "hnumform"
},
{
"t": "answer β (β s : β β β, (β i : β, numform (s i)) β§ Tendsto s atTop (π (Real.sqrt 2)))",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1990_a2",
"tags": [
"analysis"
]
} |
Consider a paper punch that can be centered at any point of the plane and that, when operated, removes from the plane precisely those points whose distance from the center is irrational. How many punches are needed to remove every point? | 3 | Show that three punches are needed. | open Filter Topology Nat | [] | @Eq Nat answer (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3))) | β | [
{
"t": "sInf {n : β | β S : Set (EuclideanSpace β (Fin 2)), S.encard = n β§ β Q : EuclideanSpace β (Fin 2), β P β S, Irrational (dist P Q)} = answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1990_a4",
"tags": [
"set_theory",
"number_theory"
]
} |
If $\mathbf{A}$ and $\mathbf{B}$ are square matrices of the same size such that $\mathbf{ABAB}=\mathbf{0}$, does it follow that $\mathbf{BABA}=\mathbf{0}$? | False | Show that the answer is no. | open Filter Topology Nat | [] | @Eq Prop answer False | Prop | [
{
"t": "answer β\n (β n β₯ 1, β A B : Matrix (Fin n) (Fin n) β,\n A * B * A * B = 0 β B * A * B * A = 0)",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1990_a5",
"tags": [
"linear_algebra"
]
} |
If $X$ is a finite set, let $|X|$ denote the number of elements in $X$. Call an ordered pair $(S,T)$ of subsets of $\{1,2,\dots,n\}$ \emph{admissible} if $s>|T|$ for each $s \in S$, and $t>|S|$ for each $t \in T$. How many admissible ordered pairs of subsets of $\{1,2,\dots,10\}$ are there? Prove your answer. | 17711 | Show that the number of admissible ordered pairs of subsets of $\{1,2,\dots,10\}$ equals the $22$nd Fibonacci number $F_{22}=17711$. | open Filter Topology Nat | [] | @Eq Nat answer (@OfNat.ofNat Nat (nat_lit 17711) (instOfNatNat (nat_lit 17711))) | β | [
{
"t": "((Finset.univ : Finset <| Finset (Set.Icc 1 10) Γ Finset (Set.Icc 1 10)).filter\n fun β¨S, Tβ© β¦ (β s β S, T.card < s) β§ (β t β T, S.card < t)).card =\n answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1990_a6",
"tags": [
"algebra"
]
} |
Find all real-valued continuously differentiable functions $f$ on the real line such that for all $x$, $(f(x))^2=\int_0^x [(f(t))^2+(f'(t))^2]\,dt+1990$. | $\{\sqrt{1990} e^x, -\sqrt{1990} e^x\}$ | Show that there are two such functions, namely $f(x)=\sqrt{1990}e^x$, and $f(x)=-\sqrt{1990}e^x$. | open Filter Topology Nat | [] | @Eq (Set (Real β Real)) answer
(@Insert.insert (Real β Real) (Set (Real β Real)) (@Set.instInsert (Real β Real))
(fun (x : Real) =>
@HMul.hMul Real Real Real (@instHMul Real Real.instMul)
(Real.sqrt
(@OfNat.ofNat Real (nat_lit 1990)
(@instOfNatAtLeastTwo Real (nat_lit 1990) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 1988) (instOfNatNat (nat_lit 1988)))))))
(Real.exp x))
(@Singleton.singleton (Real β Real) (Set (Real β Real)) (@Set.instSingletonSet (Real β Real)) fun (x : Real) =>
@HMul.hMul Real Real Real (@instHMul Real Real.instMul)
(@Neg.neg Real Real.instNeg
(Real.sqrt
(@OfNat.ofNat Real (nat_lit 1990)
(@instOfNatAtLeastTwo Real (nat_lit 1990) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 1988) (instOfNatNat (nat_lit 1988))))))))
(Real.exp x))) | Set (β β β) | [
{
"t": "(β β β) β Prop",
"v": null,
"name": "P"
},
{
"t": "β f, P f β β x,\n (f x) ^ 2 = (β« t in (0 : β)..x, (f t) ^ 2 + (deriv f t) ^ 2) + 1990",
"v": null,
"name": "P_def"
},
{
"t": "β β β",
"v": null,
"name": "f"
},
{
"t": "f β answer β (ContDiff β 1 f β§ P f)",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1990_b1",
"tags": [
"analysis"
]
} |
Let $G$ be a finite group of order $n$ generated by $a$ and $b$. Prove or disprove: there is a sequence $g_1,g_2,g_3,\dots,g_{2n}$ such that
\begin{itemize}
\item[(1)] every element of $G$ occurs exactly twice, and
\item[(2)] $g_{i+1}$ equals $g_ia$ or $g_ib$ for $i=1,2,\dots,2n$. (Interpret $g_{2n+1}$ as $g_1$.)
\end{itemize} | True | Show that such a sequence does exist. | open Filter Topology Nat | [] | @Eq Prop answer True | Prop | [
{
"t": "β (G : Type*) (_ : Fintype G) (_ : Group G) (n : β) (a b : G), (n = Fintype.card G β§ G = Subgroup.closure {a, b} β§ G β Subgroup.closure {a} β§ G β Subgroup.closure {b}) β (β g : β β G, (β x : G, {i : Fin (2 * n) | g i = x}.encard = 2)\n β§ (β i : Fin (2 * n), (g ((i + 1) % (2 * n)) = g i * a) β¨ (g ((i + 1) % (2 * n)) = g i * b))) β answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1990_b4",
"tags": [
"abstract_algebra"
]
} |
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? | True | Show that the answer is yes, such an infinite sequence exists. | open Filter Polynomial Topology Nat | [] | @Eq Prop answer True | Prop | [
{
"t": "β β β",
"v": null,
"name": "a"
},
{
"t": "β i, a i β 0",
"v": null,
"name": "h_nonzero"
},
{
"t": "β n β₯ 1, (β i in Finset.Iic n, a i β’ Polynomial.X ^ i : Polynomial β).roots.toFinset.card = n",
"v": null,
"name": "h_roots"
},
{
"t": "answer β (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’ Polynomial.X ^ i : Polynomial β).roots.toFinset.card = n))",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1990_b5",
"tags": [
"algebra",
"analysis"
]
} |
Let $\mathbf{A}$ and $\mathbf{B}$ be different $n \times n$ matrices with real entries. If $\mathbf{A}^3=\mathbf{B}^3$ and $\mathbf{A}^2\mathbf{B}=\mathbf{B}^2\mathbf{A}$, can $\mathbf{A}^2+\mathbf{B}^2$ be invertible? | False | Show that the answer is no. | open Filter Topology | [] | @Eq Prop answer False | Prop | [
{
"t": "β",
"v": null,
"name": "n"
},
{
"t": "1 β€ n",
"v": null,
"name": "hn"
},
{
"t": "answer β (β A B : Matrix (Fin n) (Fin n) β,\n A β B β§ A ^ 3 = B ^ 3 β§\n A ^ 2 * B = B ^ 2 * A β§\n Nonempty (Invertible (A ^ 2 + B ^ 2)))",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1991_a2",
"tags": [
"linear_algebra"
]
} |
Find all real polynomials $p(x)$ of degree $n \geq 2$ for which there exist real numbers $r_1<r_2<\cdots<r_n$ such that
\begin{enumerate}
\item $p(r_i)=0, \qquad i=1,2,\dots,n$, and
\item $p'(\frac{r_i+r_{i+1}}{2})=0 \qquad i=1,2,\dots,n-1$,
\end{enumerate}
where $p'(x)$ denotes the derivative of $p(x)$. | the set of all real polynomials of degree 2 with two distinct real roots | Show that the real polynomials with the required property are exactly those that are of degree $2$ with $2$ distinct real zeros. | open Filter Topology | [] | @Eq (Set (@Polynomial Real Real.semiring)) answer
(@setOf (@Polynomial Real Real.semiring) fun (p_1 : @Polynomial Real Real.semiring) =>
And
(@Eq (WithBot Nat) (@Polynomial.degree Real Real.semiring p_1)
(@OfNat.ofNat (WithBot Nat) (nat_lit 2)
(@instOfNatAtLeastTwo (WithBot Nat) (nat_lit 2)
(@AddMonoidWithOne.toNatCast (WithBot Nat) (@WithBot.addMonoidWithOne Nat Nat.instAddMonoidWithOne))
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0)))))))
(@Exists Real fun (r1 : Real) =>
@Exists Real fun (r2 : Real) =>
And (@Ne Real r1 r2)
(And
(@Eq Real (@Polynomial.eval Real Real.semiring r1 p_1)
(@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)))
(@Eq Real (@Polynomial.eval Real Real.semiring r2 p_1)
(@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)))))) | Set (Polynomial β) | [
{
"t": "Polynomial β",
"v": null,
"name": "p"
},
{
"t": "β",
"v": null,
"name": "n"
},
{
"t": "n = p.degree",
"v": null,
"name": "hn"
},
{
"t": "n β₯ 2",
"v": null,
"name": "hge"
},
{
"t": "p β answer β\n (β r : β β β, (β i : Fin (n - 1), r i < r (i + 1)) β§\n (β i : Fin n, p.eval (r i) = 0) β§\n (β i : Fin (n - 1), (Polynomial.derivative p).eval ((r i + r (i + 1)) / 2) = 0))",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1991_a3",
"tags": [
"algebra",
"analysis"
]
} |
Does there exist an infinite sequence of closed discs $D_1,D_2,D_3,\dots$ in the plane, with centers $c_1,c_2,c_3,\dots$, respectively, such that
\begin{enumerate}
\item the $c_i$ have no limit point in the finite plane,
\item the sum of the areas of the $D_i$ is finite, and
\item every line in the plane intersects at least one of the $D_i$?
\end{enumerate} | True | Show that the answer is yes, such a sequence of closed discs exists. | open Filter Metric Topology | [] | @Eq Prop answer True | Prop | [
{
"t": "(β (c : β β EuclideanSpace β (Fin 2)) (r : β β β),\n (Β¬ β p, MapClusterPt p atTop c) β§\n (Summable <| fun i β¦ r i ^ 2) β§\n (β L : AffineSubspace β (EuclideanSpace β (Fin 2)),\n Module.finrank β L.direction = 1 β β i, (βL β© closedBall (c i) (r i)).Nonempty)) β\n answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1991_a4",
"tags": [
"geometry",
"analysis"
]
} |
Find the maximum value of $\int_0^y \sqrt{x^4+(y-y^2)^2}\,dx$ for $0 \leq y \leq 1$. | $\frac{1}{3}$ | Show that the maximum value of the integral is $1/3$. | open Filter Topology | [] | @Eq Real answer
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))
(@OfNat.ofNat Real (nat_lit 3)
(@instOfNatAtLeastTwo Real (nat_lit 3) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))))) | β | [
{
"t": "Set.Icc (0 : β) 1 β β",
"v": null,
"name": "f"
},
{
"t": "β y : Set.Icc 0 1, f y = β« x in Set.Ioo 0 y, Real.sqrt (x ^ 4 + (y - y ^ 2) ^ 2)",
"v": null,
"name": "hf"
},
{
"t": "IsGreatest (f '' (Set.Icc 0 1)) answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1991_a5",
"tags": [
"analysis"
]
} |
For each integer $n \geq 0$, let $S(n)=n-m^2$, where $m$ is the greatest integer with $m^2 \leq n$. Define a sequence $(a_k)_{k=0}^\infty$ by $a_0=A$ and $a_{k+1}=a_k+S(a_k)$ for $k \geq 0$. For what positive integers $A$ is this sequence eventually constant? | the set of positive integers $A$ such that $A$ is a perfect square | Show that this sequence is eventually constant if and only if $A$ is a perfect square. | open Filter Topology | [] | @Eq (Set Int) answer
(@setOf Int fun (A_1 : Int) =>
@Exists Int fun (x : Int) =>
And (@GT.gt Int Int.instLTInt x (@OfNat.ofNat Int (nat_lit 0) (@instOfNat (nat_lit 0))))
(@Eq Int A_1
(@HPow.hPow Int Nat Int (@instHPow Int Nat (@Monoid.toNatPow Int Int.instMonoid)) x
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))) | Set β€ | [
{
"t": "β€ β β€",
"v": null,
"name": "m"
},
{
"t": "β€ β β€",
"v": null,
"name": "S"
},
{
"t": "β€",
"v": null,
"name": "A"
},
{
"t": "β β β€",
"v": null,
"name": "a"
},
{
"t": "β n, 0 β€ n β (m n) ^ 2 β€ n β§ (β m' : β€, m' ^ 2 β€ n β m' β€ m n)",
"v": null,
"name": "hm"
},
{
"t": "β n, 0 β€ n β S n = n - (m n) ^ 2",
"v": null,
"name": "hS"
},
{
"t": "a 0 = A β§ (β k, a (k + 1) = a k + S (a k))",
"v": null,
"name": "ha"
},
{
"t": "A > 0",
"v": null,
"name": "hA"
},
{
"t": "(β (K : β) (c : β), β k β₯ K, a k = c) β A β answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1991_b1",
"tags": [
"algebra"
]
} |
Let $p$ be an odd prime and let $\mathbb{Z}_p$ denote (the field of) integers modulo $p$. How many elements are in the set $\{x^2:x \in \mathbb{Z}_p\} \cap \{y^2+1:y \in \mathbb{Z}_p\}$? | $\lceil p / 4 \rceil$ | Show that the number of elements in the intersection is $\lceil p/4 \rceil$. | open Filter Topology | [] | @Eq (Nat β Nat) answer fun (p_1 : Nat) =>
@Nat.ceil Real Real.orderedSemiring (@FloorRing.toFloorSemiring Real Real.instLinearOrderedRing Real.instFloorRing)
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@Nat.cast Real Real.instNatCast p_1)
(@OfNat.ofNat Real (nat_lit 4)
(@instOfNatAtLeastTwo Real (nat_lit 4) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))))) | β β β | [
{
"t": "β",
"v": null,
"name": "p"
},
{
"t": "Odd p",
"v": null,
"name": "podd"
},
{
"t": "Prime p",
"v": null,
"name": "pprime"
},
{
"t": "({z : ZMod p | β x : ZMod p, z = x ^ 2} β© {z : ZMod p | β y : ZMod p, z = y ^ 2 + 1}).encard = answer p",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1991_b5",
"tags": [
"number_theory"
]
} |
Let $a$ and $b$ be positive numbers. Find the largest number $c$, in terms of $a$ and $b$, such that $a^xb^{1-x} \leq a\frac{\sinh ux}{\sinh u}+b\frac{\sinh u(1-x)}{\sinh u}$ for all $u$ with $0<|u| \leq c$ and for all $x$, $0<x<1$. (Note: $\sinh u=(e^u-e^{-u})/2$.) | (fun a b : β => |Real.log (a / b)|) | Show that the largest $c$ for which the inequality holds for $0<|u| \leq c$ is $c=|\ln(a/b)|$. | open Filter Topology | [] | @Eq (Real β Real β Real) answer fun (a_1 b_1 : Real) =>
@abs Real Real.lattice Real.instAddGroup
(Real.log (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) a_1 b_1)) | β β β β β | [
{
"t": "β",
"v": null,
"name": "a"
},
{
"t": "β",
"v": null,
"name": "b"
},
{
"t": "a > 0 β§ b > 0",
"v": null,
"name": "abpos"
},
{
"t": "IsGreatest {c | β u, (0 < |u| β§ |u| β€ c) β (β x β Set.Ioo 0 1, a ^ x * b ^ (1 - x) β€ a * (Real.sinh (u * x) / Real.sinh u) + b * (Real.sinh (u * (1 - x)) / Real.sinh u))} (answer a b)",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1991_b6",
"tags": [
"analysis"
]
} |
Define $C(\alpha)$ to be the coefficient of $x^{1992}$ in the power series about $x=0$ of $(1 + x)^\alpha$. Evaluate
\[
\int_0^1 \left( C(-y-1) \sum_{k=1}^{1992} \frac{1}{y+k} \right)\,dy.
