informal_problem string | informal_answer string | header string | formal_answer string | formal_answer_type string | metainfo dict | independent_variables list | hypotheses list | conclusions list |
|---|---|---|---|---|---|---|---|---|
Find all differentiable functions $f:(0,\infty) \to (0,\infty)$ for which there is a positive real number $a$ such that $f'(\frac{a}{x})=\frac{x}{f(x)}$ for all $x>0$. | the set of functions $f$ such that $f(x) = c x^d$ for some $c > 0$ and $d > 0$, with $d = 1$ implying $c = 1$ | open Nat Set | answer =
({f : ℝ → ℝ |
∃ c > (0 : ℝ),
∃ d > (0 : ℝ),
(d = (1 : ℝ) → c = (1 : ℝ)) ∧ Set.EqOn f (fun (x : ℝ) => (c * (x ^ d : ℝ) : ℝ)) (Set.Ioi (0 : ℝ) : Set ℝ)} :
Set (ℝ → ℝ)) | Set (ℝ → ℝ) | {
"benchmark": "putnam_solving",
"index": 201
} | [
{
"t": "ℝ → ℝ",
"v": null,
"name": "f",
"t_type": "Type"
}
] | [
{
"t": "∀ x > 0, 0 < f x",
"v": null,
"name": "hf",
"t_type": "Prop"
},
{
"t": "DifferentiableOn ℝ f (Ioi 0)",
"v": null,
"name": "hf'",
"t_type": "Prop"
}
] | [
"((∃ a > 0, ∀ x > 0, deriv f (a / x) = x / f x) ↔ f ∈ answer)"
] |
Find the volume of the region of points $(x,y,z)$ such that
\[
(x^2 + y^2 + z^2 + 8)^2 \leq 36(x^2 + y^2).
\] | 6 * Real.pi ^ 2 | open Real | answer = ((6 : ℝ) * ((π : ℝ) ^ (2 : ℕ) : ℝ) : ℝ) | ℝ | {
"benchmark": "putnam_solving",
"index": 202
} | [] | [] | [
"((MeasureTheory.volume {a : ℝ × ℝ × ℝ | (a.1 ^ 2 + a.2.1 ^ 2 + a.2.2 ^ 2 + 8) ^ 2 ≤ 36 * (a.1 ^ 2 + a.2.1 ^ 2)}).toReal = answer)"
] |
Let $S=\{1,2,\dots,n\}$ for some integer $n>1$. Say a permutation $\pi$ of $S$ has a \emph{local maximum} at $k \in S$ if
\begin{enumerate}
\item[(i)] $\pi(k)>\pi(k+1)$ for $k=1$;
\item[(ii)] $\pi(k-1)<\pi(k)$ and $\pi(k)>\pi(k+1)$ for $1<k<n$;
\item[(iii)] $\pi(k-1)<\pi(k)$ for $k=n$.
\end{enumerate}
(For example, if ... | (fun n : ℕ => (n + 1) / 3) | answer = fun (n : ℕ) => (((↑n : ℝ) + (1 : ℝ) : ℝ) / (3 : ℝ) : ℝ) | ℕ → ℝ | {
"benchmark": "putnam_solving",
"index": 203
} | [
{
"t": "ℕ",
"v": null,
"name": "n",
"t_type": "Type"
}
] | [
{
"t": "Equiv.Perm (Fin n) → (ℕ → ℕ)",
"v": null,
"name": "pnat",
"t_type": "Type"
},
{
"t": "Equiv.Perm (Fin n) → ℕ",
"v": null,
"name": "pcount",
"t_type": "Type"
},
{
"t": "n > 1",
"v": null,
"name": "ngt1",
"t_type": "Prop"
},
{
"t": "∀ p : Equiv.P... | [
"((∑ p : Equiv.Perm (Fin n), pcount p) / {p : Equiv.Perm (Fin n) | true}.ncard = answer n)"
] | |
Let $n$ be a positive odd integer and let $\theta$ be a real number such that $\theta/\pi$ is irrational. Set $a_k=\tan(\theta+k\pi/n)$, $k=1,2,\dots,n$. Prove that $\frac{a_1+a_2+\cdots+a_n}{a_1a_2 \cdots a_n}$ is an integer, and determine its value. | (fun n : ℕ => if (n ≡ 1 [MOD 4]) then n else -n) | answer = fun (n : ℕ) => (if n ≡ (1 : ℕ) [MOD (4 : ℕ)] then (↑n : ℤ) else (-(↑n : ℤ) : ℤ) : ℤ) | ℕ → ℤ | {
"benchmark": "putnam_solving",
"index": 204
} | [
{
"t": "ℕ",
"v": null,
"name": "n",
"t_type": "Type"
}
] | [
{
"t": "ℝ",
"v": null,
"name": "theta",
"t_type": "Type"
},
{
"t": "Set.Icc 1 n → ℝ",
"v": null,
"name": "a",
"t_type": "Type"
},
{
"t": "Odd n",
"v": null,
"name": "nodd",
"t_type": "Prop"
},
{
"t": "Irrational (theta / Real.pi)",
"v": null,
"... | [
"((∑ k : Set.Icc 1 n, a k) / (∏ k : Set.Icc 1 n, a k) = answer n)"
] | |
Show that the curve $x^3 + 3xy + y^3 = 1$ contains only one set of three distinct points, $A$, $B$, and $C$, which are vertices of an equilateral triangle, and find its area. | 3 * Real.sqrt 3 / 2 | answer = (((3 : ℝ) * (√(3 : ℝ) : ℝ) : ℝ) / (2 : ℝ) : ℝ) | ℝ | {
"benchmark": "putnam_solving",
"index": 205
} | [] | [
{
"t": "Set (ℝ × ℝ)",
"v": null,
"name": "curve",
"t_type": "Type"
},
{
"t": "curve = {c | c.1 ^ 3 + 3 * c.1 * c.2 + c.2 ^ 3 = 1}",
"v": null,
"name": "hcurve",
"t_type": "Prop"
},
{
"t": "Set (ℝ × ℝ) → Prop",
"v": null,
"name": "equilateral",
"t_type": "Type"... | [
"((∃! S : Set (ℝ × ℝ), S ⊆ curve ∧ equilateral S) ∧ (∃ S : Set (ℝ × ℝ), S ⊆ curve ∧ equilateral S ∧ (MeasureTheory.volume (convexHull ℝ S)).toReal = answer))"
] | |
Let $S$ be a finite set of points in the plane. A linear partition of $S$ is an unordered pair $\{A,B\}$ of subsets of $S$ such that $A \cup B=S$, $A \cap B=\emptyset$, and $A$ and $B$ lie on opposite sides of some straight line disjoint from $S$ ($A$ or $B$ may be empty). Let $L_S$ be the number of linear partitions o... | $\binom{n}{2} + 1$ | answer = fun (n : ℕ) => ((n.choose (2 : ℕ) : ℕ) + (1 : ℕ) : ℕ) | ℕ → ℕ | {
"benchmark": "putnam_solving",
"index": 206
} | [
{
"t": "ℕ",
"v": null,
"name": "n",
"t_type": "Type"
}
] | [
{
"t": "Finset (Fin 2 → ℝ) → Finset (Finset (Fin 2 → ℝ)) → Prop",
"v": null,
"name": "IsLinearPartition",
"t_type": "Type"
},
{
"t": "∀ S AB, IsLinearPartition S AB ↔\n (AB.card = 2 ∧ ∃ A ∈ AB, ∃ B ∈ AB,\n A ≠ B ∧ (A ∪ B = S) ∧ (A ∩ B = ∅) ∧\n (∃ m b : ℝ,\n (∀ p ∈... | [
"(IsGreatest {L S | (S) (hS : S.card = n)} (answer n))"
] | |
Let $Z$ denote the set of points in $\mathbb{R}^n$ whose coordinates are $0$ or $1$. (Thus $Z$ has $2^n$ elements, which are the vertices of a unit hypercube in $\mathbb{R}^n$.) Given a vector subspace $V$ of $\mathbb{R}^n$, let $Z(V)$ denote the number of members of $Z$ that lie in $V$. Let $k$ be given, $0 \leq k \le... | $2^k$ | answer = fun (k : ℕ) => ((2 : ℕ) ^ k : ℕ) | ℕ → ℕ | {
"benchmark": "putnam_solving",
"index": 207
} | [
{
"t": "ℕ",
"v": null,
"name": "k",
"t_type": "Type"
}
] | [
{
"t": "ℕ",
"v": null,
"name": "n",
"t_type": "Type"
},
{
"t": "n > 0",
"v": null,
"name": "npos",
"t_type": "Prop"
},
{
"t": "k ≤ n",
"v": null,
"name": "hk",
"t_type": "Prop"
},
{
"t": "Set (Fin n → ℝ)",
"v": null,
"name": "Z",
"t_type": ... | [
"(IsGreatest\n {y | ∃ V : Subspace ℝ (Fin n → ℝ), Module.rank ℝ V = k ∧ (Z ∩ V).ncard = y}\n (answer k))"
] | |
For each continuous function $f: [0,1] \to \mathbb{R}$, let $I(f) = \int_0^1 x^2 f(x)\,dx$ and $J(x) = \int_0^1 x \left(f(x)\right)^2\,dx$. Find the maximum value of $I(f) - J(f)$ over all such functions $f$. | $\frac{1}{16}$ | open Set | answer = (1 / 16 : ℝ) | ℝ | {
"benchmark": "putnam_solving",
"index": 208
} | [] | [
{
"t": "(ℝ → ℝ) → ℝ",
"v": null,
"name": "I",
"t_type": "Type"
},
{
"t": "(ℝ → ℝ) → ℝ",
"v": null,
"name": "J",
"t_type": "Type"
},
{
"t": "I = fun f ↦ ∫ x in (0)..1, x ^ 2 * (f x)",
"v": null,
"name": "hI",
"t_type": "Prop"
},
{
"t": "J = fun f ↦ ∫ x ... | [
"(IsGreatest {y | ∃ f : ℝ → ℝ, ContinuousOn f (Icc 0 1) ∧ I f - J f = y} answer)"
] |
Let $k$ be an integer greater than 1. Suppose $a_0 > 0$, and define \[ a_{n+1} = a_n + \frac{1}{\sqrt[k]{a_n}} \] for $n > 0$. Evaluate \[\lim_{n \to \infty} \frac{a_n^{k+1}}{n^k}.\] | fun k => ((k+1)/k)^k | open Set Topology Filter | answer = fun (k : ℕ) => ((((↑k : ℝ) + (1 : ℝ) : ℝ) / (↑k : ℝ) : ℝ) ^ k : ℝ) | ℕ → ℝ | {
"benchmark": "putnam_solving",
"index": 209
} | [
{
"t": "ℕ",
"v": null,
"name": "k",
"t_type": "Type"
}
] | [
{
"t": "k > 1",
"v": null,
"name": "hk",
"t_type": "Prop"
},
{
"t": "ℕ → ℝ",
"v": null,
"name": "a",
"t_type": "Type"
},
{
"t": "a 0 > 0",
"v": null,
"name": "ha0",
"t_type": "Prop"
},
{
"t": "∀ n : ℕ, a (n + 1) = a n + 1/((a n)^((1 : ℝ)/k))",
"v":... | [
"(Tendsto (fun n => (a n)^(k+1)/(n ^ k)) atTop (𝓝 (answer k)))"
] |
Find all values of $\alpha$ for which the curves $y = \alpha*x^2 + \alpha*x + 1/24$ and $x = \alpha*y^2 + \alpha*y + 1/24$ are tangent to each other. | {2 / 3, 3 / 2, (13 + √601) / 12, (13 - √601) / 12} | answer =
(Insert.insert (2 / 3 : ℝ)
(Insert.insert (3 / 2 : ℝ)
(Insert.insert (((13 : ℝ) + (√(601 : ℝ) : ℝ) : ℝ) / (12 : ℝ) : ℝ)
({(((13 : ℝ) - (√(601 : ℝ) : ℝ) : ℝ) / (12 : ℝ) : ℝ)} : Set ℝ) :
Set ℝ) :
Set ℝ) :
Set ℝ) | Set ℝ | {
"benchmark": "putnam_solving",
"index": 210
} | [] | [
{
"t": "ℝ",
"v": null,
"name": "α",
"t_type": "Type"
},
{
"t": "(ℝ → ℝ) → Prop",
"v": null,
"name": "P",
"t_type": "Type"
},
{
"t": "∀ f, P f ↔ ∃ x y, f x = y ∧ f y = x ∧ deriv f x * deriv f y = 1",
"v": null,
"name": "P_def",
"t_type": "Prop"
}
] | [
"(P (fun t ↦ α * t ^ 2 + α * t + 1 / 24) ↔ α ∈ answer)"
] | |
Find the least possible area of a convex set in the plane that intersects both branches of the hyperbola $xy=1$ and both branches of the hyperbola $xy=-1$. (A set $S$ in the plane is called \emph{convex} if for any two points in $S$ the line segment connecting them is contained in $S$.) | 4 | open MeasureTheory | answer = (4 : ENNReal) | ENNReal | {
"benchmark": "putnam_solving",
"index": 211
} | [] | [] | [
"(IsLeast\n {y | ∃ S : Set (Fin 2 → ℝ),\n Convex ℝ S ∧\n (∃ p ∈ S, p 0 > 0 ∧ p 1 > 0 ∧ p 0 * p 1 = 1) ∧\n (∃ p ∈ S, p 0 < 0 ∧ p 1 < 0 ∧ p 0 * p 1 = 1) ∧\n (∃ p ∈ S, p 0 < 0 ∧ p 1 > 0 ∧ p 0 * p 1 = -1) ∧\n (∃ p ∈ S, p 0 > 0 ∧ p 1 < 0 ∧ p 0 * p 1 = -1) ∧\n volume S = y} answer)"
] |
Let $k$ be a positive integer. Suppose that the integers $1, 2, 3, \dots, 3k+1$ are written down in random order. What is the probability that at no time during this process, the sum of the integers that have been written up to that time is a positive integer divisible by $3$? Your answer should be in closed form, but ... | fun k ↦ (k)! * (k + 1)! / ((3 * k + 1) * (2 * k)!) | open Set Nat | answer = fun (k : ℕ) =>
(((↑(k ! : ℕ) : ℚ) * (↑((k + (1 : ℕ) : ℕ)! : ℕ) : ℚ) : ℚ) /
((((3 : ℚ) * (↑k : ℚ) : ℚ) + (1 : ℚ) : ℚ) * (↑(((2 : ℕ) * k : ℕ)! : ℕ) : ℚ) : ℚ) :
ℚ) | ℕ → ℚ | {
"benchmark": "putnam_solving",
"index": 212
} | [
{
"t": "ℕ",
"v": null,
"name": "k",
"t_type": "Type"
}
] | [
{
"t": "k > 0",
"v": null,
"name": "kpos",
"t_type": "Prop"
},
{
"t": "Set (Fin (3 * k + 1) → ℤ)",
"v": null,
"name": "perms",
"t_type": "Type"
},
{
"t": "Set (Fin (3 * k + 1) → ℤ)",
"v": null,
"name": "goodperms",
"t_type": "Type"
},
{
"t": "goodperms... | [
"(goodperms.ncard = perms.ncard * (answer k))"
] |
A \emph{repunit} is a positive integer whose digits in base 10 are all ones. Find all polynomials $f$ with real coefficients such that if $n$ is a repunit, then so is $f(n)$. | the set of polynomials of the form $f(n) = \frac{1}{9} \left(10^c (9n + 1)^d - 1\right)$ where $d \in \mathbb{N}$ and $c \geq 1 - d$ | open Set Nat | answer =
({f : Polynomial ℝ |
∃ (d : ℕ),
∃ c ≥ ((1 : ℤ) - (↑d : ℤ) : ℤ),
∀ (n : ℝ),
(Polynomial.eval n f : ℝ) =
((1 / 9 : ℝ) * ((((10 : ℝ) ^ c : ℝ) * ((((9 : ℝ) * n : ℝ) + (1 : ℝ) : ℝ) ^ d : ℝ) : ℝ) - (1 : ℝ) : ℝ) :
ℝ)} :
Set (Polynomial ℝ)) | Set (Polynomial ℝ) | {
"benchmark": "putnam_solving",
"index": 213
} | [] | [
{
"t": "Set (Polynomial ℝ)",
"v": null,
"name": "S",
"t_type": "Type"
},
{
"t": "ℝ → Prop",
"v": null,
"name": "repunit",
"t_type": "Type"
},
{
"t": "∀ x, repunit x ↔ x > 0 ∧ x = floor x ∧ ∀ d ∈ (digits 10 (floor x)), d = 1",
"v": null,
"name": "hrepunit",
"t_... | [
"(S = answer)"
] |
Let $x_0 = 1$ and for $n \geq 0$, let $x_{n+1} = 3x_n + \lfloor x_n \sqrt{5} \rfloor$. In particular, $x_1 = 5$, $x_2 = 26$, $x_3 = 136$, $x_4 = 712$. Find a closed-form expression for $x_{2007}$. ($\lfloor a \rfloor$ means the largest integer $\leq a$.) | $\frac{2^{2006}}{\sqrt{5}} \left( \left( \frac{1 + \sqrt{5}}{2} \right)^{3997} - \left( \frac{1 + \sqrt{5}}{2} \right)^{-3997} \right)$ | open Set Nat Function | answer =
((((2 : ℝ) ^ (2006 : ℕ) : ℝ) / (√(5 : ℝ) : ℝ) : ℝ) *
(((((1 : ℝ) + (√(5 : ℝ) : ℝ) : ℝ) / (2 : ℝ) : ℝ) ^ (3997 : ℕ) : ℝ) -
((((1 : ℝ) + (√(5 : ℝ) : ℝ) : ℝ) / (2 : ℝ) : ℝ) ^ (-3997 : ℤ) : ℝ) :
ℝ) :
ℝ) | ℝ | {
"benchmark": "putnam_solving",
"index": 214
} | [] | [
{
"t": "ℕ → ℝ",
"v": null,
"name": "x",
"t_type": "Type"
},
{
"t": "x 0 = 1",
"v": null,
"name": "hx0",
"t_type": "Prop"
},
{
"t": "∀ n : ℕ, x (n + 1) = 3 * (x n) + ⌊(x n) * Real.sqrt 5⌋",
"v": null,
"name": "hx",
"t_type": "Prop"
}
] | [
"(x 2007 = answer)"
] |
Let $n$ be a positive integer. Find the number of pairs $P, Q$ of polynomials with real coefficients such that
\[
(P(X))^2 + (Q(X))^2 = X^{2n} + 1
\]
and $\deg P > \deg Q$. | fun n ↦ 2 ^ (n + 1) | open Set Nat Function | answer = fun (n : ℕ) => ((2 : ℕ) ^ (n + (1 : ℕ) : ℕ) : ℕ) | ℕ → ℕ | {
"benchmark": "putnam_solving",
"index": 215
} | [
{
"t": "ℕ",
"v": null,
"name": "n",
"t_type": "Type"
}
] | [
{
"t": "n > 0",
"v": null,
"name": "npos",
"t_type": "Prop"
}
] | [
"({a : (Polynomial ℝ) × (Polynomial ℝ) | a.1 ^ 2 + a.2 ^ 2 = Polynomial.X ^ (2 * n) + 1 ∧ a.1.degree > a.2.degree}.ncard = answer n)"