\] | 1992 | Prove that the integral evaluates to $1992$. | open Topology Filter | [] | @Eq Real answer
(@OfNat.ofNat Real (nat_lit 1992)
(@instOfNatAtLeastTwo Real (nat_lit 1992) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 1990) (instOfNatNat (nat_lit 1990)))))) | β | [
{
"t": "β β β",
"v": null,
"name": "C"
},
{
"t": "C = fun Ξ± β¦ taylorCoeffWithin (fun x β¦ (1 + x) ^ Ξ±) 1992 Set.univ 0",
"v": null,
"name": "hC"
},
{
"t": "β« y in (0)..1, C (-y - 1) * β k in Finset.Icc (1 : β) 1992, 1 / (y + k) = answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1992_a2",
"tags": [
"analysis",
"algebra"
]
} |
For a given positive integer $m$, find all triples $(n, x, y)$ of positive integers, with $n$ relatively prime to $m$, which satisfy
\[
(x^2 + y^2)^m = (xy)^n.
\] | $\emptyset$ if $m$ is odd, otherwise $\{(m + 1, 2^{m/2}, 2^{m/2})\}$ | Prove that if $m$ is odd, there are no solutions, and if $m$ is even, the only solution is
$(n, x, y) = (m + 1, 2 ^ {m/2}, 2 ^{m/2})$. | open Topology Filter Nat | [] | @Eq (Nat β Set (Prod Nat (Prod Nat Nat))) answer fun (m_1 : Nat) =>
@ite (Set (Prod Nat (Prod Nat Nat))) (@Odd Nat Nat.instSemiring m_1) (Nat.instDecidablePredOdd m_1)
(@EmptyCollection.emptyCollection (Set (Prod Nat (Prod Nat Nat)))
(@Set.instEmptyCollection (Prod Nat (Prod Nat Nat))))
(@Singleton.singleton (Prod Nat (Prod Nat Nat)) (Set (Prod Nat (Prod Nat Nat)))
(@Set.instSingletonSet (Prod Nat (Prod Nat Nat)))
(@Prod.mk Nat (Prod Nat Nat)
(@HAdd.hAdd Nat Nat Nat (@instHAdd Nat instAddNat) m_1
(@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))
(@Prod.mk Nat Nat
(@HPow.hPow Nat Nat Nat (@instHPow Nat Nat (@Monoid.toNatPow Nat Nat.instMonoid))
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))
(@HDiv.hDiv Nat Nat Nat (@instHDiv Nat Nat.instDiv) m_1
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))
(@HPow.hPow Nat Nat Nat (@instHPow Nat Nat (@Monoid.toNatPow Nat Nat.instMonoid))
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))
(@HDiv.hDiv Nat Nat Nat (@instHDiv Nat Nat.instDiv) m_1
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))))) | β β Set (β Γ β Γ β) | [
{
"t": "β",
"v": null,
"name": "m"
},
{
"t": "m > 0",
"v": null,
"name": "mpos"
},
{
"t": "Set (β Γ β Γ β)",
"v": null,
"name": "S"
},
{
"t": "β n x y : β, (n, x, y) β S β n > 0 β§ x > 0 β§ y > 0 β§ Coprime n m β§ (x ^ 2 + y ^ 2) ^ m = (x * y) ^ n",
"v": null,
"name": "hS"
},
{
"t": "S = answer m",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1992_a3",
"tags": [
"algebra",
"number_theory"
]
} |
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 $f^{(k)}(0), k = 1, 2, 3, \dots$. | fun k β¦ ite (Even k) ((-1) ^ (k / 2) * factorial k) 0 | Prove that
\[
f^{(k)}(0) =
\begin{cases}
(-1)^{k/2}k! & \text{if $k$ is even;} \\
0 & \text{if $k$ is odd.} \\
\end{cases}
\] | open Topology Filter Nat Function | [] | @Eq (Nat β Real) answer fun (k : Nat) =>
@ite Real (@Even Nat instAddNat k) (Nat.instDecidablePredEven k)
(@HMul.hMul Real Real Real (@instHMul Real Real.instMul)
(@HPow.hPow Real Nat Real (@instHPow Real Nat (@Monoid.toNatPow Real Real.instMonoid))
(@Neg.neg Real Real.instNeg (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)))
(@HDiv.hDiv Nat Nat Nat (@instHDiv Nat Nat.instDiv) k
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))
(@Nat.cast Real Real.instNatCast (Nat.factorial k)))
(@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) | β β β | [
{
"t": "β β β",
"v": null,
"name": "f"
},
{
"t": "ContDiff β β€ f",
"v": null,
"name": "hfdiff"
},
{
"t": "β n : β, n > 0 β f (1 / n) = n ^ 2 / (n ^ 2 + 1)",
"v": null,
"name": "hf"
},
{
"t": "β k : β, k > 0 β iteratedDeriv k f 0 = answer k",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1992_a4",
"tags": [
"analysis"
]
} |
Let $S$ be a set of $n$ distinct real numbers. Let $A_S$ be the set of numbers that occur as averages of two distinct elements of $S$. For a given $n \geq 2$, what is the smallest possible number of elements in $A_S$? | fun n β¦ 2 * n - 3 | Show that the answer is $2n - 3$. | open Topology Filter Nat Function | [] | @Eq (Nat β Int) answer fun (n_1 : Nat) =>
@HSub.hSub Int Int Int (@instHSub Int Int.instSub)
(@HMul.hMul Int Int Int (@instHMul Int Int.instMul) (@OfNat.ofNat Int (nat_lit 2) (@instOfNat (nat_lit 2)))
(@Nat.cast Int instNatCastInt n_1))
(@OfNat.ofNat Int (nat_lit 3) (@instOfNat (nat_lit 3))) | β β β€ | [
{
"t": "β",
"v": null,
"name": "n"
},
{
"t": "n β₯ 2",
"v": null,
"name": "hn"
},
{
"t": "Finset β β Set β",
"v": null,
"name": "A"
},
{
"t": "A = fun S β¦ {x | β a β S, β b β S, a β b β§ (a + b) / 2 = x}",
"v": null,
"name": "hA"
},
{
"t": "IsLeast {k : β€ | β S : Finset β, S.card = n β§ k = (A S).ncard} (answer n)",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1992_b1",
"tags": [
"algebra"
]
} |
For any pair $(x,y)$ of real numbers, a sequence $(a_n(x,y))_{n \geq 0}$ is defined as follows:
\begin{align*}
a_0(x,y)&=x, \\
a_{n+1}(x,y)&=\frac{(a_n(x,y))^2+y^2}{2},\text{ for $n \geq 0$.}
\end{align*}
Find the area of the region $\{(x,y) \mid (a_n(x,y))_{n \geq 0}\text{ converges}\}$. | 4 + Ο | Show that the area is $4+\pi$. | open Topology Filter Nat Function Polynomial | [] | @Eq Real answer
(@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd)
(@OfNat.ofNat Real (nat_lit 4)
(@instOfNatAtLeastTwo Real (nat_lit 4) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))))
Real.pi) | β | [
{
"t": "(Fin 2 β β) β (β β β)",
"v": null,
"name": "a"
},
{
"t": "β p, (a p) 0 = p 0 β§ (β n, (a p) (n + 1) = (((a p) n) ^ 2 + (p 1) ^ 2) / 2)",
"v": null,
"name": "ha"
},
{
"t": "answer = (MeasureTheory.volume {p | β L, Tendsto (a p) atTop (π L)}).toReal",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1992_b3",
"tags": [
"geometry",
"analysis"
]
} |
Let $p(x)$ be a nonzero polynomial of degree less than $1992$ having no nonconstant factor in common with $x^3 - x$. Let
\[
\frac{d^{1992}}{dx^{1992}} \left( \frac{p(x)}{x^3 - x} \right) = \frac{f(x)}{g(x)}
\]
for polynomials $f(x)$ and $g(x)$. Find the smallest possible degree of $f(x)$. | 3984 | Show that the minimum degree is $3984$. | open Topology Filter Nat Function Polynomial | [] | @Eq Nat answer (@OfNat.ofNat Nat (nat_lit 3984) (instOfNatNat (nat_lit 3984))) | β | [
{
"t": "Polynomial β β Prop",
"v": null,
"name": "IsValid"
},
{
"t": "Polynomial β β Polynomial β β Prop",
"v": null,
"name": "pair"
},
{
"t": "β p, IsValid p β p β 0 β§ p.degree < 1992 β§ IsCoprime p (X ^ 3 - X)",
"v": null,
"name": "IsValid_def"
},
{
"t": "β p f, pair p f β β g : Polynomial β, iteratedDeriv 1992 (fun x β¦ p.eval x / (x ^ 3 - x)) = fun x β¦ f.eval x / g.eval x",
"v": null,
"name": "hpair"
},
{
"t": "IsLeast {k : β | β p f, IsValid p β§ pair p f β§ k = f.degree} answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1992_b4",
"tags": [
"algebra"
]
} |
Let $D_n$ denote the value of the $(n-1) \times (n-1)$ determinant
\[
\left[
\begin{array}{cccccc}
3 & 1 & 1 & 1 & \cdots & 1 \\
1 & 4 & 1 & 1 & \cdots & 1 \\
1 & 1 & 5 & 1 & \cdots & 1 \\
1 & 1 & 1 & 6 & \cdots & 1 \\
\vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\
1 & 1 & 1 & 1 & \cdots & n+1
\end{array}
\right].