] |
Define $f : \mathbb{R} \to \mathbb{R} by $f(x) = x$ if $x \leq e$ and $f(x) = x * f(\ln(x))$ if $x > e$. Does $\sum_{n=1}^{\infty} 1/(f(n))$ converge? | False | open Filter Topology | answer = False | Prop | {
"benchmark": "putnam_solving",
"index": 216
} | [] | [
{
"t": "ℝ → ℝ",
"v": null,
"name": "f",
"t_type": "Type"
},
{
"t": "f = fun x => if x ≤ Real.exp 1 then x else x * (f (Real.log x))",
"v": null,
"name": "hf",
"t_type": "Prop"
}
] | [
"((∃ r : ℝ, Tendsto (fun N : ℕ => ∑ n in Finset.range N, 1/(f (n + 1))) atTop (𝓝 r)) ↔ answer)"
] |
What is the maximum number of rational points that can lie on a circle in $\mathbb{R}^2$ whose center is not a rational point? (A \emph{rational point} is a point both of whose coordinates are rational numbers.) | 2 | open Filter Topology Set | answer = (2 : ℕ) | ℕ | {
"benchmark": "putnam_solving",
"index": 217
} | [] | [
{
"t": "EuclideanSpace ℝ (Fin 2) → Prop",
"v": null,
"name": "is_rational_point",
"t_type": "Type"
},
{
"t": "∀ p : EuclideanSpace ℝ (Fin 2), is_rational_point p ↔ ∃ (a b : ℚ), a = p 0 ∧ b = p 1",
"v": null,
"name": "h_rational_point",
"t_type": "Prop"
},
{
"t": "Euclidea... | [
"(IsGreatest {n : ℕ | ∃ (c : EuclideanSpace ℝ (Fin 2)) (r : ℝ), ¬ is_rational_point c ∧ (Set.ncard {p : EuclideanSpace ℝ (Fin 2) | p ∈ real_circle c r ∧ is_rational_point p} = n)} answer)"
] |
Let $F_0(x)=\ln x$. For $n \geq 0$ and $x>0$, let $F_{n+1}(x)=\int_0^x F_n(t)\,dt$. Evaluate $\lim_{n \to \infty} \frac{n!F_n(1)}{\ln n}$. | -1 | open Filter Topology Set Nat | answer = (-1 : ℝ) | ℝ | {
"benchmark": "putnam_solving",
"index": 218
} | [] | [
{
"t": "ℕ → ℝ → ℝ",
"v": null,
"name": "F",
"t_type": "Type"
},
{
"t": "∀ x : ℝ, F 0 x = Real.log x",
"v": null,
"name": "hF0",
"t_type": "Prop"
},
{
"t": "∀ n : ℕ, ∀ x > 0, F (n + 1) x = ∫ t in Set.Ioo 0 x, F n t",
"v": null,
"name": "hFn",
"t_type": "Prop"
... | [
"(Tendsto (fun n : ℕ => ((n)! * F n 1) / Real.log n) atTop (𝓝 answer))"
] |
What is the largest possible radius of a circle contained in a $4$-dimensional hypercube of side length $1$? | √2 / 2 | open Metric Filter Topology Set Nat | answer = ((√(2 : ℝ) : ℝ) / (2 : ℝ) : ℝ) | ℝ | {
"benchmark": "putnam_solving",
"index": 219
} | [] | [
{
"t": "Set (EuclideanSpace ℝ (Fin 4))",
"v": null,
"name": "H",
"t_type": "Type"
},
{
"t": "H = {P : Fin 4 → ℝ | ∀ i : Fin 4, |P i| ≤ 1 / 2}",
"v": null,
"name": "H_def",
"t_type": "Prop"
},
{
"t": "ℝ → Prop",
"v": null,
"name": "contains",
"t_type": "Type"
... | [
"(IsGreatest contains answer)"
] |
Find all continuously differentiable functions f : \mathbb{R} \to \mathbb{R} such that for every rational number $q$, the number $f(q)$ is rational and has the same denominator as $q$. | {fun (x : ℝ) => x + n | n : ℤ} ∪ {fun (x : ℝ) => -x + n | n : ℤ} | open Filter Topology Set Nat | answer =
(({x : ℝ → ℝ | ∃ (n : ℤ), (fun (x : ℝ) => (x + (↑n : ℝ) : ℝ)) = x} : Set (ℝ → ℝ)) ∪
({x : ℝ → ℝ | ∃ (n : ℤ), (fun (x : ℝ) => ((-x : ℝ) + (↑n : ℝ) : ℝ)) = x} : Set (ℝ → ℝ)) :
Set (ℝ → ℝ)) | Set (ℝ → ℝ) | {
"benchmark": "putnam_solving",
"index": 220
} | [] | [
{
"t": "(ℝ → ℝ) → ℚ → Prop",
"v": null,
"name": "fqsat",
"t_type": "Type"
},
{
"t": "∀ f q, fqsat f q ↔ ContDiff ℝ 1 f ∧ (∃ p : ℚ, p = f q ∧ p.den = q.den)",
"v": null,
"name": "hfqsat",
"t_type": "Prop"
}
] | [
"(∀ f : (ℝ → ℝ), (∀ q : ℚ, fqsat f q) ↔ f ∈ answer)"
] |
Let $f$ be a real-valued function on the plane such that for every square $ABCD$ in the plane, $f(A)+f(B)+f(C)+f(D)=0$. Does it follow that $f(P)=0$ for all points $P$ in the plane? | True | open Topology MvPolynomial Filter | answer = True | Prop | {
"benchmark": "putnam_solving",
"index": 221
} | [] | [
{
"t": "ℝ × ℝ → ℝ",
"v": null,
"name": "f",
"t_type": "Type"
},
{
"t": "∀ O v : ℝ × ℝ, v ≠ (0, 0) → f (O.1, O.2) + f (O.1 + v.1, O.2 + v.2) + f (O.1 + v.1 - v.2, O.2 + v.2 + v.1) + f (O.1 - v.2, O.2 + v.1) = 0",
"v": null,
"name": "h_square",
"t_type": "Prop"
}
] | [
"((∀ P : ℝ × ℝ, f P = 0) ↔ answer)"
] |
Functions $f,g,h$ are differentiable on some open interval around $0$
and satisfy the equations and initial conditions
\begin{gather*}
f' = 2f^2gh+\frac{1}{gh},\quad f(0)=1, \\
g'=fg^2h+\frac{4}{fh}, \quad g(0)=1, \\
h'=3fgh^2+\frac{1}{fg}, \quad h(0)=1.
\end{gather*}
Find an explicit formula for $f(x)$, valid in some ... | $2^{-1/12} \left(\frac{\sin(6x+\pi/4)}{\cos^2(6x+\pi/4)}\right)^{1/6}$ | open Topology MvPolynomial Filter Set | answer = fun (x : ℝ) =>
(((2 : ℝ) ^ (-1 / 12 : ℝ) : ℝ) *
(((Real.sin (((6 : ℝ) * x : ℝ) + ((Real.pi : ℝ) / (4 : ℝ) : ℝ) : ℝ) : ℝ) /
((Real.cos (((6 : ℝ) * x : ℝ) + ((Real.pi : ℝ) / (4 : ℝ) : ℝ) : ℝ) : ℝ) ^ (2 : ℕ) : ℝ) :
ℝ) ^
(1 / 6 : ℝ) :
ℝ) :
ℝ) | ℝ → ℝ | {
"benchmark": "putnam_solving",
"index": 222
} | [] | [
{
"t": "ℝ → ℝ",
"v": null,
"name": "f",
"t_type": "Type"
},
{
"t": "ℝ → ℝ",
"v": null,
"name": "g",
"t_type": "Type"
},
{
"t": "ℝ → ℝ",
"v": null,
"name": "h",
"t_type": "Type"
},
{
"t": "ℝ",
"v": null,
"name": "a",
"t_type": "Type"
},
... | [
"(∃ c d : ℝ, 0 ∈ Ioo c d ∧ ∀ x ∈ Ioo c d, f x = answer x)"
] |
Let $d_n$ be the determinant of the $n \times n$ matrix whose entries, from left to right and then from top to bottom, are $\cos 1, \cos 2, \dots, \cos n^2$. (For example,\[ d_3 = \left|\begin{matrix} \cos 1 & \cos 2 & \cos 3 \\ \cos 4 & \cos 5 & \cos 6 \\ \cos 7 & \cos 8 & \cos 9 \end{matrix} \right|. \]The argument ... | 0 | open Topology MvPolynomial Filter Set | answer = (0 : ℝ) | ℝ | {
"benchmark": "putnam_solving",
"index": 223
} | [] | [
{
"t": "(n : ℕ) → Matrix (Fin n) (Fin n) ℝ",
"v": null,
"name": "cos_matrix",
"t_type": "Type"
},
{
"t": "∀ n : ℕ, ∀ i j : Fin n, (cos_matrix n) i j = Real.cos (1 + n * i + j)",
"v": null,
"name": "hM",
"t_type": "Prop"
}
] | [
"(Tendsto (fun n => (cos_matrix n).det) atTop (𝓝 answer))"
] |
Let $S$ be a set of rational numbers such that
\begin{enumerate}
\item[(a)] $0 \in S$;
\item[(b)] If $x \in S$ then $x+1\in S$ and $x-1\in S$; and
\item[(c)] If $x\in S$ and $x\not\in\{0,1\}$, then $\frac{1}{x(x-1)}\in S$.
\end{enumerate}
Must $S$ contain all rational numbers? | False | open Topology MvPolynomial Filter Set | answer = False | Prop | {
"benchmark": "putnam_solving",
"index": 224
} | [] | [
{
"t": "Set ℚ",
"v": null,
"name": "S",
"t_type": "Type"
},
{
"t": "0 ∈ S",
"v": null,
"name": "h0",
"t_type": "Prop"
},
{
"t": "∀ x ∈ S, x + 1 ∈ S ∧ x - 1 ∈ S",
"v": null,
"name": "h1",
"t_type": "Prop"
},
{
"t": "∀ x ∈ S, x ∉ ({0, 1} : Set ℚ) → 1 / (... | [
"((∀ r : ℚ, r ∈ S) ↔ answer)"
] |
Is there a finite abelian group $G$ such that the product of the orders of all its elements is 2^{2009}? | False | open Topology MvPolynomial Filter Set | answer = False | Prop | {
"benchmark": "putnam_solving",
"index": 225
} | [] | [] | [
"(answer ↔ (∃ (G : Type*) (_ : CommGroup G) (_ : Fintype G), ∏ g : G, orderOf g = 2^2009))"
] |
A game involves jumping to the right on the real number line. If $a$ and $b$ are real numbers and $b > a$, the cost of jumping from $a$ to $b$ is $b^3-ab^2$. For what real numbers $c$ can one travel from $0$ to $1$ in a finite number of jumps with total cost exactly $c$? | the interval (1/3, 1] | open Topology MvPolynomial Filter Set | answer = (Set.Ioc (1 / 3 : ℝ) (1 : ℝ) : Set ℝ) | Set ℝ | {
"benchmark": "putnam_solving",
"index": 226
} | [] | [] | [
"({c : ℝ | ∃ s : ℕ → ℝ, s 0 = 0 ∧ StrictMono s ∧ (∃ n : ℕ, s n = 1 ∧ ((∑ i in Finset.range n, ((s (i + 1)) ^ 3 - (s i) * (s (i + 1)) ^ 2)) = c))} = answer)"
] |
Call a subset $S$ of $\{1, 2, \dots, n\}$ \emph{mediocre} if it has the following property: Whenever $a$ and $b$ are elements of $S$ whose average is an integer, that average is also an element of $S$. Let $A(n)$ be the number of mediocre subsets of $\{1,2,\dots,n\}$. [For instance, every subset of $\{1,2,3\}$ except $... | {n : ℤ | ∃ k ≥ 1, n = 2 ^ k - 1} | open Topology MvPolynomial Filter Set | answer = ({n : ℤ | ∃ k ≥ (1 : ℕ), n = (((2 : ℤ) ^ k : ℤ) - (1 : ℤ) : ℤ)} : Set ℤ) | Set ℤ | {
"benchmark": "putnam_solving",
"index": 227
} | [] | [
{
"t": "ℤ → Set ℤ → Prop",
"v": null,
"name": "mediocre",
"t_type": "Type"
},
{
"t": "∀ n S, mediocre n S ↔ (S ⊆ Icc 1 n) ∧ ∀ a ∈ S, ∀ b ∈ S, 2 ∣ a + b → (a + b) / 2 ∈ S",
"v": null,
"name": "hmediocre",
"t_type": "Prop"
},
{
"t": "ℤ → ℤ",
"v": null,
"name": "A",
... | [
"({n : ℤ | n > 0 ∧ A (n + 2) - 2 * A (n + 1) + A n = 1} = answer)"
] |
Say that a polynomial with real coefficients in two variables, $x,y$, is \emph{balanced} if the average value of the polynomial on each circle centered at the origin is $0$. The balanced polynomials of degree at most $2009$ form a vector space $V$ over $\mathbb{R}$. Find the dimension of $V$. | 2020050 | open intervalIntegral MvPolynomial Real | answer = (2020050 : ℕ) | ℕ | {
"benchmark": "putnam_solving",
"index": 228
} | [] | [
{
"t": "MvPolynomial (Fin 2) ℝ → Prop",
"v": null,
"name": "IsBalanced",
"t_type": "Type"
},
{
"t": "∀ P, IsBalanced P ↔ ∀ r > 0,\n (∫ t in (0 : ℝ)..(2 * π), eval ![r * cos t, r * sin t] P) / (2 * π * r) = 0",
"v": null,
"name": "IsBalanced_def",
"t_type": "Prop"
},
{
... | [
"(Module.rank ℝ V = answer)"
] |
Given a positive integer $n$, what is the largest $k$ such that the numbers $1,2,\dots,n$ can be put into $k$ boxes so that the sum of the numbers in each box is the same? [When $n=8$, the example $\{1,2,3,6\},\{4,8\},\{5,7\}$ shows that the largest $k$ is \emph{at least} $3$.] | the ceiling of n/2 | answer = fun (n : ℕ) => (⌈((↑n : ℝ) / (2 : ℝ) : ℝ)⌉₊ : ℕ) | ℕ → ℕ | {
"benchmark": "putnam_solving",
"index": 229
} | [
{
"t": "ℕ",
"v": null,
"name": "n",
"t_type": "Type"
}
] | [
{
"t": "ℕ → Prop",
"v": null,
"name": "kboxes",
"t_type": "Type"
},
{
"t": "n > 0",
"v": null,
"name": "npos",
"t_type": "Prop"
},
{
"t": "∀ k : ℕ, kboxes k =\n (∃ boxes : Finset.Icc 1 n → Fin k, ∀ i j : Fin k,\n ∑ x in Finset.univ.filter (boxes · = i), (x : ℕ) =\n ... | [
"(IsGreatest kboxes (answer n))"
] | |
Find all differentiable functions $f:\mathbb{R} \to \mathbb{R}$ such that
\[
f'(x) = \frac{f(x+n)-f(x)}{n}
\]
for all real numbers $x$ and all positive integers $n$. | {f : ℝ → ℝ | ∃ c d : ℝ, ∀ x : ℝ, f x = c*x + d} | answer = ({f : ℝ → ℝ | ∃ (c : ℝ) (d : ℝ), ∀ (x : ℝ), (f x : ℝ) = ((c * x : ℝ) + d : ℝ)} : Set (ℝ → ℝ)) | Set (ℝ → ℝ) | {
"benchmark": "putnam_solving",
"index": 230
} | [] | [] | [
"({f : ℝ → ℝ | Differentiable ℝ f ∧\n∀ x : ℝ, ∀ n : ℤ, n > 0 → deriv f x = (f (x + n) - f x)/n} = answer)"
] | |
Is there an infinite sequence of real numbers $a_1, a_2, a_3, \dots$ such that \[ a_1^m + a_2^m + a_3^m + \cdots = m \] for every positive integer $m$? | False | open Filter Topology Set | answer = False | Prop | {
"benchmark": "putnam_solving",
"index": 231
} | [] | [] | [
"(answer ↔ (∃ a : ℕ → ℝ, ∀ m : ℕ, m > 0 → ∑' i : ℕ, (a i)^m = m))"
] |
Given that $A$, $B$, and $C$ are noncollinear points in the plane with integer coordinates such that the distances $AB$, $AC$, and $BC$ are integers, what is the smallest possible value of $AB$? | 3 | open Filter Topology Set | answer = (3 : ℕ) | ℕ | {
"benchmark": "putnam_solving",
"index": 232
} | [] | [
{
"t": "EuclideanSpace ℝ (Fin 2) → EuclideanSpace ℝ (Fin 2) → EuclideanSpace ℝ (Fin 2) → Prop",
"v": null,
"name": "ABCintcoords",
"t_type": "Type"
},
{
"t": "EuclideanSpace ℝ (Fin 2) → EuclideanSpace ℝ (Fin 2) → EuclideanSpace ℝ (Fin 2) → Prop",
"v": null,
"name": "ABCintdists",
... | [
"(IsLeast {y | ∃ A B C, ABCall A B C ∧ y = dist A B} answer)"
] |
There are $2010$ boxes labeled $B_1, B_2, \dots, B_{2010}$, and $2010n$ balls have been distributed among them, for some positive integer $n$. You may redistribute the balls by a sequence of moves, each of which consists of choosing an $i$ and moving \emph{exactly} $i$ balls from box $B_i$ into any one other box. For w... | {1005} | open Filter Topology Set | answer = ({(1005 : ℕ)} : Set ℕ) | Set ℕ | {
"benchmark": "putnam_solving",
"index": 233
} | [] | [
{
"t": "ℕ",
"v": null,
"name": "n",
"t_type": "Type"
},
{
"t": "n > 0",
"v": null,
"name": "hn",
"t_type": "Prop"
},
{
"t": "(ℕ → Fin 2010 → ℕ) → ℕ → Prop",
"v": null,
"name": "trans",
"t_type": "Type"
},
{
"t": "∀ P T, trans P T ↔ ∀ t : ℕ, t < T → ∃ i... | [
"((∀ B, ∑ i, B i = 2010 * n → ∃ᵉ (P) (T), P 0 = B ∧ trans P T ∧ ∀ i, P T i = n) ↔ n ∈ answer)"
] |
Find all pairs of polynomials $p(x)$ and $q(x)$ with real coefficients for which $p(x)q(x+1)-p(x+1)q(x)=1$. | {(p, q) : Polynomial ℝ × Polynomial ℝ | p.degree ≤ 1 ∧ q.degree ≤ 1 ∧ p.coeff 0 * q.coeff 1 - p.coeff 1 * q.coeff 0 = 1} | open Filter Topology Set | answer =
({((p, q) : Polynomial ℝ × Polynomial ℝ) : Polynomial ℝ × Polynomial ℝ |
(p.degree : WithBot ℕ) ≤ (1 : WithBot ℕ) ∧
(q.degree : WithBot ℕ) ≤ (1 : WithBot ℕ) ∧
(((p.coeff (0 : ℕ) : ℝ) * (q.coeff (1 : ℕ) : ℝ) : ℝ) - ((p.coeff (1 : ℕ) : ℝ) * (q.coeff (0 : ℕ) : ℝ) : ℝ) :
ℝ) ... | Set (Polynomial ℝ × Polynomial ℝ) | {
"benchmark": "putnam_solving",
"index": 234
} | [
{