\]
Is the set $\left\{ \frac{D_n}{n!} \right\}_{n \geq 2}$ bounded? | False | Prove that the set is not bounded. | open Topology Filter Nat Function Polynomial | [] | @Eq Prop answer False | Prop | [
{
"t": "β β β",
"v": null,
"name": "D"
},
{
"t": "β n, D n = Matrix.det (fun i j : Fin (n - 1) β¦ ite (i = j) ((i : β) + 3 : β) 1)",
"v": null,
"name": "hD"
},
{
"t": "answer β (Bornology.IsBounded {x | β n β₯ 2, D n / factorial n = x})",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1992_b5",
"tags": [
"linear_algebra",
"analysis"
]
} |
The horizontal line $y=c$ intersects the curve $y=2x-3x^3$ in the first quadrant as in the figure. Find $c$ so that the areas of the two shaded regions are equal. [Figure not included. The first region is bounded by the $y$-axis, the line $y=c$ and the curve; the other lies under the curve and above the line $y=c$ between their two points of intersection.] | 4/9 | Show that the area of the two regions are equal when $c=4/9$. | null | [] | @Eq Real answer
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@OfNat.ofNat Real (nat_lit 4)
(@instOfNatAtLeastTwo Real (nat_lit 4) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))))
(@OfNat.ofNat Real (nat_lit 9)
(@instOfNatAtLeastTwo Real (nat_lit 9) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 7) (instOfNatNat (nat_lit 7))))))) | β | [
{
"t": "0 < answer β§ answer < (4 * Real.sqrt 2) / 9 β§\n (β« x in Set.Ioo 0 ((Real.sqrt 2) / 3), max (answer - (2 * x - 3 * x ^ 3)) 0) =\n (β« x in Set.Ioo 0 ((Real.sqrt 6) / 3), max ((2 * x - 3 * x ^ 3) - answer) 0)",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1993_a1",
"tags": [
"analysis",
"algebra"
]
} |
Find the smallest positive integer $n$ such that for every integer $m$ with $0<m<1993$, there exists an integer $k$ for which $\frac{m}{1993}<\frac{k}{n}<\frac{m+1}{1994}$. | 3987 | Show that the smallest positive integer $n$ satisfying the condition is $n=3987$. | null | [] | @Eq Nat answer (@OfNat.ofNat Nat (nat_lit 3987) (instOfNatNat (nat_lit 3987))) | β | [
{
"t": "IsLeast\n {n : β | 0 < n β§\n β m β Set.Ioo (0 : β€) (1993), β k : β€,\n (m / 1993 < (k : β) / n) β§ ((k : β) / n < (m + 1) / 1994) }\n answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1993_b1",
"tags": [
"algebra"
]
} |
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. | (5/4, -1/4) | Show that the limit is $(5-\pi)/4$. That is, $r=5/4$ and $s=-1/4$. | null | [] | @Eq (Prod Rat Rat) answer
(@Prod.mk Rat Rat
(@HDiv.hDiv Rat Rat Rat (@instHDiv Rat Rat.instDiv) (@OfNat.ofNat Rat (nat_lit 5) (@Rat.instOfNat (nat_lit 5)))
(@OfNat.ofNat Rat (nat_lit 4) (@Rat.instOfNat (nat_lit 4))))
(@HDiv.hDiv Rat Rat Rat (@instHDiv Rat Rat.instDiv)
(@Neg.neg Rat Rat.instNeg (@OfNat.ofNat Rat (nat_lit 1) (@Rat.instOfNat (nat_lit 1))))
(@OfNat.ofNat Rat (nat_lit 4) (@Rat.instOfNat (nat_lit 4))))) | β Γ β | [
{
"t": "(MeasureTheory.volume\n {p : Fin 2 β β | 0 < p 0 β§ p 0 < 1 β§ 0 < p 1 β§ p 1 < 1 β§ Even (round (p 0 / p 1))}\n ).toReal = answer.1 + answer.2 * Real.pi",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1993_b3",
"tags": [
"probability",
"number_theory",
"geometry"
]
} |
Find all positive integers $n$ that are within $250$ of exactly $15$ perfect squares. | the set of integers $n$ such that $315 \leq n \leq 325$ or $332 \leq n \leq 350$ | Show that an integer $n$ is within $250$ of exactly $15$ perfect squares if and only if either $315 \leq n \leq 325$ or $332 \leq n \leq 350$. | open Filter Topology | [] | @Eq (Set Int) answer
(@setOf Int fun (n_1 : Int) =>
Or
(And (@LE.le Int Int.instLEInt (@OfNat.ofNat Int (nat_lit 315) (@instOfNat (nat_lit 315))) n_1)
(@LE.le Int Int.instLEInt n_1 (@OfNat.ofNat Int (nat_lit 325) (@instOfNat (nat_lit 325)))))
(And (@LE.le Int Int.instLEInt (@OfNat.ofNat Int (nat_lit 332) (@instOfNat (nat_lit 332))) n_1)
(@LE.le Int Int.instLEInt n_1 (@OfNat.ofNat Int (nat_lit 350) (@instOfNat (nat_lit 350)))))) | Set β€ | [
{
"t": "β€",
"v": null,
"name": "n"
},
{
"t": "n β answer β (0 < n β§ {m : β | |n - m ^ 2| β€ 250}.encard = 15)",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1994_b1",
"tags": [
"algebra"
]
} |
For which real numbers $c$ is there a straight line that intersects the curve $x^4+9x^3+cx^2+9x+4$ in four distinct points? | {c : β | c < 243 / 8} | Show that there exists such a line if and only if $c<243/8$. | open Filter Topology | [] | @Eq (Set Real) answer
(@setOf Real fun (c_1 : Real) =>
@LT.lt Real Real.instLT c_1
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@OfNat.ofNat Real (nat_lit 243)
(@instOfNatAtLeastTwo Real (nat_lit 243) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 241) (instOfNatNat (nat_lit 241))))))
(@OfNat.ofNat Real (nat_lit 8)
(@instOfNatAtLeastTwo Real (nat_lit 8) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))))))) | Set β | [
{
"t": "β",
"v": null,
"name": "c"
},
{
"t": "answer = {c : β | (β m b : β,\n {x : β | m * x + b = x ^ 4 + 9 * x ^ 3 + c * x ^ 2 + 9 * x + 4}.encard = 4)}",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1994_b2",
"tags": [
"geometry",
"algebra"
]
} |
Find the set of all real numbers $k$ with the following property: For any positive, differentiable function $f$ that satisfies $f'(x)>f(x)$ for all $x$, there is some number $N$ such that $f(x)>e^{kx}$ for all $x>N$. | the set of all real numbers less than 1 | Show that the desired set is $(-\infty,1)$. | open Filter Topology | [] | @Eq (Set Real) answer
(@Set.Iio Real Real.instPreorder (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))) | Set β | [
{
"t": "{k | β f (hf : (β x, 0 < f x β§ f x < deriv f x) β§ Differentiable β f),\n β N, β x > N, Real.exp (k * x) < f x} = answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1994_b3",
"tags": [
"analysis"
]
} |
For what pairs $(a,b)$ of positive real numbers does the improper integral \[ \int_{b}^{\infty} \left( \sqrt{\sqrt{x+a}-\sqrt{x}} - \sqrt{\sqrt{x}-\sqrt{x-b}} \right)\,dx \] converge? | {x | let β¨a,bβ© := x; a = b} | Show that the solution is those pairs $(a,b)$ where $a = b$. | open Filter Topology Real | [] | @Eq (Set (Prod Real Real)) answer
(@setOf (Prod Real Real) fun (x : Prod Real Real) =>
_example.match_1 (fun (x_1 : Prod Real Real) => Prop) x fun (a b : Real) => @Eq Real a b) | Set (β Γ β) | [
{
"t": "(β Γ β) β Prop",
"v": null,
"name": "habconv"
},
{
"t": "habconv = fun β¨a,bβ© => β limit : β, Tendsto (fun t : β => β« x in (Set.Icc b t), (sqrt (sqrt (x + a) - sqrt x) - sqrt (sqrt x - sqrt (x - b)))) atTop (π limit)",
"v": null,
"name": "habconv_def"
},
{
"t": "β ab : β Γ β, ab.1 > 0 β§ ab.2 > 0 β (habconv ab β ab β answer)",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1995_a2",
"tags": [
"analysis"
]
} |
Let $x_{1},x_{2},\dots,x_{n}$ be differentiable (real-valued) functions of a single variable $f$ which satisfy \begin{align*} \frac{dx_{1}}{dt} &= a_{11}x_{1} + a_{12}x_{2} + \cdots + a_{1n}x_{n} \ \frac{dx_{2}}{dt} &= a_{21}x_{1} + a_{22}x_{2} + \cdots + a_{2n}x_{n} \ \vdots && \vdots \ \frac{dx_{n}}{dt} &= a_{n1}x_{1} + a_{n2}x_{2} + \cdots + a_{nn}x_{n} \end{align*} for some constants $a_{ij}>0$. Suppose that for all $i$, $x_{i}(t) \to 0$ as $t \to \infty$. Are the functions $x_{1},x_{2},\dots,x_{n}$ necessarily linearly dependent? | True | Show that the answer is yes, the functions must be linearly dependent. | open Filter Topology Real | [] | @Eq Prop answer True | Prop | [
{
"t": "β",
"v": null,
"name": "n"
},
{
"t": "Fin n β (β β β)",
"v": null,
"name": "x"
},
{
"t": "Fin n β Fin n β β",
"v": null,
"name": "a"
},
{
"t": "0 < n",
"v": null,
"name": "npos"
},
{
"t": "β i, Differentiable β (x i)",
"v": null,
"name": "hdiff"
},
{
"t": "β i j, a i j > 0",
"v": null,
"name": "hpos"
},
{
"t": "β t i, (deriv (x i)) t = β j : Fin n, (a i j) * ((x j) t)",
"v": null,
"name": "hsys"
},
{
"t": "β i, Tendsto (x i) atTop (π 0)",
"v": null,
"name": "htendsto"
},
{
"t": "answer β Β¬(β b : Fin n β β, (β t : β, β i : Fin n, (b i) * ((x i) t) = 0) β (β i, b i = 0))",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1995_a5",
"tags": [
"linear_algebra",
"analysis"
]
} |
To each positive integer with $n^{2}$ decimal digits, we associate the determinant of the matrix obtained by writing the digits in order across the rows. For example, for $n=2$, to the integer 8617 we associate $\det \left( \begin{array}{cc} 8 & 6 \ 1 & 7 \end{array} \right) = 50$. Find, as a function of $n$, the sum of all the determinants associated with $n^{2}$-digit integers. (Leading digits are assumed to be nonzero; for example, for $n=2$, there are 9000 determinants.) | 45 if n = 1, 10 * 45Β² if n = 2, and 0 otherwise | Show that the solution is $45$ if $n = 1$, $45^2*10$ if $n = 2$, and $0$ if $n$ is greater than 2. | open Filter Topology Real Nat | [] | @Eq (Nat β Int) answer fun (n_1 : Nat) =>
@ite Int (@Eq Nat n_1 (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))
(instDecidableEqNat n_1 (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))
(@OfNat.ofNat Int (nat_lit 45) (@instOfNat (nat_lit 45)))
(@ite Int (@Eq Nat n_1 (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))
(instDecidableEqNat n_1 (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))
(@HMul.hMul Int Int Int (@instHMul Int Int.instMul) (@OfNat.ofNat Int (nat_lit 10) (@instOfNat (nat_lit 10)))
(@HPow.hPow Int Nat Int (@instHPow Int Nat (@Monoid.toNatPow Int Int.instMonoid))
(@OfNat.ofNat Int (nat_lit 45) (@instOfNat (nat_lit 45)))
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))
(@OfNat.ofNat Int (nat_lit 0) (@instOfNat (nat_lit 0)))) | β β β€ | [
{
"t": "β",
"v": null,
"name": "n"
},
{
"t": "n > 0",
"v": null,
"name": "hn"
},
{
"t": "Set (β β β)",
"v": null,
"name": "digits_set"
},
{
"t": "digits_set = {f | f 0 β 0 β§ (β i : Fin (n ^ 2), f i β€ 9) β§ (β i β₯ n ^ 2, f i = 0)}",
"v": null,
"name": "hdigits_set"
},
{
"t": "(β β β) β Matrix (Fin n) (Fin n) β€",
"v": null,
"name": "digits_to_matrix"
},
{
"t": "digits_to_matrix = fun f => (fun i j => f (i.1 * n + j.1))",
"v": null,
"name": "hdigits_to_matrix"
},
{
"t": "β' f : digits_set, (digits_to_matrix f).det = answer n",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1995_b3",
"tags": [
"linear_algebra"
]
} |
Evaluate \[ \sqrt[8]{2207 - \frac{1}{2207-\frac{1}{2207-\dots}}}. \] Express your answer in the form $\frac{a+b\sqrt{c}}{d}$, where $a,b,c,d$ are integers. | β¨3,1,5,2β© | Show that the solution is $(3 + 1*\sqrt{5})/2. | open Filter Topology Real Nat | [] | @Eq (Prod Int (Prod Int (Prod Int Int))) answer
(@Prod.mk Int (Prod Int (Prod Int Int)) (@OfNat.ofNat Int (nat_lit 3) (@instOfNat (nat_lit 3)))
(@Prod.mk Int (Prod Int Int) (@OfNat.ofNat Int (nat_lit 1) (@instOfNat (nat_lit 1)))
(@Prod.mk Int Int (@OfNat.ofNat Int (nat_lit 5) (@instOfNat (nat_lit 5)))
(@OfNat.ofNat Int (nat_lit 2) (@instOfNat (nat_lit 2)))))) | β€ Γ β€ Γ β€ Γ β€ | [
{
"t": "β",
"v": null,
"name": "contfrac"
},
{
"t": "contfrac = 2207 - 1 / contfrac",
"v": null,
"name": "hcontfrac"
},
{
"t": "1 < contfrac",
"v": null,
"name": "hcontfrac'"
},
{
"t": "let β¨a, b, c, dβ© := answer;\n contfrac ^ ((1 : β) / 8) = (a + b * sqrt c) / d",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1995_b4",
"tags": [
"algebra"
]
} |
Let $C_1$ and $C_2$ be circles whose centers are $10$ units apart, and whose radii are $1$ and $3$. Find, with proof, the locus of all points $M$ for which there exist points $X$ on $C_1$ and $Y$ on $C_2$ such that $M$ is the midpoint of the line segment $XY$. | the set of points $p$ such that the distance from $p$ to the midpoint of $O_1$ and $O_2$ is between 1 and 2 | Let $O_1$ and $O_2$ be the centers of $C_1$ and $C_2$, respectively. Then show that the desired locus is an annulus centered at the midpoint $O$ of $O_1O_2$, with inner radius $1$ and outer radius $2$. | open Metric | [] | @Eq
(EuclideanSpace Real (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) β
EuclideanSpace Real (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) β
Set (EuclideanSpace Real (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))))
answer fun (O1_1 O2_1 : EuclideanSpace Real (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))) =>
@setOf (EuclideanSpace Real (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))
fun (p : EuclideanSpace Real (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))) =>
And
(@GE.ge Real Real.instLE
(@Dist.dist (EuclideanSpace Real (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))
(@PiLp.instDist
(@OfNat.ofNat ENNReal (nat_lit 2)
(@instOfNatAtLeastTwo ENNReal (nat_lit 2)
(@AddMonoidWithOne.toNatCast ENNReal