"t": "Polynomial ℝ",
"v": null,
"name": "p",
"t_type": "Type"
},
{
"t": "Polynomial ℝ",
"v": null,
"name": "q",
"t_type": "Type"
}
] | [] | [
"((∀ x : ℝ, p.eval x * q.eval (x + 1) - p.eval (x + 1) * q.eval x = 1) ↔ (p, q) ∈ answer)"
] |
Is there a strictly increasing function $f: \mathbb{R} \to \mathbb{R}$ such that $f'(x) = f(f(x))$ for all $x$? | False | open Filter Topology Set | answer = False | Prop | {
"benchmark": "putnam_solving",
"index": 235
} | [] | [] | [
"((∃ f : ℝ → ℝ, StrictMono f ∧ Differentiable ℝ f ∧ (∀ x : ℝ, deriv f x = f (f x))) ↔ answer)"
] |
Define a \emph{growing spiral} in the plane to be a sequence of points with integer coordinates $P_0=(0,0),P_1,\dots,P_n$ such that $n \geq 2$ and:
\begin{itemize}
\item the directed line segments $P_0P_1,P_1P_2,\dots,P_{n-1}P_n$ are in the successive coordinate directions east (for $P_0P_1$), north, west, south, east,... | 10053 | answer = (10053 : ℕ) | ℕ | {
"benchmark": "putnam_solving",
"index": 236
} | [] | [
{
"t": "List (Fin 2 → ℤ) → Prop",
"v": null,
"name": "IsSpiral",
"t_type": "Type"
},
{
"t": "∀ P, IsSpiral P ↔ P.length ≥ 3 ∧ P[0]! = 0 ∧\n (∃ l : Fin (P.length - 1) → ℕ, l > 0 ∧ StrictMono l ∧ (∀ i : Fin (P.length - 1),\n (i.1 % 4 = 0 → (P[i] 0 + l i = P[i.1 + 1]! 0 ∧ P[i] 1 = P[i.1 + 1... | [
"({p | 0 ≤ p 0 ∧ p 0 ≤ 2011 ∧ 0 ≤ p 1 ∧ p 1 ≤ 2011 ∧ ¬∃ spiral, IsSpiral spiral ∧ spiral.getLast! = p}.encard = answer)"
] | |
Let $a_1,a_2,\dots$ and $b_1,b_2,\dots$ be sequences of positive real numbers such that $a_1 = b_1 = 1$ and $b_n = b_{n-1} a_n - 2$ for$n=2,3,\dots$. Assume that the sequence $(b_j)$ is bounded. Prove tha \[ S = \sum_{n=1}^\infty \frac{1}{a_1...a_n} \] converges, and evaluate $S$. | 3/2 | open Topology Filter | answer = (3 / 2 : ℝ) | ℝ | {
"benchmark": "putnam_solving",
"index": 237
} | [] | [
{
"t": "ℕ → ℝ",
"v": null,
"name": "a",
"t_type": "Type"
},
{
"t": "ℕ → ℝ",
"v": null,
"name": "b",
"t_type": "Type"
},
{
"t": "∀ n : ℕ, a n > 0 ∧ b n > 0",
"v": null,
"name": "habn",
"t_type": "Prop"
},
{
"t": "a 0 = 1 ∧ b 0 = 1",
"v": null,
"... | [
"(Tendsto (fun n => ∑ i : Fin n, 1/(∏ j : Fin (i + 1), (a j))) atTop (𝓝 answer))"
] |
Find a real number $c$ and a positive number $L$ for which $\lim_{r \to \infty} \frac{r^c \int_0^{\pi/2} x^r\sin x\,dx}{\int_0^{\pi/2} x^r\cos x\,dx}=L$. | (-1, 2 / π) | open Topology Filter | answer = (((-1 : ℝ), ((2 : ℝ) / (Real.pi : ℝ) : ℝ)) : ℝ × ℝ) | ℝ × ℝ | {
"benchmark": "putnam_solving",
"index": 238
} | [] | [
{
"t": "ℝ",
"v": null,
"name": "c",
"t_type": "Type"
},
{
"t": "ℝ",
"v": null,
"name": "L",
"t_type": "Type"
},
{
"t": "L > 0",
"v": null,
"name": "h_L_pos",
"t_type": "Prop"
},
{
"t": "Tendsto (fun r : ℝ => (r ^ c * ∫ x in Set.Ioo 0 (Real.pi / 2), x ^... | [
"(answer = (c, L))"
] |
For which positive integers $n$ is there an $n \times n$ matrix with integer entries such that every dot product of a row with itself is even, while every dot product of two different rows is odd? | the set of odd positive integers | open Topology Filter Matrix | answer = ({n : ℕ | Odd n} : Set ℕ) | Set ℕ | {
"benchmark": "putnam_solving",
"index": 239
} | [
{
"t": "ℕ",
"v": null,
"name": "n",
"t_type": "Type"
}
] | [
{
"t": "ℕ → Prop",
"v": null,
"name": "nmat",
"t_type": "Type"
},
{
"t": "∀ n, nmat n ↔\n ∃ A : Matrix (Fin n) (Fin n) ℤ,\n (∀ r, Even ((A r) ⬝ᵥ (A r))) ∧\n Pairwise fun r1 r2 ↦ Odd ((A r1) ⬝ᵥ (A r2))",
"v": null,
"name": "hnmat",
"t_type": "Prop"
},
{
... | [
"(nmat n ↔ n ∈ answer)"
] |
Let $S$ be the set of all ordered triples $(p,q,r)$ of prime numbers for which at least one rational number $x$ satisfies $px^2+qx+r=0$. Which primes appear in seven or more elements of $S$? | {2, 5} | open Topology Filter Matrix | answer = (Insert.insert (2 : ℕ) ({(5 : ℕ)} : Set ℕ) : Set ℕ) | Set ℕ | {
"benchmark": "putnam_solving",
"index": 240
} | [] | [
{
"t": "Set (Fin 3 → ℕ)",
"v": null,
"name": "S",
"t_type": "Type"
},
{
"t": "ℕ",
"v": null,
"name": "t",
"t_type": "Type"
},
{
"t": "S = {s : Fin 3 → ℕ | (s 0).Prime ∧ (s 1).Prime ∧ (s 2).Prime ∧ ∃ x : ℚ, (s 0) * x ^ 2 + (s 1) * x + (s 2) = 0}",
"v": null,
"name"... | [
"((t.Prime ∧ ({s ∈ S | ∃ i : Fin 3, s i = t}.encard ≥ 7)) ↔ t ∈ answer)"
] |
Let $f$ and $g$ be (real-valued) functions defined on an open interval containing $0$, with $g$ nonzero and continuous at $0$. If $fg$ and $f/g$ are differentiable at $0$, must $f$ be differentiable at $0$? | True | open Topology Filter Matrix | answer = True | Prop | {
"benchmark": "putnam_solving",
"index": 241
} | [] | [
{
"t": "ℝ → ℝ",
"v": null,
"name": "f",
"t_type": "Type"
},
{
"t": "ℝ → ℝ",
"v": null,
"name": "g",
"t_type": "Type"
},
{
"t": "g 0 ≠ 0",
"v": null,
"name": "h_g0",
"t_type": "Prop"
},
{
"t": "ContinuousAt g 0",
"v": null,
"name": "h_cont",
... | [
"(DifferentiableAt ℝ f 0 ↔ answer)"
] |
Let $f: [-1, 1] \to \mathbb{R}$ be a continuous function such that
\begin{itemize}
\item[(i)]
$f(x) = \frac{2-x^2}{2} f \left( \frac{x^2}{2-x^2} \right)$ for every $x$ in $[-1, 1]$,
\item[(ii)]
$f(0) = 1$, and
\item[(iii)]
$\lim_{x \to 1^-} \frac{f(x)}{\sqrt{1-x}}$ exists and is finite.
\end{itemize}
Prove that $f$ is ... | $f(x) = \sqrt{1 - x^2}$ | open Matrix Function | answer = fun (x : ℝ) => (√((1 : ℝ) - (x ^ (2 : ℕ) : ℝ) : ℝ) : ℝ) | ℝ → ℝ | {
"benchmark": "putnam_solving",
"index": 242
} | [] | [
{
"t": "Set ℝ",
"v": null,
"name": "S",
"t_type": "Type"
},
{
"t": "S = Set.Icc (-1 : ℝ) 1",
"v": null,
"name": "hS",
"t_type": "Prop"
},
{
"t": "(ℝ → ℝ) → Prop",
"v": null,
"name": "fsat",
"t_type": "Type"
},
{
"t": "fsat = fun f : ℝ → ℝ => Continuous... | [
"(fsat answer ∧ ∀ f : ℝ → ℝ, fsat f → ∀ x ∈ S, f x = answer x)"
] |
Let $\FF_p$ denote the field of integers modulo a prime $p$, and let $n$ be a positive integer. Let $v$ be a fixed vector in $\FF_p^n$, let $M$ be an $n \times n$ matrix with entries of $\FF_p$, and define $G: \FF_p^n \to \FF_p^n$ by $G(x) = v + Mx$. Let $G^{(k)}$ denote the $k$-fold composition of $G$ with itself, tha... | {q | let ⟨n, _⟩ := q; n = 1} ∪ {(2,2)} | open Matrix Function | answer =
(({((n, snd) : ℕ × ℕ) : ℕ × ℕ | n = (1 : ℕ)} : Set (ℕ × ℕ)) ∪ ({(((2 : ℕ), (2 : ℕ)) : ℕ × ℕ)} : Set (ℕ × ℕ)) :
Set (ℕ × ℕ)) | Set (ℕ × ℕ) | {
"benchmark": "putnam_solving",
"index": 243
} | [
{
"t": "ℕ",
"v": null,
"name": "n",
"t_type": "Type"
}
] | [
{
"t": "ℕ",
"v": null,
"name": "p",
"t_type": "Type"
},
{
"t": "n > 0",
"v": null,
"name": "hn",
"t_type": "Prop"
},
{
"t": "Nat.Prime p",
"v": null,
"name": "hp",
"t_type": "Prop"
},
{
"t": "Type*",
"v": null,
"name": "F",
"t_type": "Type"... | [
"((n, p) ∈ answer ↔\n ∃ᵉ (M : Matrix (Fin n) (Fin n) F) (v : (Fin n → F)),\n ¬(∃ i j : Finset.range (p^n), i ≠ j ∧ (G M v)^[i + 1] 0 = (G M v)^[j + 1] 0))"
] |
Let $f(x,y)$ be a continuous, real-valued function on $\mathbb{R}^2$. Suppose that, for every rectangular region $R$ of area $1$, the double integral of $f(x,y)$ over $R$ equals $0$.
Must $f(x,y)$ be identically $0$? | True | open Matrix Function | answer = True | Prop | {
"benchmark": "putnam_solving",
"index": 244
} | [] | [
{
"t": "((ℝ × ℝ) → ℝ) → Prop",
"v": null,
"name": "p",
"t_type": "Type"
},
{
"t": "∀ f, p f ↔\n Continuous f ∧\n ∀ x1 x2 y1 y2 : ℝ, x2 > x1 → y2 > y1\n → (x2 - x1) * (y2 - y1) = 1 → ∫ x in x1..x2, ∫ y in y1..y2, f (x, y) = 0",
"v": null,
"name": "hp",
"t_type": "... | [
"((∀ f x y, p f → f (x, y) = 0) ↔ answer)"
] |
A round-robin tournament of $2n$ teams lasted for $2n-1$ days, as follows. On each day, every team played one game against another team, with one team winning and one team losing in each of the $n$ games. Over the course of the tournament, each team played every other team exactly once. Can one necessarily choose one w... | True | open Matrix Function Real | answer = True | Prop | {
"benchmark": "putnam_solving",
"index": 245
} | [] | [
{
"t": "(n : ℕ) → (Fin (2 * n - 1) → (Fin (2 * n) → Fin (2 * n))) → Prop",
"v": null,
"name": "nmatchupsgames",
"t_type": "Type"
},
{
"t": "(n : ℕ) → (Fin (2 * n - 1) → (Fin (2 * n) → Fin (2 * n))) → Prop",
"v": null,
"name": "nmatchupsall",
"t_type": "Type"
},
{
"t": "(n... | [
"((∀ n ≥ 1, ∀ matchups wins,\n (nmatchupsgames n matchups ∧ nmatchupsall n matchups ∧ nmatchupswins n matchups wins) →\n nmatchupswinschoices n matchups wins) ↔ answer)"
] |
Suppose that $a_0 = 1$ and that $a_{n+1} = a_n + e^{-a_n}$ for $n=0,1,2,\dots$. Does $a_n - \log n$
have a finite limit as $n \to \infty$? (Here $\log n = \log_e n = \ln n$.) | True | open Matrix Function Real Topology Filter | answer = True | Prop | {
"benchmark": "putnam_solving",
"index": 246
} | [] | [
{
"t": "ℕ → ℝ",
"v": null,
"name": "a",
"t_type": "Type"
},
{
"t": "a 0 = 1",
"v": null,
"name": "ha0",
"t_type": "Prop"
},
{
"t": "∀ n : ℕ, a (n + 1) = a n + exp (-a n)",
"v": null,
"name": "han",
"t_type": "Prop"
}
] | [
"((∃ L : ℝ, Tendsto (fun n ↦ a n - Real.log n) atTop (𝓝 L)) ↔ answer)"
] |
For positive integers $n$, let the numbers $c(n)$ be determined by the rules $c(1)=1$, $c(2n)=c(n)$, and $c(2n+1)=(-1)^nc(n)$. Find the value of $\sum_{n=1}^{2013} c(n)c(n+2)$. | -1 | open Function Set | answer = (-1 : ℤ) | ℤ | {
"benchmark": "putnam_solving",
"index": 247
} | [] | [
{
"t": "ℕ → ℤ",
"v": null,
"name": "c",
"t_type": "Type"
},
{
"t": "c 1 = 1",
"v": null,
"name": "hc1",
"t_type": "Prop"
},
{
"t": "∀ n : ℕ, n > 0 → c (2 * n) = c n",
"v": null,
"name": "hceven",
"t_type": "Prop"
},
{
"t": "∀ n : ℕ, n > 0 → c (2 * n + ... | [
"((∑ n : Set.Icc 1 2013, c n * c (n.1 + 2)) = answer)"
] |
Let $C = \bigcup_{N=1}^\infty C_N$, where $C_N$ denotes the set of those `cosine polynomials' of the form
\[
f(x) = 1 + \sum_{n=1}^N a_n \cos(2 \pi n x)
\]
for which:
\begin{enumerate}
\item[(i)]
$f(x) \geq 0$ for all real $x$, and
\item[(ii)]
$a_n = 0$ whenever $n$ is a multiple of $3$.
\end{enumerate}
Determine the m... | 3 | open Function Set | answer = (3 : ℝ) | ℝ | {
"benchmark": "putnam_solving",
"index": 248
} | [] | [
{
"t": "ℕ → Set (ℝ → ℝ)",
"v": null,
"name": "CN",
"t_type": "Type"
},
{
"t": "∀ N : ℕ, CN N =\n {f : ℝ → ℝ |\n (∀ x : ℝ, f x ≥ 0) ∧\n ∃ a : List ℝ, a.length = N + 1 ∧ (∀ n : Fin (N + 1), 3 ∣ (n : ℕ) → a[n]! = 0) ∧\n ∀ x : ℝ, f x = 1 + ∑ n in Finset.Icc 1 N, a[(n : ℕ)]! * R... | [
"(IsGreatest {f 0 | f ∈ ⋃ N ∈ Ici 1, CN N} answer)"
] |
Let $A$ be the $n \times n$ matrix whose entry in the $i$-th row and $j$-th column is $\frac{1}{\min(i,j)}$ for $1 \leq i,j \leq n$. Compute $\det(A)$. | (fun n : ℕ => (-1) ^ (n - 1) / ((n - 1)! * (n)!)) | open Topology Filter Nat | answer = fun (n : ℕ) =>
(((-1 : ℝ) ^ (n - (1 : ℕ) : ℕ) : ℝ) / ((↑((n - (1 : ℕ) : ℕ)! : ℕ) : ℝ) * (↑(n ! : ℕ) : ℝ) : ℝ) : ℝ) | ℕ → ℝ | {
"benchmark": "putnam_solving",
"index": 249
} | [
{
"t": "ℕ",
"v": null,
"name": "n",
"t_type": "Type"
}
] | [
{
"t": "Matrix (Fin n) (Fin n) ℝ",
"v": null,
"name": "A",
"t_type": "Type"
},
{
"t": "n > 0",
"v": null,
"name": "npos",
"t_type": "Prop"
},
{
"t": "∀ i j : Fin n, A i j = 1 / min (i.1 + 1 : ℚ) (j.1 + 1)",
"v": null,
"name": "hA",
"t_type": "Prop"
}
] | [
"(A.det = answer n)"
] |
Let \( a_0 = \frac{5}{2} \) and \( a_k = a_{k-1}^2 - 2 \) for \( k \geq 1 \). Compute \( \prod_{k=0}^{\infty} \left(1 - \frac{1}{a_k}\right) \) in closed form. | \( \frac{3}{7} \) | open Topology Filter Nat | answer = (3 / 7 : ℝ) | ℝ | {
"benchmark": "putnam_solving",
"index": 250
} | [] | [
{
"t": "ℕ → ℝ",
"v": null,
"name": "a",
"t_type": "Type"
},
{
"t": "a 0 = 5 / 2",
"v": null,
"name": "a0",
"t_type": "Prop"
},
{
"t": "∀ k ≥ 1, a k = (a (k - 1)) ^ 2 - 2",
"v": null,
"name": "ak",
"t_type": "Prop"
}
] | [
"(Tendsto (fun n : ℕ => ∏ k in Finset.range n, (1 - 1 / a k)) atTop (𝓝 answer))"
] |
Suppose $X$ is a random variable that takes on only nonnegative integer values, with $E\left[X\right]=1$, $E\left[X^2\right]=2$, and $E\left[X^3\right]=5$. (Here $E\left[Y\right]$ denotes the expectation of the random variable $Y$.) Determine the smallest possible value of the probability of the event $X=0$. | 1/3 | open Topology Filter Nat | answer = (1 / 3 : ℝ) | ℝ | {
"benchmark": "putnam_solving",
"index": 251
} | [] | [
{
"t": "(ℕ → ℝ) → Prop",
"v": null,
"name": "Xrandvar",
"t_type": "Type"
},
{
"t": "Xrandvar = (fun X : ℕ → ℝ => (∀ n : ℕ, X n ∈ Set.Icc 0 1) ∧ ∑' n : ℕ, X n = 1)",
"v": null,
"name": "hXrandvar",
"t_type": "Prop"
},
{
"t": "(ℕ → ℝ) → (ℕ → ℝ) → ℝ",
"v": null,
"nam... | [
"(sInf {X0 : ℝ | ∃ X : ℕ → ℝ, Xrandvar X ∧ E X (fun x : ℕ => x) = 1 ∧ E X (fun x : ℕ => x ^ 2) = 2 ∧ E X (fun x : ℕ => x ^ 3) = 5 ∧ X0 = X 0} = answer)"
] |
Let \( n \) be a positive integer. What is the largest \( k \) for which there exist \( n \times n \) matrices \( M_1, \ldots, M_k \) and \( N_1, \ldots, N_k \) with real entries such that for all \( i \) and \( j \), the matrix product \( M_i N_j \) has a zero entry somewhere on its diagonal if and only if \( i \neq j... | n^n | open Topology Filter Nat | answer = fun (n : ℕ) => (n ^ n : ℕ) | ℕ → ℕ | {
"benchmark": "putnam_solving",
"index": 252
} | [
{
"t": "ℕ",
"v": null,
"name": "n",
"t_type": "Type"
}
] | [
{
"t": "ℕ → Prop",
"v": null,
"name": "kex",
"t_type": "Type"
},
{
"t": "n > 0",
"v": null,
"name": "npos",
"t_type": "Prop"
},
{
"t": "∀ k ≥ 1, kex k = ∃ M N : Fin k → Matrix (Fin n) (Fin n) ℝ, ∀ i j : Fin k, ((∃ p : Fin n, (M i * N j) p p = 0) ↔ i ≠ j)",
"v": null,
... | [
"((answer n ≥ 1 ∧ kex (answer n)) ∧ (∀ k ≥ 1, kex k → k ≤ answer n))"
] |
A \emph{base $10$ over-expansion} of a positive integer $N$ is an expression of the form
\[
N = d_k 10^k + d_{k-1} 10^{k-1} + \cdots + d_0 10^0
\]
with $d_k \neq 0$ and $d_i \in \{0,1,2,\dots,10\}$ for all $i$. For instance, the integer $N = 10$ has two base $10$ over-expansions: $10 = 10 \cdot 10^0$ and the usual base... | the set of positive integers that do not contain the digit 0 in their base 10 representation | open Topology Filter Nat | answer = ({n : ℕ | n > (0 : ℕ) ∧ ¬∃ a ∈ (Nat.digits (10 : ℕ) n : List ℕ), a = (0 : ℕ)} : Set ℕ) | Set ℕ | {
"benchmark": "putnam_solving",
"index": 253
} | [] | [
{
"t": "ℕ → List ℕ → Prop",
"v": null,
"name": "overexpansion",
"t_type": "Type"
},
{
"t": "overexpansion = fun N d ↦ N = ∑ i : Fin d.length, (d.get i) * 10 ^ i.1 ∧ d.getLastI ≠ 0 ∧ ∀ a ∈ d, a ∈ Finset.range 11",
"v": null,
"name": "hoverexpansion",
"t_type": "Prop"
},
{
... | [
"(S = answer)"
] |