(@AddCommMonoidWithOne.toAddMonoidWithOne ENNReal instENNRealAddCommMonoidWithOne))
EuclideanSpace.proof_1))
(Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))
(fun (x : Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) => Real)
(Fin.fintype (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))
fun (i : Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) =>
@PseudoMetricSpace.toDist Real Real.pseudoMetricSpace)
p
(@midpoint Real (EuclideanSpace Real (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))
(EuclideanSpace Real (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))) Real.instRing
(@invertibleTwo Real Real.instDivisionRing (@RCLike.charZero_rclike Real Real.instRCLike))
(@WithLp.instAddCommGroup
(@OfNat.ofNat ENNReal (nat_lit 2)
(@instOfNatAtLeastTwo ENNReal (nat_lit 2)
(@AddMonoidWithOne.toNatCast ENNReal
(@AddCommMonoidWithOne.toAddMonoidWithOne ENNReal instENNRealAddCommMonoidWithOne))
EuclideanSpace.proof_1))
((i : Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) β
(fun (x : Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) => Real) i)
(@Pi.addCommGroup (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))
(fun (x : Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) => Real)
fun (i : Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) => Real.instAddCommGroup))
(@WithLp.instModule
(@OfNat.ofNat ENNReal (nat_lit 2)
(@instOfNatAtLeastTwo ENNReal (nat_lit 2)
(@AddMonoidWithOne.toNatCast ENNReal
(@AddCommMonoidWithOne.toAddMonoidWithOne ENNReal instENNRealAddCommMonoidWithOne))
EuclideanSpace.proof_1))
Real
((i : Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) β
(fun (x : Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) => Real) i)
(@Ring.toSemiring Real Real.instRing)
(@Pi.addCommGroup (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))
(fun (x : Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) => Real)
fun (i : Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) => Real.instAddCommGroup)
(@Pi.Function.module (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real Real
(@Ring.toSemiring Real Real.instRing)
(@NonUnitalNonAssocSemiring.toAddCommMonoid Real
(@NonUnitalSemiring.toNonUnitalNonAssocSemiring Real
(@Semiring.toNonUnitalSemiring Real (@Ring.toSemiring Real Real.instRing))))
(@NormedSpace.toModule Real Real Real.normedField
(@NonUnitalSeminormedRing.toSeminormedAddCommGroup Real
(@NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing Real
(@SeminormedCommRing.toNonUnitalSeminormedCommRing Real
(@NormedCommRing.toSeminormedCommRing Real Real.normedCommRing))))
(@InnerProductSpace.toNormedSpace Real Real Real.instRCLike
(@NonUnitalSeminormedRing.toSeminormedAddCommGroup Real
(@NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing Real
(@SeminormedCommRing.toNonUnitalSeminormedCommRing Real
(@NormedCommRing.toSeminormedCommRing Real Real.normedCommRing))))
(@RCLike.toInnerProductSpaceReal Real Real.instRCLike)))))
(@NormedAddTorsor.toAddTorsor
(EuclideanSpace Real (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))
(EuclideanSpace Real (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))
(@PiLp.seminormedAddCommGroup
(@OfNat.ofNat ENNReal (nat_lit 2)
(@instOfNatAtLeastTwo ENNReal (nat_lit 2)
(@AddMonoidWithOne.toNatCast ENNReal
(@AddCommMonoidWithOne.toAddMonoidWithOne ENNReal instENNRealAddCommMonoidWithOne))
EuclideanSpace.proof_1))
(Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))
(fun (x : Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) => Real)
fact_one_le_two_ennreal (Fin.fintype (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))
fun (i : Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) =>
@NonUnitalSeminormedRing.toSeminormedAddCommGroup Real
(@NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing Real
(@SeminormedCommRing.toNonUnitalSeminormedCommRing Real
(@NormedCommRing.toSeminormedCommRing Real Real.normedCommRing))))
(@SeminormedAddCommGroup.toPseudoMetricSpace
(EuclideanSpace Real (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))
(@PiLp.seminormedAddCommGroup
(@OfNat.ofNat ENNReal (nat_lit 2)
(@instOfNatAtLeastTwo ENNReal (nat_lit 2)
(@AddMonoidWithOne.toNatCast ENNReal
(@AddCommMonoidWithOne.toAddMonoidWithOne ENNReal instENNRealAddCommMonoidWithOne))
EuclideanSpace.proof_1))
(Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))
(fun (x : Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) => Real)
fact_one_le_two_ennreal (Fin.fintype (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))
fun (i : Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) =>
@NonUnitalSeminormedRing.toSeminormedAddCommGroup Real
(@NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing Real
(@SeminormedCommRing.toNonUnitalSeminormedCommRing Real
(@NormedCommRing.toSeminormedCommRing Real Real.normedCommRing)))))
(@SeminormedAddCommGroup.toNormedAddTorsor
(EuclideanSpace Real (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))
(@PiLp.seminormedAddCommGroup
(@OfNat.ofNat ENNReal (nat_lit 2)
(@instOfNatAtLeastTwo ENNReal (nat_lit 2)
(@AddMonoidWithOne.toNatCast ENNReal
(@AddCommMonoidWithOne.toAddMonoidWithOne ENNReal instENNRealAddCommMonoidWithOne))
EuclideanSpace.proof_1))
(Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))
(fun (x : Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) => Real)
fact_one_le_two_ennreal (Fin.fintype (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))
fun (i : Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) =>
@NonUnitalSeminormedRing.toSeminormedAddCommGroup Real
(@NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing Real
(@SeminormedCommRing.toNonUnitalSeminormedCommRing Real
(@NormedCommRing.toSeminormedCommRing Real Real.normedCommRing))))))
O1_1 O2_1))
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)))
(@LE.le Real Real.instLE
(@Dist.dist (EuclideanSpace Real (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))
(@PiLp.instDist
(@OfNat.ofNat ENNReal (nat_lit 2)
(@instOfNatAtLeastTwo ENNReal (nat_lit 2)
(@AddMonoidWithOne.toNatCast ENNReal
(@AddCommMonoidWithOne.toAddMonoidWithOne ENNReal instENNRealAddCommMonoidWithOne))
EuclideanSpace.proof_1))
(Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))
(fun (x : Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) => Real)
(Fin.fintype (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))
fun (i : Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) =>
@PseudoMetricSpace.toDist Real Real.pseudoMetricSpace)
p
(@midpoint Real (EuclideanSpace Real (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))
(EuclideanSpace Real (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))) Real.instRing
(@invertibleTwo Real Real.instDivisionRing (@RCLike.charZero_rclike Real Real.instRCLike))
(@WithLp.instAddCommGroup
(@OfNat.ofNat ENNReal (nat_lit 2)
(@instOfNatAtLeastTwo ENNReal (nat_lit 2)
(@AddMonoidWithOne.toNatCast ENNReal
(@AddCommMonoidWithOne.toAddMonoidWithOne ENNReal instENNRealAddCommMonoidWithOne))
EuclideanSpace.proof_1))
((i : Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) β
(fun (x : Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) => Real) i)
(@Pi.addCommGroup (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))
(fun (x : Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) => Real)
fun (i : Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) => Real.instAddCommGroup))
(@WithLp.instModule
(@OfNat.ofNat ENNReal (nat_lit 2)
(@instOfNatAtLeastTwo ENNReal (nat_lit 2)
(@AddMonoidWithOne.toNatCast ENNReal
(@AddCommMonoidWithOne.toAddMonoidWithOne ENNReal instENNRealAddCommMonoidWithOne))
EuclideanSpace.proof_1))
Real
((i : Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) β
(fun (x : Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) => Real) i)
(@Ring.toSemiring Real Real.instRing)
(@Pi.addCommGroup (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))
(fun (x : Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) => Real)
fun (i : Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) => Real.instAddCommGroup)
(@Pi.Function.module (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real Real
(@Ring.toSemiring Real Real.instRing)
(@NonUnitalNonAssocSemiring.toAddCommMonoid Real
(@NonUnitalSemiring.toNonUnitalNonAssocSemiring Real
(@Semiring.toNonUnitalSemiring Real (@Ring.toSemiring Real Real.instRing))))
(@NormedSpace.toModule Real Real Real.normedField
(@NonUnitalSeminormedRing.toSeminormedAddCommGroup Real
(@NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing Real
(@SeminormedCommRing.toNonUnitalSeminormedCommRing Real
(@NormedCommRing.toSeminormedCommRing Real Real.normedCommRing))))
(@InnerProductSpace.toNormedSpace Real Real Real.instRCLike
(@NonUnitalSeminormedRing.toSeminormedAddCommGroup Real
(@NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing Real
(@SeminormedCommRing.toNonUnitalSeminormedCommRing Real
(@NormedCommRing.toSeminormedCommRing Real Real.normedCommRing))))
(@RCLike.toInnerProductSpaceReal Real Real.instRCLike)))))
(@NormedAddTorsor.toAddTorsor
(EuclideanSpace Real (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))
(EuclideanSpace Real (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))
(@PiLp.seminormedAddCommGroup
(@OfNat.ofNat ENNReal (nat_lit 2)
(@instOfNatAtLeastTwo ENNReal (nat_lit 2)
(@AddMonoidWithOne.toNatCast ENNReal
(@AddCommMonoidWithOne.toAddMonoidWithOne ENNReal instENNRealAddCommMonoidWithOne))
EuclideanSpace.proof_1))
(Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))
(fun (x : Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) => Real)
fact_one_le_two_ennreal (Fin.fintype (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))
fun (i : Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) =>
@NonUnitalSeminormedRing.toSeminormedAddCommGroup Real
(@NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing Real
(@SeminormedCommRing.toNonUnitalSeminormedCommRing Real
(@NormedCommRing.toSeminormedCommRing Real Real.normedCommRing))))
(@SeminormedAddCommGroup.toPseudoMetricSpace
(EuclideanSpace Real (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))
(@PiLp.seminormedAddCommGroup
(@OfNat.ofNat ENNReal (nat_lit 2)
(@instOfNatAtLeastTwo ENNReal (nat_lit 2)
(@AddMonoidWithOne.toNatCast ENNReal
(@AddCommMonoidWithOne.toAddMonoidWithOne ENNReal instENNRealAddCommMonoidWithOne))
EuclideanSpace.proof_1))
(Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))
(fun (x : Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) => Real)
fact_one_le_two_ennreal (Fin.fintype (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))
fun (i : Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) =>
@NonUnitalSeminormedRing.toSeminormedAddCommGroup Real
(@NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing Real
(@SeminormedCommRing.toNonUnitalSeminormedCommRing Real
(@NormedCommRing.toSeminormedCommRing Real Real.normedCommRing)))))
(@SeminormedAddCommGroup.toNormedAddTorsor
(EuclideanSpace Real (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))
(@PiLp.seminormedAddCommGroup
(@OfNat.ofNat ENNReal (nat_lit 2)
(@instOfNatAtLeastTwo ENNReal (nat_lit 2)
(@AddMonoidWithOne.toNatCast ENNReal
(@AddCommMonoidWithOne.toAddMonoidWithOne ENNReal instENNRealAddCommMonoidWithOne))
EuclideanSpace.proof_1))
(Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))
(fun (x : Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) => Real)
fact_one_le_two_ennreal (Fin.fintype (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))
fun (i : Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) =>
@NonUnitalSeminormedRing.toSeminormedAddCommGroup Real
(@NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing Real
(@SeminormedCommRing.toNonUnitalSeminormedCommRing Real
(@NormedCommRing.toSeminormedCommRing Real Real.normedCommRing))))))
O1_1 O2_1))
(@OfNat.ofNat Real (nat_lit 2)
(@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))))))) | (EuclideanSpace β (Fin 2)) β (EuclideanSpace β (Fin 2)) β Set (EuclideanSpace β (Fin 2)) | [
{
"t": "EuclideanSpace β (Fin 2)",
"v": null,
"name": "O1"
},
{
"t": "EuclideanSpace β (Fin 2)",
"v": null,
"name": "O2"
},
{
"t": "Set (EuclideanSpace β (Fin 2))",
"v": null,
"name": "C1"
},
{
"t": "Set (EuclideanSpace β (Fin 2))",
"v": null,
"name": "C2"
},
{
"t": "C1 = Metric.sphere O1 1",
"v": null,
"name": "hC1"
},
{
"t": "C2 = Metric.sphere O2 3",
"v": null,
"name": "hC2"
},
{
"t": "dist O1 O2 = 10",
"v": null,
"name": "hO1O2"
},
{
"t": "{M : EuclideanSpace β (Fin 2) | β X Y, X β C1 β§ Y β C2 β§ M = midpoint β X Y} = answer O1 O2",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1996_a2",