Suppose that \( f \) is a function on the interval \([1,3]\) such that \(-1 \leq f(x) \leq 1\) for all \( x \) and \( \int_{1}^{3} f(x) \, dx = 0 \). How large can \(\int_{1}^{3} \frac{f(x)}{x} \, dx \) be? | \( \log(4/3) \) | open Topology Filter Nat | answer = (Real.log (4 / 3 : ℝ) : ℝ) | ℝ | {
"benchmark": "putnam_solving",
"index": 254
} | [] | [] | [
"(IsGreatest {t | ∃ f : ℝ → ℝ,\n (∀ x : Set.Icc (1 : ℝ) 3, -1 ≤ f x ∧ f x ≤ 1) ∧\n (∫ x in Set.Ioo 1 3, f x = 0) ∧\n (∫ x in Set.Ioo 1 3, (f x) / x) = t}\n answer)"
] |
Let $a_0=1$, $a_1=2$, and $a_n=4a_{n-1}-a_{n-2}$ for $n \geq 2$. Find an odd prime factor of $a_{2015}$. | 181 | answer = (181 : ℕ) | ℕ | {
"benchmark": "putnam_solving",
"index": 255
} | [] | [
{
"t": "ℕ → ℤ",
"v": null,
"name": "a",
"t_type": "Type"
},
{
"t": "a 0 = 1 ∧ a 1 = 2",
"v": null,
"name": "abase",
"t_type": "Prop"
},
{
"t": "∀ n ≥ 2, a n = 4 * a (n - 1) - a (n - 2)",
"v": null,
"name": "arec",
"t_type": "Prop"
}
] | [
"(Odd answer ∧ answer.Prime ∧ ((answer : ℤ) ∣ a 2015))"
] | |
Compute $\log_2 \left( \prod_{a=1}^{2015}\prod_{b=1}^{2015}(1+e^{2\pi iab/2015}) \right)$. Here $i$ is the imaginary unit (that is, $i^2=-1$). | 13725 | answer = (13725 : ℂ) | ℂ | {
"benchmark": "putnam_solving",
"index": 256
} | [] | [] | [
"(Complex.log (∏ a : Fin 2015, ∏ b : Fin 2015, (1 + Complex.exp (2 * Real.pi * Complex.I * (a.1 + 1) * (b.1 + 1) / 2015))) / Complex.log 2 = answer)"
] | |
For each real number $x$, let
\[
f(x) = \sum_{n\in S_x} \frac{1}{2^n},
\]
where $S_x$ is the set of positive integers $n$ for which $\lfloor nx \rfloor$ is even. What is the largest real number $L$ such that $f(x) \geq L$ for all $x \in [0,1)$? (As usual, $\lfloor z \rfloor$ denotes the greatest integer less than or eq... | 4/7 | answer = (4 / 7 : ℝ) | ℝ | {
"benchmark": "putnam_solving",
"index": 257
} | [] | [
{
"t": "ℝ → Set ℤ",
"v": null,
"name": "S",
"t_type": "Type"
},
{
"t": "ℝ → ℝ",
"v": null,
"name": "f",
"t_type": "Type"
},
{
"t": "S = fun (x : ℝ) ↦ {n : ℤ | n > 0 ∧ Even ⌊n * x⌋}",
"v": null,
"name": "hS",
"t_type": "Prop"
},
{
"t": "f = fun (x : ℝ) ... | [
"(IsGreatest {l : ℝ | ∀ x ∈ Set.Ico 0 1, f x ≥ l} answer)"
] | |
Given a list of the positive integers $1,2,3,4,\dots$, take the first three numbers $1,2,3$ and their sum $6$ and cross all four numbers off the list. Repeat with the three smallest remaining numbers $4,5,7$ and their sum $16$. Continue in this way, crossing off the three smallest remaining numbers and their sum, and c... | True | answer = True | Prop | {
"benchmark": "putnam_solving",
"index": 258
} | [] | [
{
"t": "ℕ → Set ℕ",
"v": null,
"name": "sets",
"t_type": "Type"
},
{
"t": "Set ℕ → (Fin 3 → ℕ)",
"v": null,
"name": "Smin3",
"t_type": "Type"
},
{
"t": "ℕ → ℕ",
"v": null,
"name": "sums",
"t_type": "Type"
},
{
"t": "sets 0 = Set.Ici 1",
"v": null,
... | [
"((∃ n : ℕ, List.IsPrefix [5, 1, 0, 2] (Nat.digits 10 (sums n))) ↔ answer)"
] | |
Let $S$ be the set of all $2 \times 2$ real matrices $M=\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ whose entries $a,b,c,d$ (in that order) form an arithmetic progression. Find all matrices $M$ in $S$ for which there is some integer $k>1$ such that $M^k$ is also in $S$. | the set of all matrices of the form α * I or β * A, where I is the identity matrix and A is a specific matrix | answer =
({A : Matrix (Fin (2 : ℕ)) (Fin (2 : ℕ)) ℝ |
(∃ (α : ℝ), ∀ (i j : Fin (2 : ℕ)), (A i j : ℝ) = (α * (1 : ℝ) : ℝ)) ∨
∃ (β : ℝ),
(A (0 : Fin (2 : ℕ)) (0 : Fin (2 : ℕ)) : ℝ) = (β * (-3 : ℝ) : ℝ) ∧
(A (0 : Fin (2 : ℕ)) (1 : Fin (2 : ℕ)) : ℝ) = (β * (-1 : ℝ) : ℝ) ∧
... | Set (Matrix (Fin 2) (Fin 2) ℝ) | {
"benchmark": "putnam_solving",
"index": 259
} | [] | [
{
"t": "Matrix (Fin 2) (Fin 2) ℝ",
"v": null,
"name": "M",
"t_type": "Type"
},
{
"t": "Set (Matrix (Fin 2) (Fin 2) ℝ)",
"v": null,
"name": "S",
"t_type": "Type"
},
{
"t": "S = {M' | (M' 0 1 - M' 0 0 = M' 1 0 - M' 0 1) ∧ (M' 1 0 - M' 0 1 = M' 1 1 - M' 1 0)}",
"v": null... | [
"(M ∈ S ∧ (∃ k > 1, M ^ k ∈ S) ↔ M ∈ answer)"
] | |
Let $T$ be the set of all triples $(a,b,c)$ of positive integers for which there exist triangles with side lengths $a,b,c$. Express
\[
\sum_{(a,b,c) \in T} \frac{2^a}{3^b 5^c}
\]
as a rational number in lowest terms. | (17, 21) | answer = (((17 : ℤ), (21 : ℕ)) : ℤ × ℕ) | ℤ × ℕ | {
"benchmark": "putnam_solving",
"index": 260
} | [] | [
{
"t": "ℚ → (ℤ × ℕ)",
"v": null,
"name": "quotientof",
"t_type": "Type"
},
{
"t": "∀ q : ℚ, quotientof q = (q.num, q.den)",
"v": null,
"name": "hquotientof",
"t_type": "Prop"
}
] | [
"(quotientof (∑' t : (Fin 3 → ℤ), if (∀ n : Fin 3, t n > 0) ∧ t 0 < t 1 + t 2 ∧ t 1 < t 2 + t 0 ∧ t 2 < t 0 + t 1\n then 2^(t 0)/(3^(t 1)*5^(t 2)) else 0) = answer)"
] | |
Let $P_n$ be the number of permutations $\pi$ of $\{1,2,\dots,n\}$ such that
\[
|i-j| = 1 \mbox{ implies } |\pi(i) -\pi(j)| \leq 2
\]
for all $i,j$ in $\{1,2,\dots,n\}$. Show that for $n \geq 2$, the quantity
\[
P_{n+5} - P_{n+4} - P_{n+3} + P_n
\]
does not depend on $n$, and find its value. | 4 | open Function | answer = (4 : ℕ) | ℕ | {
"benchmark": "putnam_solving",
"index": 261
} | [] | [
{
"t": "ℕ → ℕ",
"v": null,
"name": "P",
"t_type": "Type"
},
{
"t": "P = fun n ↦ {pi : Finset.Icc 1 n → Finset.Icc 1 n | Bijective pi ∧ ∀ i j : Finset.Icc 1 n, Nat.dist i j = 1 → Nat.dist (pi i) (pi j) ≤ 2}.ncard",
"v": null,
"name": "hP",
"t_type": "Prop"
}
] | [
"(∀ n : ℕ, n ≥ 2 → (P (n + 5) : ℤ) - (P (n + 4) : ℤ) - (P (n + 3) : ℤ) + (P n : ℤ) = answer)"
] |
For each positive integer $k$, let $A(k)$ be the number of odd divisors of $k$ in the interval $[1,\sqrt{2k})$. Evaluate $\sum_{k=1}^\infty (-1)^{k-1}\frac{A(k)}{k}$. | π² / 16 | open Filter Topology | answer = (((Real.pi : ℝ) ^ (2 : ℕ) : ℝ) / (16 : ℝ) : ℝ) | ℝ | {
"benchmark": "putnam_solving",
"index": 262
} | [] | [
{
"t": "ℕ → ℕ",
"v": null,
"name": "A",
"t_type": "Type"
},
{
"t": "∀ k > 0, A k = {j : ℕ | Odd j ∧ j ∣ k ∧ j < Real.sqrt (2 * k)}.encard",
"v": null,
"name": "hA",
"t_type": "Prop"
}
] | [
"(Tendsto (fun K : ℕ ↦ ∑ k in Finset.Icc 1 K, (-1 : ℝ) ^ ((k : ℝ) - 1) * (A k / (k : ℝ))) atTop (𝓝 answer))"
] |
Find the smallest positive integer $j$ such that for every polynomial $p(x)$ with integer coefficients and for every integer $k$, the integer \[ p^{(j)}(k) = \left. \frac{d^j}{dx^j} p(x) \right|_{x=k} \] (the $j$-th derivative of $p(x)$ at $k$) is divisible by 2016. | 8 | open Polynomial Filter Topology Real Set Nat | answer = (8 : ℕ) | ℕ | {
"benchmark": "putnam_solving",
"index": 263
} | [] | [] | [
"(IsLeast {j : ℕ | 0 < j ∧ ∀ P : ℤ[X], ∀ k : ℤ, 2016 ∣ (derivative^[j] P).eval k} answer)"
] |
Given a positive integer $n$, let $M(n)$ be the largest integer $m$ such that
\[
\binom{m}{n-1} > \binom{m-1}{n}.
\]
Evaluate
\[
\lim_{n \to \infty} \frac{M(n)}{n}.
\] | (3 + √5) / 2 | open Polynomial Filter Topology Real Set Nat | answer = (((3 : ℝ) + (√(5 : ℝ) : ℝ) : ℝ) / (2 : ℝ) : ℝ) | ℝ | {
"benchmark": "putnam_solving",
"index": 264
} | [] | [
{
"t": "ℕ → ℕ",
"v": null,
"name": "M",
"t_type": "Type"
},
{
"t": "∀ n > 0, IsGreatest {m | 0 < m ∧ (m - 1).choose n < m.choose (n - 1)} (M n)",
"v": null,
"name": "hM",
"t_type": "Prop"
}
] | [
"(Tendsto (fun n ↦ M n / (n : ℝ)) atTop (𝓝 answer))"
] |
Suppose that $f$ is a function from $\mathbb{R}$ to $\mathbb{R}$ such that
\[
f(x) + f\left( 1 - \frac{1}{x} \right) = \arctan x
\]
for all real $x \neq 0$. (As usual, $y = \arctan x$ means $-\pi/2 < y < \pi/2$ and $\tan y = x$.) Find
\[
\int_0^1 f(x)\,dx.