"tags": [
"geometry"
]
} |
Suppose that each of 20 students has made a choice of anywhere from 0 to 6 courses from a total of 6 courses offered. Prove or disprove: there are 5 students and 2 courses such that all 5 have chosen both courses or all 5 have chosen neither course. | False | Show that the solution is that the statement is false. | null | [] | @Eq Prop answer False | Prop | [
{
"t": "Fin 20 β Set (Fin 6)",
"v": null,
"name": "choices"
},
{
"t": "answer β (β (students : Finset (Fin 20)) (courses : Finset (Fin 6)),\n students.card = 5 β§\n courses.card = 2 β§\n (βcourses β β s β students, choices s β¨ βcourses β β s β students, (choices s)αΆ))",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1996_a3",
"tags": [
"combinatorics"
]
} |
Let $c>0$ be a constant. Give a complete description, with proof, of the set of all continuous functions $f:\mathbb{R} \to \mathbb{R}$ such that $f(x)=f(x^2+c)$ for all $x \in \mathbb{R}$. | the set of functions described by the condition | Show that if $c \leq 1/4$ then $f$ must be constant, and if $c>1/4$ then $f$ can be defined on $[0,c]$ as any continuous function with equal values on the endpoints, then extended to $x>c$ by the relation $f(x)=f(x^2+c)$, then extended further to $x<0$ by the relation $f(x)=f(-x)$. | open Function | [] | @Eq (Real β Set (Real β Real)) answer fun (c_1 : Real) =>
@ite (Set (Real β Real))
(@LE.le Real Real.instLE c_1
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))
(@OfNat.ofNat Real (nat_lit 4)
(@instOfNatAtLeastTwo Real (nat_lit 4) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))))))
(Real.decidableLE c_1
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))
(@OfNat.ofNat Real (nat_lit 4)
(@instOfNatAtLeastTwo Real (nat_lit 4) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))))))
(@setOf (Real β Real) fun (f_1 : Real β Real) => @Exists Real fun (d : Real) => β (x : Real), @Eq Real (f_1 x) d)
(@setOf (Real β Real) fun (f_1 : Real β Real) =>
And
(@ContinuousOn Real Real
(@UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace))
(@UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) f_1
(@Set.Icc Real Real.instPreorder (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) c_1))
(And (@Eq Real (f_1 (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero))) (f_1 c_1))
(And
(β (x : Real),
@GT.gt Real Real.instLT x (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) β
@Eq Real (f_1 x)
(f_1
(@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd)
(@HPow.hPow Real Nat Real (@instHPow Real Nat (@Monoid.toNatPow Real Real.instMonoid)) x
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))
c_1)))
(β (x : Real),
@LT.lt Real Real.instLT x (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) β
@Eq Real (f_1 x) (f_1 (@Neg.neg Real Real.instNeg x)))))) | β β Set (β β β) | [
{
"t": "β",
"v": null,
"name": "c"
},
{
"t": "β β β",
"v": null,
"name": "f"
},
{
"t": "c > 0",
"v": null,
"name": "cgt0"
},
{
"t": "(Continuous f β§ β x : β, f x = f (x ^ 2 + c)) β f β answer c",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1996_a6",
"tags": [
"analysis",
"algebra"
]
} |
Define a \emph{selfish} set to be a set which has its own cardinality (number of elements) as an element. Find, with proof, the number of subsets of $\{1,2,\ldots,n\}$ which are \emph{minimal} selfish sets, that is, selfish sets none of whose proper subsets is selfish. | the nth Fibonacci number | Show that the number of subsets is $F_n$, the $n$th Fibonacci number. | open Function | [] | @Eq (Nat β Nat) answer Nat.fib | β β β | [
{
"t": "Finset β β Prop",
"v": null,
"name": "selfish"
},
{
"t": "β",
"v": null,
"name": "n"
},
{
"t": "β s : Finset β, selfish s = (s.card β s)",
"v": null,
"name": "hselfish"
},
{
"t": "n β₯ 1",
"v": null,
"name": "npos"
},
{
"t": "{s : Finset β | (s : Set β) β Set.Icc 1 n β§ selfish s β§ (β ss : Finset β, ss β s β Β¬selfish ss)}.encard = answer n",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1996_b1",
"tags": [
"algebra"
]
} |
Given that $\{x_1,x_2,\ldots,x_n\}=\{1,2,\ldots,n\}$, find, with proof, the largest possible value, as a function of $n$ (with $n \geq 2$), of $x_1x_2+x_2x_3+\cdots+x_{n-1}x_n+x_nx_1$. | $(2n^3 + 3n^2 - 11n + 18) / 6$ | Show that the maximum is $(2n^3+3n^2-11n+18)/6$. | open Function | [] | @Eq (Nat β Nat) answer fun (n_1 : Nat) =>
@HDiv.hDiv Nat Nat Nat (@instHDiv Nat Nat.instDiv)
(@HAdd.hAdd Nat Nat Nat (@instHAdd Nat instAddNat)
(@HSub.hSub Nat Nat Nat (@instHSub Nat instSubNat)
(@HAdd.hAdd Nat Nat Nat (@instHAdd Nat instAddNat)
(@HMul.hMul Nat Nat Nat (@instHMul Nat instMulNat) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))
(@HPow.hPow Nat Nat Nat (@instHPow Nat Nat (@Monoid.toNatPow Nat Nat.instMonoid)) n_1
(@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))))
(@HMul.hMul Nat Nat Nat (@instHMul Nat instMulNat) (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))
(@HPow.hPow Nat Nat Nat (@instHPow Nat Nat (@Monoid.toNatPow Nat Nat.instMonoid)) n_1
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))))
(@HMul.hMul Nat Nat Nat (@instHMul Nat instMulNat) (@OfNat.ofNat Nat (nat_lit 11) (instOfNatNat (nat_lit 11)))
n_1))
(@OfNat.ofNat Nat (nat_lit 18) (instOfNatNat (nat_lit 18))))
(@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) | β β β | [
{
"t": "β",
"v": null,
"name": "n"
},
{
"t": "n β₯ 2",
"v": null,
"name": "hn"
},
{
"t": "IsGreatest\n {k | β x : β β β€,\n (x '' (Finset.range n) = Set.Icc (1 : β€) n) β§\n β i : Fin n, x i * x ((i + 1) % n) = k}\n (answer n)",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1996_b3",
"tags": [
"algebra"
]
} |
For any square matrix $A$, we can define $\sin A$ by the usual power series: $\sin A=\sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!}A^{2n+1}$. Prove or disprove: there exists a $2 \times 2$ matrix $A$ with real entries such that $\sin A=\begin{pmatrix} 1 & 1996 \\ 0 & 1 \end{pmatrix}$. | False | Show that there does not exist such a matrix $A$. | open Function Nat | [] | @Eq Prop answer False | Prop | [
{
"t": "Matrix (Fin 2) (Fin 2) β β Matrix (Fin 2) (Fin 2) β",
"v": null,
"name": "matsin"
},
{
"t": "Matrix (Fin 2) (Fin 2) β",
"v": null,
"name": "mat1996"
},
{
"t": "β A, matsin A = β' n : β, ((-(1 : β)) ^ n / (2 * n + 1)!) β’ A ^ (2 * n + 1)",
"v": null,
"name": "hmatsin"
},
{
"t": "mat1996 0 0 = 1 β§ mat1996 0 1 = 1996 β§ mat1996 1 0 = 0 β§ mat1996 1 1 = 1",
"v": null,
"name": "hmat1996"
},
{
"t": "(β A, matsin A = mat1996) β answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1996_b4",
"tags": [
"linear_algebra"
]
} |
Given a finite string $S$ of symbols $X$ and $O$, we write $\Delta(S)$ for the number of $X$'s in $S$ minus the number of $O$'s. For example, $\Delta(XOOXOOX)=-1$. We call a string $S$ \emph{balanced} if every substring $T$ of (consecutive symbols of) $S$ has $-2 \leq \Delta(T) \leq 2$. Thus, $XOOXOOX$ is not balanced, since it contains the substring $OOXOO$. Find, with proof, the number of balanced strings of length $n$. | $2^{\lfloor (n + 2) / 2 \rfloor} + 2^{\lfloor (n + 1) / 2 \rfloor} - 2$ | Show that the number of balanced strings of length $n$ is $3 \cdot 2^{n/2}-2$ if $n$ is even, and $2^{(n+1)/2}-2$ if $n$ is odd. | open Function Nat | [] | @Eq (Nat β Nat) answer fun (n_1 : Nat) =>
@HSub.hSub Nat Nat Nat (@instHSub Nat instSubNat)
(@HAdd.hAdd Nat Nat Nat (@instHAdd Nat instAddNat)
(@HPow.hPow Nat Nat Nat (@instHPow Nat Nat (@Monoid.toNatPow Nat Nat.instMonoid))
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))
(@Nat.floor Nat Nat.instOrderedSemiring instFloorSemiringNat
(@HDiv.hDiv Nat Nat Nat (@instHDiv Nat Nat.instDiv)
(@HAdd.hAdd Nat Nat Nat (@instHAdd Nat instAddNat) n_1
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))))
(@HPow.hPow Nat Nat Nat (@instHPow Nat Nat (@Monoid.toNatPow Nat Nat.instMonoid))
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))
(@Nat.floor Nat Nat.instOrderedSemiring instFloorSemiringNat
(@HDiv.hDiv Nat Nat Nat (@instHDiv Nat Nat.instDiv)
(@HAdd.hAdd Nat Nat Nat (@instHAdd Nat instAddNat) n_1
(@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))))
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) | β β β | [
{
"t": "β",
"v": null,
"name": "n"
},
{
"t": "(Fin n β β€Λ£) β Fin n β Fin n β β€",
"v": null,
"name": "Ξ"
},
{
"t": "(Fin n β β€Λ£) β Prop",
"v": null,
"name": "balanced"
},
{
"t": "β S, β a b, a β€ b β Ξ S a b = β i in Finset.Icc a b, (S i : β€)",
"v": null,
"name": "hΞ"
},
{
"t": "β S, balanced S β β a b, a β€ b β |Ξ S a b| β€ 2",
"v": null,
"name": "hbalanced"
},
{
"t": "{S : Fin n β β€Λ£ | balanced S}.ncard = answer n",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1996_b5",
"tags": [
"algebra"
]
} |
Evaluate \begin{gather*} \int_0^\infty \left(x-\frac{x^3}{2}+\frac{x^5}{2\cdot 4}-\frac{x^7}{2\cdot 4\cdot 6}+\cdots\right) \\ \left(1+\frac{x^2}{2^2}+\frac{x^4}{2^2\cdot 4^2}+\frac{x^6}{2^2\cdot 4^2 \cdot 6^2}+\cdots\right)\,dx. \end{gather*} | \(\sqrt{e}\) | Show that the solution is $\sqrt{e}$. | open Filter Topology | [] | @Eq Real answer (Real.sqrt (Real.exp (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)))) | β | [
{
"t": "β β β",
"v": null,
"name": "series1"
},
{
"t": "β β β",
"v": null,
"name": "series2"
},
{
"t": "series1 = fun x => β' n : β, (-1)^n * x^(2*n + 1)/(β i : Finset.range n, 2 * ((i : β) + 1))",
"v": null,
"name": "hseries1"
},
{
"t": "series2 = fun x => β' n : β, x^(2*n)/(β i : Finset.range n, (2 * ((i : β) + 1))^2)",
"v": null,
"name": "hseries2"
},
{
"t": "Tendsto (fun t => β« x in Set.Icc 0 t, series1 x * series2 x) atTop (π answer)",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1997_a3",
"tags": [
"analysis"
]
} |
Let $N_n$ denote the number of ordered $n$-tuples of positive integers $(a_1,a_2,\ldots,a_n)$ such that $1/a_1 + 1/a_2 +\ldots + 1/a_n=1$. Determine whether $N_{10}$ is even or odd. | True | Show that $N_{10}$ is odd. | open Filter Topology | [] | @Eq Prop answer True | Prop | [
{
"t": "(n : β+) β Set (Fin n β β+)",
"v": null,
"name": "N"
},
{
"t": "N = fun (n : β+) => {t : Fin n β β+ | (β i j : Fin n, i < j β t i <= t j) β§ (β i : Fin n, (1 : β)/(t i) = 1) }",
"v": null,
"name": "hN"
},
{
"t": "Odd (N 10).ncard β answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1997_a5",
"tags": [
"number_theory"
]
} |
For a positive integer $n$ and any real number $c$, define $x_k$ recursively by $x_0=0$, $x_1=1$, and for $k\geq 0$, \[x_{k+2}=\frac{cx_{k+1}-(n-k)x_k}{k+1}.\] Fix $n$ and then take $c$ to be the largest value for which $x_{n+1}=0$. Find $x_k$ in terms of $n$ and $k$, $1\leq k\leq n$. | fun n k => Nat.choose (n.toNat-1) (k.toNat-1) | Show that the solution is that $x_k = {n - 1 \choose k - 1}$. | open Filter Topology | [] | @Eq (Int β Int β Real) answer fun (n_1 k : Int) =>
@Nat.cast Real Real.instNatCast
(Nat.choose
(@HSub.hSub Nat Nat Nat (@instHSub Nat instSubNat) (Int.toNat n_1)
(@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))
(@HSub.hSub Nat Nat Nat (@instHSub Nat instSubNat) (Int.toNat k)
(@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))) | β€ β β€ β β | [
{
"t": "β€",
"v": null,
"name": "n"
},
{
"t": "n > 0",
"v": null,
"name": "hn"
},
{
"t": "β β (β€ β β)",
"v": null,
"name": "x"
},
{
"t": "β c, x c 0 = 0",
"v": null,
"name": "hx0"
},
{
"t": "β c, x c 1 = 1",
"v": null,
"name": "hx1"
},
{
"t": "β c, β k β₯ 0, x c (k + 2) = (c*(x c (k + 1)) - (n - k)*(x c k))/(k + 1)",
"v": null,
"name": "hxk"
},
{
"t": "Set β",
"v": null,
"name": "S"
},
{
"t": "S = {c : β | x c (n + 1) = 0}",
"v": null,
"name": "hS"
},
{
"t": "β k : Set.Icc 1 n, x (sSup S) k = answer n k",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1997_a6",
"tags": [
"algebra"
]
} |
Let $\{x\}$ denote the distance between the real number $x$ and the nearest integer. For each positive integer $n$, evaluate \[F_n=\sum_{m=1}^{6n-1} \min(\{\frac{m}{6n}\},\{\frac{m}{3n}\}).\] (Here $\min(a,b)$ denotes the minimum of $a$ and $b$.) | fun n => n | Show that the solution is $n$. | open Filter Topology | [
{
"t": "Real",
"v": null,
"name": "r"
}
] | @Eq (Nat β Real) answer fun (n : Nat) => @Nat.cast Real Real.instNatCast n | β β β | [
{
"t": "β β β",
"v": null,
"name": "dist_to_int"
},
{
"t": "dist_to_int r = |r - round r|",
"v": null,
"name": "h_dist_to_int"
},
{
"t": "β β β",
"v": null,
"name": "F"
},
{
"t": "F = fun (n : β) => β m in Finset.Icc 1 (6 * n - 1), min (dist_to_int (m/(6*n)) ) (dist_to_int (m/(3*n)))",
"v": null,
"name": "hF"
},
{
"t": "β n, n > 0 β F n = answer n",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1997_b1",
"tags": [
"algebra"
]
} |
For each positive integer $n$, write the sum $\sum_{m=1}^n 1/m$ in the form $p_n/q_n$, where $p_n$ and $q_n$ are relatively prime positive integers. Determine all $n$ such that 5 does not divide $q_n$. | {n | (1 β€ n β§ n β€ 4) β¨ (20 β€ n β§ n β€ 24) β¨ (100 β€ n β§ n β€ 104) β¨ (120 β€ n β§ n β€ 124)} | Show that the solution is the set of natural numbers which are between $1$ and $4$, or between $20$ and $24$, or between $100$ and $104$, or between $120$ and $124$. | open Filter Topology Bornology Set | [] | @Eq (Set Nat) answer
(@setOf Nat fun (n_1 : Nat) =>
Or
(And (@LE.le Nat instLENat (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))) n_1)
(@LE.le Nat instLENat n_1 (@OfNat.ofNat Nat (nat_lit 4) (instOfNatNat (nat_lit 4)))))
(Or
(And (@LE.le Nat instLENat (@OfNat.ofNat Nat (nat_lit 20) (instOfNatNat (nat_lit 20))) n_1)
(@LE.le Nat instLENat n_1 (@OfNat.ofNat Nat (nat_lit 24) (instOfNatNat (nat_lit 24)))))
(Or
(And (@LE.le Nat instLENat (@OfNat.ofNat Nat (nat_lit 100) (instOfNatNat (nat_lit 100))) n_1)
(@LE.le Nat instLENat n_1 (@OfNat.ofNat Nat (nat_lit 104) (instOfNatNat (nat_lit 104)))))
(And (@LE.le Nat instLENat (@OfNat.ofNat Nat (nat_lit 120) (instOfNatNat (nat_lit 120))) n_1)
(@LE.le Nat instLENat n_1 (@OfNat.ofNat Nat (nat_lit 124) (instOfNatNat (nat_lit 124)))))))) | Set β | [
{
"t": "β",
"v": null,
"name": "n"
},
{
"t": "n > 0",
"v": null,
"name": "hn"
},
{
"t": "n β answer β Β¬5 β£ (β m in Finset.Icc 1 n, 1/m : β).den",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1997_b3",
"tags": [
"number_theory"
]
} |
Let $A_1=0$ and $A_2=1$. For $n>2$, the number $A_n$ is defined by concatenating the decimal expansions of $A_{n-1}$ and $A_{n-2}$ from left to right. For example $A_3=A_2 A_1=10$, $A_4=A_3 A_2 = 101$, $A_5=A_4 A_3 = 10110$, and so forth. Determine all $n$ such that $11$ divides $A_n$. | all $n$ such that $n \equiv 1 \pmod{6}$ | Show that the solution is those n for which n can be written as 6k+1 for some integer k. | null | [] | @Eq (Set Nat) answer
(@setOf Nat fun (n : Nat) =>
Nat.ModEq (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) n
(@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))) | Set β | [
{
"t": "β β List β",
"v": null,
"name": "A"
},
{
"t": "A 1 = [0]",
"v": null,
"name": "hA1"
},
{
"t": "A 2 = [1]",
"v": null,
"name": "hA2"
},
{
"t": "β n > 0, A (n + 2) = A (n + 1) ++ A n",
"v": null,
"name": "hA"
},
{
"t": "{n | 1 β€ n β§ 11 β£ Nat.ofDigits 10 (A n).reverse} = answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1998_a4",
"tags": [
"algebra"
]
} |
Find the minimum value of \[\frac{(x+1/x)^6-(x^6+1/x^6)-2}{(x+1/x)^3+(x^3+1/x^3)}\] for $x>0$. | 6 | Show that the minimum value is 6. | open Set Function Metric | [] | @Eq Real answer
(@OfNat.ofNat Real (nat_lit 6)
(@instOfNatAtLeastTwo Real (nat_lit 6) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 4) (instOfNatNat (nat_lit 4)))))) | β | [
{
"t": "sInf {((x + 1/x)^6 - (x^6 + 1/x^6) - 2)/((x + 1/x)^3 + (x^3 + 1/x^3)) | x > (0 : β)} = answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1998_b1",
"tags": [
"algebra"
]
} |
Given a point $(a,b)$ with $0<b<a$, determine the minimum perimeter of a triangle with one vertex at $(a,b)$, one on the $x$-axis, and one on the line $y=x$. You may assume that a triangle of minimum perimeter exists. | $\sqrt{2a^2 + 2b^2}$ if $a > b$, otherwise $0$ | Show that the solution is $\sqrt{2a^2 + 2b^2}. | open Set Function Metric | [] | @Eq (Real β Real β Real) answer fun (a_1 b_1 : Real) =>
@ite Real (@GT.gt Real Real.instLT a_1 b_1) (Real.decidableLT b_1 a_1)
(Real.sqrt
(@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd)
(@HMul.hMul Real Real Real (@instHMul Real Real.instMul)
(@OfNat.ofNat Real (nat_lit 2)
(@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))))))
(@HPow.hPow Real Nat Real (@instHPow Real Nat (@Monoid.toNatPow Real Real.instMonoid)) a_1
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))
(@HMul.hMul Real Real Real (@instHMul Real Real.instMul)
(@OfNat.ofNat Real (nat_lit 2)
(@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))))))
(@HPow.hPow Real Nat Real (@instHPow Real Nat (@Monoid.toNatPow Real Real.instMonoid)) b_1
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))))
(@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) | β β β β β | [
{
"t": "β",
"v": null,
"name": "a"
},
{
"t": "β",
"v": null,
"name": "b"
},
{
"t": "0 < b β§ b < a",
"v": null,
"name": "hab"
},
{
"t": "sInf {d : β | β (c : β) (x : β), d = Real.sqrt ((a - c)^2 + (b - 0)^2) + Real.sqrt ((c - x)^2 + (0 - x)^2) + Real.sqrt ((a - x)^2 + (b - x)^2) β§\n Real.sqrt ((a - c)^2 + (b - 0)^2) + Real.sqrt ((c - x)^2 + (0 - x)^2) > Real.sqrt ((a - x)^2 + (b - x)^2) β§\n Real.sqrt ((a - c)^2 + (b - 0)^2) + Real.sqrt ((a - x)^2 + (b - x)^2) > Real.sqrt ((c - x)^2 + (0 - x)^2) β§\n Real.sqrt ((c - x)^2 + (0 - x)^2) + Real.sqrt ((a - x)^2 + (b - x)^2) > Real.sqrt ((a - c)^2 + (b - 0)^2)} = answer a b",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1998_b2",
"tags": [
"geometry",
"algebra"
]
} |
Find necessary and sufficient conditions on positive integers $m$ and $n$ so that \[\sum_{i=0}^{mn-1} (-1)^{\lfloor i/m \rfloor +\lfloor i/n\rfloor}=0.\] | {nm | let β¨n,mβ© := nm; multiplicity 2 n β multiplicity 2 m} | Show that the sum is 0 if and only if the largest powers of $2$ dividing $m$ and $n$ are different. | open Set Function Metric | [] | @Eq (Set (Prod Nat Nat)) answer
(@setOf (Prod Nat Nat) fun (nm : Prod Nat Nat) =>
_example.match_1 (fun (nm_1 : Prod Nat Nat) => Prop) nm fun (n_1 m_1 : Nat) =>
@Ne Nat (@multiplicity Nat Nat.instMonoid (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) n_1)
(@multiplicity Nat Nat.instMonoid (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) m_1)) | Set (β Γ β) | [
{
"t": "β β β β β€",
"v": null,
"name": "quantity"
},
{
"t": "quantity = fun n m => β i in Finset.range (m * n), (-1)^(i/m + i/n)",
"v": null,
"name": "hquantity"
},
{
"t": "β",
"v": null,
"name": "n"
},
{
"t": "β",
"v": null,
"name": "m"
},
{
"t": "n > 0 β§ m > 0",
"v": null,
"name": "hnm"
},
{
"t": "quantity n m = 0 β β¨n, mβ© β answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1998_b4",
"tags": [
"number_theory"
]
} |
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 $\sqrt N$. | 1 | Show that the thousandth digit is 1. | open Set Function Metric | [] | @Eq Nat answer (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))) | β | [
{
"t": "β",
"v": null,
"name": "N"
},
{
"t": "N = β i in Finset.range 1998, 10^i",
"v": null,
"name": "hN"
},
{
"t": "answer = (Nat.floor (10^1000 * Real.sqrt N)) % 10",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1998_b5",
"tags": [
"number_theory"
]
} |
Find polynomials $f(x)$,$g(x)$, and $h(x)$, if they exist, such that for all $x$, \[|f(x)|-|g(x)|+h(x) = \begin{cases} -1 & \mbox{if $x<-1$} \\3x+2 & \mbox{if $-1 \leq x \leq 0$} \\-2x+2 & \mbox{if $x>0$.}\end{cases}\]? | True | Show that the answer is such functions do exist. | null | [] | @Eq Prop answer True | Prop | [
{
"t": "answer β β f g h : Polynomial β, β x : β, |f.eval x| - |g.eval x| + h.eval x = if x < -1 then -1 else (if (x β€ 0) then 3 * x + 2 else -2 * x + 2)",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1999_a1",
"tags": [
"algebra"
]
} |
Sum the series \[\sum_{m=1}^\infty \sum_{n=1}^\infty \frac{m^2 n}{3^m(n3^m+m3^n)}.\] | 9/32 | Show that the solution is 9/32. | open Filter Topology Metric | [] | @Eq Real answer
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@OfNat.ofNat Real (nat_lit 9)
(@instOfNatAtLeastTwo Real (nat_lit 9) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 7) (instOfNatNat (nat_lit 7))))))
(@OfNat.ofNat Real (nat_lit 32)
(@instOfNatAtLeastTwo Real (nat_lit 32) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 30) (instOfNatNat (nat_lit 30))))))) | β | [
{
"t": "Tendsto (fun i => β m in Finset.range i, β' n : β, (((m + 1)^2*(n+1))/(3^(m + 1) * ((n+1)*3^(m + 1) + (m + 1)*3^(n+1))) : β)) atTop (π answer)",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1999_a4",
"tags": [
"number_theory"
]
} |
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 \[\lim_{(x,y)\rightarrow (1,1), (x,y)\in A} (1-xy^2)(1-x^2y)S(x,y).\] | 3 | Show that the answer is 3. | open Filter Topology Metric | [] | @Eq Real answer
(@OfNat.ofNat Real (nat_lit 3)
(@instOfNatAtLeastTwo Real (nat_lit 3) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))) | β | [
{
"t": "Set (β Γ β)",
"v": null,
"name": "A"
},
{
"t": "A = {xy | 0 β€ xy.1 β§ xy.1 < 1 β§ 0 β€ xy.2 β§ xy.2 < 1}",
"v": null,
"name": "hA"
},
{
"t": "β β β β β",
"v": null,
"name": "S"
},
{
"t": "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",
"v": null,
"name": "hS"
},
{
"t": "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 answer)",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1999_b3",
"tags": [
"algebra"
]
} |
For an integer $n\geq 3$, let $\theta=2\pi/n$. Evaluate the determinant of the $n\times n$ matrix $I+A$, where $I$ is the $n\times n$ identity matrix and $A=(a_{jk})$ has entries $a_{jk}=\cos(j\theta+k\theta)$ for all $j,k$. | fun n => 1 - n^2/4 | Show that the answer is $(1 - n^2)/4$. | open Filter Topology Metric | [] | @Eq (Nat β Real) answer fun (n_1 : Nat) =>
@HSub.hSub Real Real Real (@instHSub Real Real.instSub)
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@HPow.hPow Real Nat Real (@instHPow Real Nat (@Monoid.toNatPow Real Real.instMonoid))
(@Nat.cast Real Real.instNatCast n_1) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))
(@OfNat.ofNat Real (nat_lit 4)
(@instOfNatAtLeastTwo Real (nat_lit 4) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))))) | β β β | [
{
"t": "β",
"v": null,
"name": "n"
},
{
"t": "n β₯ 3",
"v": null,
"name": "hn"
},
{
"t": "β",
"v": null,
"name": "theta"
},
{
"t": "theta = 2 * Real.pi / n",
"v": null,
"name": "htheta"
},
{
"t": "Matrix (Fin n) (Fin n) β",
"v": null,
"name": "A"
},
{
"t": "A = fun j k => Real.cos ((j.1 + 1) * theta + (k.1 + 1) * theta)",
"v": null,
"name": "hA"
},
{
"t": "(1 + A).det = answer n",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1999_b5",
"tags": [
"linear_algebra"
]
} |
Let $A$ be a positive real number. What are the possible values of $\sum_{j=0}^\infty x_j^2$, given that $x_0,x_1,\ldots$ are positive numbers for which $\sum_{j=0}^\infty x_j=A$? | the open interval $(0, A^2)$ | Show that the possible values comprise the interval $(0,A^2)$. | open Topology Filter | [] | @Eq (Real β Set Real) answer fun (A_1 : Real) =>
@Set.Ioo Real Real.instPreorder (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero))
(@HPow.hPow Real Nat Real (@instHPow Real Nat (@Monoid.toNatPow Real Real.instMonoid)) A_1
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) | β β Set β | [
{
"t": "β",
"v": null,
"name": "A"
},
{
"t": "A > 0",
"v": null,
"name": "Apos"
},
{
"t": "{S : β |\n β x : β β β,\n (β j : β, x j > 0) β§ \n (β' j : β, x j) = A β§ \n (β' j : β, (x j) ^ 2) = S} \n = answer A",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2000_a1",
"tags": [
"analysis"
]
} |
For each integer $m$, consider the polynomial
\[P_m(x)=x^4-(2m+4)x^2+(m-2)^2.\] For what values of $m$ is $P_m(x)$
the product of two non-constant polynomials with integer coefficients? | {m : β€ | β k : β€, k^2 = m β¨ 2*k^2 = m} | $P_m(x)$ factors into two nonconstant polynomials over
the integers if and only if $m$ is either a square or twice a square. | open Topology Filter Polynomial Set | [] | @Eq (Set Int) answer
(@setOf Int fun (m : Int) =>
@Exists Int fun (k : Int) =>
Or
(@Eq Int
(@HPow.hPow Int Nat Int (@instHPow Int Nat (@Monoid.toNatPow Int Int.instMonoid)) k
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))
m)
(@Eq Int
(@HMul.hMul Int Int Int (@instHMul Int Int.instMul) (@OfNat.ofNat Int (nat_lit 2) (@instOfNat (nat_lit 2)))
(@HPow.hPow Int Nat Int (@instHPow Int Nat (@Monoid.toNatPow Int Int.instMonoid)) k
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))
m)) | Set β€ | [
{
"t": "β€ β Polynomial β€",
"v": null,
"name": "P"
},
{
"t": "P = fun m : β€ => (Polynomial.X)^4 - (Polynomial.C (2*m + 4))*(Polynomial.X)^2 + Polynomial.C ((m - 2)^2)",
"v": null,
"name": "hP"
},
{
"t": "{m : β€ | β a b, P m = a * b β§\n(β n β Ici 1, a.coeff n β 0) β§ (β n β Ici 1, b.coeff n β 0)} = answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2001_a3",
"tags": [
"algebra"
]
} |
Find all pairs of real numbers $(x,y)$ satisfying the system of equations
\begin{align*}
\frac{1}{x}+\frac{1}{2y}&=(x^2+3y^2)(3x^2+y^2) \\
\frac{1}{x}-\frac{1}{2y}&=2(y^4-x^4).
\end{align*} | the set containing the pair $\left(\frac{3^{1/5} + 1}{2}, \frac{3^{1/5} - 1}{2}\right)$ | Show that $x=(3^{1/5}+1)/2$ and $y=(3^{1/5}-1)/2$ is the unique solution satisfying the given equations. | open Topology Filter Polynomial Set | [] | @Eq (Set (Prod Real Real)) answer
(@Singleton.singleton (Prod Real Real) (Set (Prod Real Real)) (@Set.instSingletonSet (Prod Real Real))
(@Prod.mk Real Real
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd)
(@HPow.hPow Real Real Real (@instHPow Real Real Real.instPow)
(@OfNat.ofNat Real (nat_lit 3)
(@instOfNatAtLeastTwo Real (nat_lit 3) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))))
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))
(@OfNat.ofNat Real (nat_lit 5)
(@instOfNatAtLeastTwo Real (nat_lit 5) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3))))))))
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)))
(@OfNat.ofNat Real (nat_lit 2)
(@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0)))))))
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@HSub.hSub Real Real Real (@instHSub Real Real.instSub)
(@HPow.hPow Real Real Real (@instHPow Real Real Real.instPow)
(@OfNat.ofNat Real (nat_lit 3)
(@instOfNatAtLeastTwo Real (nat_lit 3) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))))
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))
(@OfNat.ofNat Real (nat_lit 5)
(@instOfNatAtLeastTwo Real (nat_lit 5) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3))))))))
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)))
(@OfNat.ofNat Real (nat_lit 2)
(@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))))))))) | Set (β Γ β) | [
{
"t": "β",
"v": null,
"name": "x"
},
{
"t": "β",
"v": null,
"name": "y"
},
{
"t": "x β 0",
"v": null,
"name": "hx"
},
{
"t": "y β 0",
"v": null,
"name": "hy"
},
{
"t": "1 / x + 1 / (2 * y) = (x ^ 2 + 3 * y ^ 2) * (3 * x ^ 2 + y ^ 2)",
"v": null,
"name": "eq1"
},
{
"t": "1 / x - 1 / (2 * y) = 2 * (y ^ 4 - x ^ 4)",
"v": null,
"name": "eq2"
},
{
"t": "(x, y) β answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2001_b2",
"tags": [
"algebra"
]
} |
For any positive integer $n$, let $\langle n \rangle$ denote the closest integer to $\sqrt{n}$. Evaluate $\sum_{n=1}^\infty \frac{2^{\langle n \rangle}+2^{-\langle n \rangle}}{2^n}$. | 3 | Show that the sum is $3$. | open Topology Filter Polynomial Set | [] | @Eq Real answer
(@OfNat.ofNat Real (nat_lit 3)
(@instOfNatAtLeastTwo Real (nat_lit 3) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))) | β | [
{
"t": "β' n : Set.Ici 1, ((2 : β) ^ (round (Real.sqrt n)) + (2 : β) ^ (-round (Real.sqrt n))) / 2 ^ (n : β) = answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2001_b3",
"tags": [
"analysis"
]
} |
Assume that $(a_n)_{n \geq 1}$ is an increasing sequence of positive real numbers such that $\lim a_n/n=0$. Must there exist infinitely many positive integers $n$ such that $a_{n-i}+a_{n+i}<2a_n$ for $i=1,2,\ldots,n-1$? | True | Show that the answer is yes, there must exist infinitely many such $n$. | open Topology Filter Polynomial Set | [] | @Eq Prop answer True | Prop | [
{
"t": "β€ β β",
"v": null,
"name": "a"
},
{
"t": "β n β₯ 1, a n > 0 β§ a n < a (n + 1)",
"v": null,
"name": "h_pos_inc"
},
{
"t": "Tendsto (fun n : β€ => a (n + 1) / (n + 1)) atTop (π 0)",
"v": null,
"name": "h_limit"
},
{
"t": "{n : β€ | n > 0 β§ (β i β Set.Icc 1 (n - 1), a (n - i) + a (n + i) < 2 * a n)}.Infinite β answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2001_b6",
"tags": [
"analysis"
]
} |
Let $k$ be a fixed positive integer. The $n$-th derivative of $\frac{1}{x^k-1}$ has the form $\frac{P_n(x)}{(x^k-1)^{n+1}}$ where $P_n(x)$ is a polynomial. Find $P_n(1)$. | (-k)^n * n! | Show that $P_n(1)=(-k)^nn!$ for all $n \geq 0$. | open Nat | [] | @Eq (Nat β Nat β Real) answer fun (k_1 n : Nat) =>
@HMul.hMul Real Real Real (@instHMul Real Real.instMul)
(@HPow.hPow Real Nat Real (@instHPow Real Nat (@Monoid.toNatPow Real Real.instMonoid))
(@Neg.neg Real Real.instNeg (@Nat.cast Real Real.instNatCast k_1)) n)
(@Nat.cast Real Real.instNatCast (Nat.factorial n)) | β β β β β | [
{
"t": "β",
"v": null,
"name": "k"
},
{
"t": "β β Polynomial β",
"v": null,
"name": "P"
},
{
"t": "k > 0",
"v": null,
"name": "kpos"
},
{
"t": "β n x, iteratedDeriv n (fun x' : β => 1 / (x' ^ k - 1)) x = ((P n).eval x) / ((x ^ k - 1) ^ (n + 1))",
"v": null,
"name": "Pderiv"
},
{
"t": "β n, (P n).eval 1 = answer k n",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2002_a1",
"tags": [
"analysis",
"algebra"
]
} |
Fix an integer $b \geq 2$. Let $f(1) = 1$, $f(2) = 2$, and for each
$n \geq 3$, define $f(n) = n f(d)$, where $d$ is the number of
base-$b$ digits of $n$. For which values of $b$ does
\[
\sum_{n=1}^\infty \frac{1}{f(n)}
\]
converge? | {2} | The sum converges for $b=2$ and diverges for $b \geq 3$. | open Nat Set Topology Filter | [] | @Eq (Set Nat) answer
(@Singleton.singleton Nat (Set Nat) (@Set.instSingletonSet Nat)
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) | Set β | [
{
"t": "β β β β β",
"v": null,
"name": "f"
},
{
"t": "β b : β, f b 1 = 1 β§ f b 2 = 2 β§ β n β Ici 3, f b n = n * f b (Nat.digits b n).length",
"v": null,
"name": "hf"
},
{
"t": "{b β Ici 2 | β L : β, Tendsto (fun m : β => β n in Finset.Icc 1 m, 1/(f b n)) atTop (π L)} = answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2002_a6",
"tags": [
"analysis",
"number_theory"
]
} |
Let $n$ be a fixed positive integer. How many ways are there to write $n$ as a sum of positive integers, \[ n = a_1 + a_2 + \dots + a_k, \] with $k$ an arbitrary positive integer and $a_1 \leq a_2 \leq \dots \leq a_k \leq a_1 + 1$? For example, with $n = 4$, there are four ways: $4, 2 + 2, 1 + 1 + 2, 1 + 1 + 1 + 1$ | $n$ | Show that there are $n$ such sums. | open MvPolynomial | [] | @Eq (Nat β Nat) answer fun (n_1 : Nat) => n_1 | β β β | [
{
"t": "β",
"v": null,
"name": "n"
},
{
"t": "n > 0",
"v": null,
"name": "hn"
},
{
"t": "Set.encard {a : β β β€ |\n β k > 0, (β i : Fin k, a i = n) β§\n (β i : Fin k, a i > 0) β§\n (β i : Fin (k - 1), a i β€ a (i + 1)) β§\n a (k - 1) β€ a 0 + 1 β§ (β i β₯ k, a i = 0)} = answer n",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2003_a1",
"tags": [
"algebra"
]
} |
Find the minimum value of $|\sin x+\cos x+\tan x+\cot x+\sec x+\csc x|$ for real numbers $x$. | $2\sqrt{2} - 1$ | Show that the minimum is $2\sqrt{2}-1$. | open Set | [] | @Eq Real answer
(@HSub.hSub Real Real Real (@instHSub Real Real.instSub)
(@HMul.hMul Real Real Real (@instHMul Real Real.instMul)
(@OfNat.ofNat Real (nat_lit 2)
(@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))))))
(Real.sqrt
(@OfNat.ofNat Real (nat_lit 2)
(@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))))))))
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))) | β | [
{
"t": "β β β",
"v": null,
"name": "f"
},
{
"t": "β x : β, f x = |Real.sin x + Real.cos x + Real.tan x + 1 / Real.tan x + 1 / Real.cos x + 1 / Real.sin x|",
"v": null,
"name": "hf"
},
{
"t": "IsLeast (Set.range f) answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2003_a3",
"tags": [
"analysis"
]
} |
For a set $S$ of nonnegative integers, let $r_S(n)$ denote the number of ordered pairs $(s_1,s_2)$ such that $s_1 \in S$, $s_2 \in S$, $s_1 \ne s_2$, and $s_1+s_2=n$. Is it possible to partition the nonnegative integers into two sets $A$ and $B$ in such a way that $r_A(n)=r_B(n)$ for all $n$? | True | Show that such a partition is possible. | open MvPolynomial Set | [] | @Eq Prop answer True | Prop | [
{
"t": "Set β β β β β",
"v": null,
"name": "r"
},
{
"t": "β S n, r S n = β' s1 : S, β' s2 : S, if (s1 β s2 β§ s1 + s2 = n) then 1 else 0",
"v": null,
"name": "hr"
},
{
"t": "(β A B : Set β, A βͺ B = β β§ A β© B = β
β§ (β n : β, r A n = r B n)) β answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2003_a6",
"tags": [
"algebra"
]
} |
Do there exist polynomials $a(x), b(x), c(y), d(y)$ such that \[ 1 + xy + x^2y^2 = a(x)c(y) + b(x)d(y)\] holds identically? | False | Show that no such polynomials exist. | open MvPolynomial Set | [] | @Eq Prop answer False | Prop | [
{
"t": "answer = (β a b c d : Polynomial β, (β x y : β, 1 + x * y + x ^ 2 * y ^ 2 = a.eval x * c.eval y + b.eval x * d.eval y))",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2003_b1",
"tags": [
"linear_algebra",
"algebra"
]
} |
Basketball star Shanille O'Keal's team statistician keeps track of the number, $S(N)$, of successful free throws she has made in her first $N$ attempts of the season. Early in the season, $S(N)$ was less than $80\%$ of $N$, but by the end of the season, $S(N)$ was more than $80\%$ of $N$. Was there necessarily a moment in between when $S(N)$ was exactly $80\%$ of $N$? | True | Show that the answer is yes. | open Nat Topology Filter | [] | @Eq Prop answer True | Prop | [
{
"t": "(β β Fin 2) β β β β",
"v": null,
"name": "S"
},
{
"t": "β attempts, β N β₯ 1, S attempts N = (β i : Fin N, (attempts i).1) / N",
"v": null,
"name": "hS"
},
{
"t": "answer β (β attempts a b,\n (1 β€ a β§ a < b β§ S attempts a < 0.8 β§ S attempts b > 0.8) β\n (β c : β, a < c β§ c < b β§ S attempts c = 0.8))",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2004_a1",
"tags": [
"probability"
]
} |
Let $n$ be a positive integer, $n \ge 2$, and put $\theta = 2 \pi / n$. Define points $P_k = (k,0)$ in the $xy$-plane, for $k = 1, 2, \dots, n$. Let $R_k$ be the map that rotates the plane counterclockwise by the angle $\theta$ about the point $P_k$. Let $R$ denote the map obtained by applying, in order, $R_1$, then $R_2, \dots$, then $R_n$. For an arbitrary point $(x,y)$, find, and simplify, the coordinates of $R(x,y)$. | fun n z β¦ z + n | Show that $R(x, y) = (x + n, y)$. | open Nat Topology Filter | [] | @Eq (Nat β Complex β Complex) answer fun (n_1 : Nat) (z : Complex) =>
@HAdd.hAdd Complex Complex Complex (@instHAdd Complex Complex.instAdd) z
(@Nat.cast Complex
(@AddMonoidWithOne.toNatCast Complex (@AddGroupWithOne.toAddMonoidWithOne Complex Complex.addGroupWithOne)) n_1) | β β β β β | [
{
"t": "β",
"v": null,
"name": "n"
},
{
"t": "n β₯ 2",
"v": null,
"name": "nge2"
},
{
"t": "β β β β β",
"v": null,
"name": "R"
},
{
"t": "β β β β β",
"v": null,
"name": "Rk"
},
{
"t": "R 0 = id β§ β k : β, R (k + 1) = Rk (k + 1) β R k",
"v": null,
"name": "hR"
},
{
"t": "Rk = fun (k : β) (Q : β) β¦ k + Complex.exp (Complex.I * 2 * Real.pi / n) * (Q - k)",
"v": null,
"name": "hRk"
},
{
"t": "R n = answer n",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2004_b4",
"tags": [
"geometry"
]
} |
Evaluate $\lim_{x \to 1^-} \prod_{n=0}^\infty \left(\frac{1+x^{n+1}}{1+x^n}\right)^{x^n}$. | 2 / e | Show that the desired limit is $2/e$. | open Nat Topology Filter | [] | @Eq Real answer
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@OfNat.ofNat Real (nat_lit 2)
(@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))))))
(Real.exp (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)))) | β | [
{
"t": "β β β",
"v": null,
"name": "xprod"
},
{
"t": "β x β Set.Ioo 0 1,\n Tendsto (fun N β¦ β n in Finset.range N, ((1 + x ^ (n + 1)) / (1 + x ^ n)) ^ (x ^ n))\n atTop (π (xprod x))",
"v": null,
"name": "hxprod"
},
{
"t": "Tendsto xprod (π[<] 1) (π answer)",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2004_b5",
"tags": [
"analysis"
]
} |
Let $\mathbf{S} = \{(a,b) | a = 1, 2, \dots,n, b = 1,2,3\}$.
A \emph{rook tour} of $\mathbf{S}$ is a polygonal path made up of line segments connecting points $p_1, p_2, \dots, p_{3n}$ in sequence such that
\begin{enumerate}
\item[(i)] $p_i \in \mathbf{S}$,
\item[(ii)] $p_i$ and $p_{i+1}$ are a unit distance apart, for
$1 \leq i <3n$,
\item[(iii)] for each $p \in \mathbf{S}$ there is a unique $i$ such that
$p_i = p$.
\end{enumerate}
How many rook tours are there that begin at $(1,1)$
and end at $(n,1)$? | fun n β¦ if n = 1 then 0 else 2 ^ (n - 2) | Show that the number of rook tours is $0$ if $n = 1$ and $2 ^ {n - 2}$ if $n \geq 2$. | open Nat Set | [] | @Eq (Nat β Nat) answer fun (n_1 : Nat) =>
@ite Nat (@Eq Nat n_1 (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))
(instDecidableEqNat n_1 (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))
(@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0)))
(@HPow.hPow Nat Nat Nat (@instHPow Nat Nat (@Monoid.toNatPow Nat Nat.instMonoid))
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))
(@HSub.hSub Nat Nat Nat (@instHSub Nat instSubNat) n_1 (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))) | β β β | [
{
"t": "β",
"v": null,
"name": "n"
},
{
"t": "n > 0",
"v": null,
"name": "npos"
},
{
"t": "Set (β€ Γ β€)",
"v": null,
"name": "S"
},
{
"t": "β€ Γ β€ β β€ Γ β€ β Prop",
"v": null,
"name": "unit"
},
{
"t": "(β β β€ Γ β€) β Prop",
"v": null,
"name": "rooktour"
},
{
"t": "S = prod (Icc 1 (n : β€)) (Icc 1 3)",
"v": null,
"name": "hS"
},
{
"t": "unit = fun (a, b) (c, d) β¦ a = c β§ |d - b| = 1 β¨ b = d β§ |c - a| = 1",
"v": null,
"name": "hunit"
},
{
"t": "rooktour = fun p β¦ (β P β S, β! i, i β Icc 1 (3 * n) β§ p i = P) β§ (β i β Icc 1 (3 * n - 1), unit (p i) (p (i + 1))) β§ p 0 = 0 β§ β i > 3 * n, p i = 0",
"v": null,
"name": "hrooktour"
},
{
"t": "{p : β β β€ Γ β€ | rooktour p β§ p 1 = (1, 1) β§ p (3 * n) = ((n : β€), 1)}.encard = answer n",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2005_a2",
"tags": [
"combinatorics"
]
} |
Evaluate $\int_0^1 \frac{\ln(x+1)}{x^2+1}\,dx$. | $\frac{\pi \ln 2}{8}$ | Show that the solution is $\pi / 8 * \log 2$. | open Nat Set | [] | @Eq Real answer
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@HMul.hMul Real Real Real (@instHMul Real Real.instMul) Real.pi
(Real.log
(@OfNat.ofNat Real (nat_lit 2)
(@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))))))))
(@OfNat.ofNat Real (nat_lit 8)
(@instOfNatAtLeastTwo Real (nat_lit 8) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))))))) | β | [
{
"t": "β« x in (0:β)..1, (Real.log (x+1))/(x^2 + 1) = answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2005_a5",
"tags": [
"analysis"
]
} |
Find a nonzero polynomial $P(x,y)$ such that $P(\lfloor a \rfloor,\lfloor 2a \rfloor)=0$ for all real numbers $a$. (Note: $\lfloor \nu \rfloor$ is the greatest integer less than or equal to $\nu$.) | (y - 2x)(y - 2x - 1) | Show that $P(x,y)=(y-2x)(y-2x-1)$ works. | open Nat Set | [] | @Eq (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real Real.instCommSemiring) answer
(@HMul.hMul (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real Real.instCommSemiring)
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real Real.instCommSemiring)
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real Real.instCommSemiring)
(@instHMul
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real Real.instCommSemiring)
(@Distrib.toMul
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real Real.instCommSemiring)
(@NonUnitalNonAssocSemiring.toDistrib
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real Real.instCommSemiring)
(@NonUnitalNonAssocCommSemiring.toNonUnitalNonAssocSemiring
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real Real.instCommSemiring)
(@NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real Real.instCommSemiring)
(@NonUnitalCommRing.toNonUnitalNonAssocCommRing
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real
Real.instCommSemiring)
(@CommRing.toNonUnitalCommRing
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real
Real.instCommSemiring)
(@MvPolynomial.instCommRingMvPolynomial Real
(Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real.commRing))))))))
(@HSub.hSub
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real Real.instCommSemiring)
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real Real.instCommSemiring)
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real Real.instCommSemiring)
(@instHSub
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real Real.instCommSemiring)
(@SubNegMonoid.toSub
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real Real.instCommSemiring)
(@AddGroup.toSubNegMonoid
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real Real.instCommSemiring)
(@AddGroupWithOne.toAddGroup
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real Real.instCommSemiring)
(@Ring.toAddGroupWithOne
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real
Real.instCommSemiring)
(@CommRing.toRing
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real
Real.instCommSemiring)
(@MvPolynomial.instCommRingMvPolynomial Real
(Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real.commRing)))))))
(@MvPolynomial.X Real (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real.instCommSemiring
(@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) (nat_lit 1)
(@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))
(@Nat.instNeZeroSucc (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))) (nat_lit 1))))
(@HMul.hMul
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real Real.instCommSemiring)
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real Real.instCommSemiring)
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real Real.instCommSemiring)
(@instHMul
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real Real.instCommSemiring)
(@Distrib.toMul
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real Real.instCommSemiring)
(@NonUnitalNonAssocSemiring.toDistrib
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real Real.instCommSemiring)
(@NonUnitalNonAssocCommSemiring.toNonUnitalNonAssocSemiring
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real
Real.instCommSemiring)
(@NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real
Real.instCommSemiring)
(@NonUnitalCommRing.toNonUnitalNonAssocCommRing
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real
Real.instCommSemiring)
(@CommRing.toNonUnitalCommRing
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real
Real.instCommSemiring)
(@MvPolynomial.instCommRingMvPolynomial Real
(Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real.commRing))))))))
(@OfNat.ofNat
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real Real.instCommSemiring)
(nat_lit 2)
(@instOfNatAtLeastTwo
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real Real.instCommSemiring)
(nat_lit 2)
(@AddMonoidWithOne.toNatCast
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real Real.instCommSemiring)
(@AddGroupWithOne.toAddMonoidWithOne
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real
Real.instCommSemiring)
(@Ring.toAddGroupWithOne
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real
Real.instCommSemiring)
(@CommRing.toRing
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real
Real.instCommSemiring)
(@MvPolynomial.instCommRingMvPolynomial Real
(Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real.commRing)))))
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))))))
(@MvPolynomial.X Real (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real.instCommSemiring
(@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) (nat_lit 0)
(@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))
(@Nat.instNeZeroSucc (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))) (nat_lit 0))))))
(@HSub.hSub
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real Real.instCommSemiring)
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real Real.instCommSemiring)
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real Real.instCommSemiring)
(@instHSub
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real Real.instCommSemiring)
(@SubNegMonoid.toSub
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real Real.instCommSemiring)
(@AddGroup.toSubNegMonoid
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real Real.instCommSemiring)
(@AddGroupWithOne.toAddGroup
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real Real.instCommSemiring)
(@Ring.toAddGroupWithOne
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real
Real.instCommSemiring)
(@CommRing.toRing
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real
Real.instCommSemiring)
(@MvPolynomial.instCommRingMvPolynomial Real
(Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real.commRing)))))))
(@HSub.hSub
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real Real.instCommSemiring)
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real Real.instCommSemiring)
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real Real.instCommSemiring)
(@instHSub
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real Real.instCommSemiring)
(@SubNegMonoid.toSub
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real Real.instCommSemiring)
(@AddGroup.toSubNegMonoid
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real Real.instCommSemiring)
(@AddGroupWithOne.toAddGroup
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real
Real.instCommSemiring)
(@Ring.toAddGroupWithOne
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real
Real.instCommSemiring)
(@CommRing.toRing
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real
Real.instCommSemiring)
(@MvPolynomial.instCommRingMvPolynomial Real
(Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real.commRing)))))))
(@MvPolynomial.X Real (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real.instCommSemiring
(@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) (nat_lit 1)
(@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))
(@Nat.instNeZeroSucc (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))) (nat_lit 1))))
(@HMul.hMul
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real Real.instCommSemiring)
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real Real.instCommSemiring)
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real Real.instCommSemiring)
(@instHMul
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real Real.instCommSemiring)
(@Distrib.toMul
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real Real.instCommSemiring)
(@NonUnitalNonAssocSemiring.toDistrib
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real
Real.instCommSemiring)
(@NonUnitalNonAssocCommSemiring.toNonUnitalNonAssocSemiring
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real
Real.instCommSemiring)
(@NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real
Real.instCommSemiring)
(@NonUnitalCommRing.toNonUnitalNonAssocCommRing
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real
Real.instCommSemiring)
(@CommRing.toNonUnitalCommRing
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real
Real.instCommSemiring)
(@MvPolynomial.instCommRingMvPolynomial Real
(Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real.commRing))))))))
(@OfNat.ofNat
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real Real.instCommSemiring)
(nat_lit 2)
(@instOfNatAtLeastTwo
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real Real.instCommSemiring)
(nat_lit 2)
(@AddMonoidWithOne.toNatCast
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real
Real.instCommSemiring)
(@AddGroupWithOne.toAddMonoidWithOne
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real
Real.instCommSemiring)
(@Ring.toAddGroupWithOne
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real
Real.instCommSemiring)
(@CommRing.toRing
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real
Real.instCommSemiring)
(@MvPolynomial.instCommRingMvPolynomial Real
(Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real.commRing)))))
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))))))
(@MvPolynomial.X Real (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real.instCommSemiring
(@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) (nat_lit 0)
(@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))
(@Nat.instNeZeroSucc (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))) (nat_lit 0))))))
(@OfNat.ofNat
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real Real.instCommSemiring)
(nat_lit 1)
(@One.toOfNat1
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real Real.instCommSemiring)
(@AddMonoidWithOne.toOne
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real Real.instCommSemiring)
(@AddGroupWithOne.toAddMonoidWithOne
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real Real.instCommSemiring)
(@Ring.toAddGroupWithOne
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real
Real.instCommSemiring)
(@CommRing.toRing
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real
Real.instCommSemiring)
(@MvPolynomial.instCommRingMvPolynomial Real
(Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real.commRing))))))))) | MvPolynomial (Fin 2) β | [
{
"t": "answer β 0",
"v": null,
"name": "h_nonzero"
},
{
"t": "β a : β, MvPolynomial.eval (fun n : Fin 2 => if (n = 0) then (Int.floor a : β) else (Int.floor (2 * a))) answer = 0",
"v": null,
"name": "h_eval"
}
] | {
"problem_name": "putnam_2005_b1",
"tags": [
"algebra"
]
} |
Find all positive integers $n,k_1,\dots,k_n$ such that $k_1+\cdots+k_n=5n-4$ and $\frac{1}{k_1}+\cdots+\frac{1}{k_n}=1$. | {(n, k) : β Γ (β β β€) | (n = 1 β§ k 0 = 1) β¨ (n = 3 β§ (k '' {0, 1, 2} = {2, 3, 6})) β¨ (n = 4 β§ (β i : Fin 4, k i = 4))} | Show that the solutions are $n=1$ and $k_1=1$, $n=3$ and $(k_1,k_2,k_3)$ is a permutation of $(2,3,6)$, and $n=4$ and $(k_1,k_2,k_3,k_4)=(4,4,4,4)$. | open Nat Set | [] | @Eq (Set (Prod Nat (Nat β Int))) answer
(@setOf (Prod Nat (Nat β Int)) fun (x : Prod Nat (Nat β Int)) =>
_example.match_1 (fun (x_1 : Prod Nat (Nat β Int)) => Prop) x fun (n : Nat) (k : Nat β Int) =>
Or
(And (@Eq Nat n (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))
(@Eq Int (k (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))))
(@OfNat.ofNat Int (nat_lit 1) (@instOfNat (nat_lit 1)))))
(Or
(And (@Eq Nat n (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3))))
(@Eq (Set Int)
(@Set.image Nat Int k
(@Insert.insert Nat (Set Nat) (@Set.instInsert Nat)
(@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0)))
(@Insert.insert Nat (Set Nat) (@Set.instInsert Nat)
(@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))
(@Singleton.singleton Nat (Set Nat) (@Set.instSingletonSet Nat)
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))))
(@Insert.insert Int (Set Int) (@Set.instInsert Int)
(@OfNat.ofNat Int (nat_lit 2) (@instOfNat (nat_lit 2)))
(@Insert.insert Int (Set Int) (@Set.instInsert Int)
(@OfNat.ofNat Int (nat_lit 3) (@instOfNat (nat_lit 3)))
(@Singleton.singleton Int (Set Int) (@Set.instSingletonSet Int)
(@OfNat.ofNat Int (nat_lit 6) (@instOfNat (nat_lit 6))))))))
(And (@Eq Nat n (@OfNat.ofNat Nat (nat_lit 4) (instOfNatNat (nat_lit 4))))
(β (i : Fin (@OfNat.ofNat Nat (nat_lit 4) (instOfNatNat (nat_lit 4)))),
@Eq Int (k (@Fin.val (@OfNat.ofNat Nat (nat_lit 4) (instOfNatNat (nat_lit 4))) i))
(@OfNat.ofNat Int (nat_lit 4) (@instOfNat (nat_lit 4))))))) | Set (β Γ (β β β€)) | [
{
"t": "{((n : β), (k : β β β€)) | (n > 0) β§ (β i β Finset.range n, k i > 0) β§ (β i in Finset.range n, k i = 5 * n - 4) β§ (β i : Finset.range n, (1 : β) / (k i) = 1)} = answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2005_b2",
"tags": [
"algebra"
]
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.