\] | 3π/8 | open Polynomial Filter Topology Real Set Nat | answer = (((3 : ℝ) * (π : ℝ) : ℝ) / (8 : ℝ) : ℝ) | ℝ | {
"benchmark": "putnam_solving",
"index": 265
} | [] | [
{
"t": "ℝ → ℝ",
"v": null,
"name": "f",
"t_type": "Type"
},
{
"t": "∀ x : ℝ, x ≠ 0 → f x + f (1 - 1 / x) = Real.arctan x",
"v": null,
"name": "hf",
"t_type": "Prop"
}
] | [
"(∫ x in (0)..1, f x = answer)"
] |
Find the smallest constant $C$ such that for every real polynomial $P(x)$ of degree $3$ that has a root in the interval $[0,1]$,
\[
\int_0^1 \left| P(x) \right|\,dx \leq C \max_{x \in [0,1]} \left| P(x) \right|.
\] | 5 / 6 | open Polynomial Filter Topology Real Set Nat List | answer = (5 / 6 : ℝ) | ℝ | {
"benchmark": "putnam_solving",
"index": 266
} | [] | [
{
"t": "ℝ → Prop",
"v": null,
"name": "p",
"t_type": "Type"
},
{
"t": "∀ c, p c ↔\n ∀ P : Polynomial ℝ, P.degree = 3 →\n (∃ x ∈ Icc 0 1, P.eval x = 0) →\n ∫ x in (0)..1, |P.eval x| ≤ c * (sSup {y | ∃ x ∈ Icc 0 1, y = |P.eval x|})",
"v": null,
"name": "hp",
"t... | [
"(IsLeast p answer)"
] |
Let $x_0,x_1,x_2,\dots$ be the sequence such that $x_0=1$ and for $n \geq 0$,
\[
x_{n+1} = \ln(e^{x_n} - x_n)
\]
(as usual, the function $\ln$ is the natural logarithm). Show that the infinite series
\[
x_0 + x_1 + x_2 + \cdots
\]
converges and find its sum. | exp 1 - 1 | open Polynomial Filter Topology Real Set Nat List | answer = ((rexp (1 : ℝ) : ℝ) - (1 : ℝ) : ℝ) | ℝ | {
"benchmark": "putnam_solving",
"index": 267
} | [] | [
{
"t": "ℕ → ℝ",
"v": null,
"name": "x",
"t_type": "Type"
},
{
"t": "x 0 = 1",
"v": null,
"name": "hx0",
"t_type": "Prop"
},
{
"t": "∀ n : ℕ, x (n + 1) = log (exp (x n) - (x n))",
"v": null,
"name": "hxn",
"t_type": "Prop"
}
] | [
"(∑' n : ℕ, x n = answer)"
] |
Define a positive integer $n$ to be \emph{squarish} if either $n$ is itself a perfect square or the distance from $n$ to the nearest perfect square is a perfect square. For example, $2016$ is squarish, because the nearest perfect square to $2016$ is $45^2 = 2025$ and $2025 - 2016 = 9$ is a perfect square. (Of the posit... | (3/4, 4/3) | open Classical Polynomial Filter Topology Real Set Nat List | answer = (((3 / 4 : ℝ), (4 / 3 : ℝ)) : ℝ × ℝ) | ℝ × ℝ | {
"benchmark": "putnam_solving",
"index": 268
} | [] | [
{
"t": "ℤ → Prop",
"v": null,
"name": "squarish",
"t_type": "Type"
},
{
"t": "∀ n, squarish n ↔ IsSquare n ∨ ∃ w : ℤ, IsSquare |n - w ^ 2| ∧ ∀ v : ℕ, |n - w ^ 2| ≤ |n - v ^ 2|",
"v": null,
"name": "hsquarish",
"t_type": "Prop"
},
{
"t": "ℤ → ℕ",
"v": null,
"name":... | [
"((∀ α β : ℝ, ((α, β) = answer ↔ p α β)) ∨ ¬∃ α β : ℝ, p α β)"
] |
Let $A$ be a $2n \times 2n$ matrix, with entries chosen independently at random. Every entry is chosen to be $0$ or $1$, each with probability $1/2$. Find the expected value of $\det(A-A^t)$ (as a function of $n$), where $A^t$ is the transpose of $A$. | (fun n : ℕ => (2 * n)! / (4 ^ n * (n)!)) | open Real Set Nat | answer = fun (n : ℕ) => ((↑(((2 : ℕ) * n : ℕ)! : ℕ) : ℝ) / (((4 : ℝ) ^ n : ℝ) * (↑(n ! : ℕ) : ℝ) : ℝ) : ℝ) | ℕ → ℝ | {
"benchmark": "putnam_solving",
"index": 269
} | [
{
"t": "ℕ",
"v": null,
"name": "n",
"t_type": "Type"
}
] | [
{
"t": "n ≥ 1",
"v": null,
"name": "npos",
"t_type": "Prop"
},
{
"t": "Set (Matrix (Fin (2 * n)) (Fin (2 * n)) ℝ)",
"v": null,
"name": "mats01",
"t_type": "Type"
},
{
"t": "mats01 = {A | ∀ i j : Fin (2 * n), A i j = 0 ∨ A i j = 1}",
"v": null,
"name": "hmats01",
... | [
"((∑' A : mats01, (A.1 - (Matrix.transpose A)).det) / mats01.ncard = answer n)"
] |
Find all functions $f$ from the interval $(1,\infty)$ to $(1,\infty)$ with the following property: if $x,y \in (1,\infty)$ and $x^2 \leq y \leq x^3$, then $(f(x))^2 \leq f(y) \leq (f(x))^3$. | the set of all functions of the form f(x) = x^c for some c > 0 | open Polynomial Filter Topology Real Set Nat List | answer =
({f : ↑(Set.Ioi (1 : ℝ) : Set ℝ) → ↑(Set.Ioi (1 : ℝ) : Set ℝ) |
∃ c > (0 : ℝ),
∀ (x : ↑(Set.Ioi (1 : ℝ) : Set ℝ)), (↑(f x : ↑(Set.Ioi (1 : ℝ) : Set ℝ)) : ℝ) = ((↑x : ℝ) ^ c : ℝ)} :
Set (↑(Set.Ioi (1 : ℝ) : Set ℝ) → ↑(Set.Ioi (1 : ℝ) : Set ℝ))) | Set (Set.Ioi (1 : ℝ) → Set.Ioi (1 : ℝ)) | {
"benchmark": "putnam_solving",
"index": 270
} | [
{
"t": "Set.Ioi (1 : ℝ) → Set.Ioi (1 : ℝ)",
"v": null,
"name": "f",
"t_type": "Type"
}
] | [] | [
"(f ∈ answer ↔\n (∀ x y : Set.Ioi (1 : ℝ), ((x : ℝ) ^ 2 ≤ y ∧ y ≤ (x : ℝ) ^ 3) → ((f x : ℝ) ^ 2 ≤ f y ∧ f y ≤ (f x : ℝ) ^ 3)))"
] |
Evaluate $\sum_{k=1}^\infty \frac{(-1)^{k-1}}{k} \sum_{n=0}^\infty \frac{1}{k2^n+1}$. | 1 | open Polynomial Filter Topology Real Set Nat List | answer = (1 : ℝ) | ℝ | {
"benchmark": "putnam_solving",
"index": 271
} | [] | [] | [
"(∑' k : ℕ, ((-1 : ℝ) ^ ((k + 1 : ℤ) - 1) / (k + 1 : ℝ)) * ∑' n : ℕ, (1 : ℝ) / ((k + 1) * (2 ^ n) + 1) = answer)"
] |
Let $S$ be the smallest set of positive integers such that (a) $2$ is in $S$, (b) $n$ is in $S$ whenever $n^2$ is in $S$, and (c) $(n+5)^2$ is in $S$ whenever $n$ is in $S$. Which positive integers are not in $S$?. | {x : ℤ | x > 0 ∧ (x = 1 ∨ 5 ∣ x)} | answer = ({x : ℤ | x > (0 : ℤ) ∧ (x = (1 : ℤ) ∨ (5 : ℤ) ∣ x)} : Set ℤ) | Set ℤ | {
"benchmark": "putnam_solving",
"index": 272
} | [] | [
{
"t": "Set ℤ → Prop",
"v": null,
"name": "IsQualifying",
"t_type": "Type"
},
{
"t": "∀ S, IsQualifying S ↔\n (∀ n ∈ S, 0 < n) ∧\n 2 ∈ S ∧\n (∀ n > 0, n ^ 2 ∈ S → n ∈ S) ∧\n (∀ n ∈ S, (n + 5) ^ 2 ∈ S)",
"v": null,
"name": "IsQualifying_def",
"t_type": "Prop"
},
{
"t":... | [
"(Sᶜ ∩ {n | 0 < n} = answer)"
] | |
Suppose that a positive integer $N$ can be expressed as the sum of $k$ consecutive positive integers
\[
N = a + (a+1) +(a+2) + \cdots + (a+k-1)
\]
for $k=2017$ but for no other values of $k>1$. Considering all positive integers $N$ with this property, what is the smallest positive integer $a$ that occurs in any of thes... | 16 | open Topology Filter | answer = (16 : ℕ) | ℕ | {
"benchmark": "putnam_solving",
"index": 273
} | [] | [
{
"t": "ℤ → ℕ → ℤ",
"v": null,
"name": "S",
"t_type": "Type"
},
{
"t": "ℤ → ℕ → Prop",
"v": null,
"name": "p",
"t_type": "Type"
},
{
"t": "ℤ → Prop",
"v": null,
"name": "q",
"t_type": "Type"
},
{
"t": "S = fun (a : ℤ) k ↦ ∑ i in Finset.range k, (a + i)... | [
"(IsLeast {a : ℤ | q (S a 2017)} answer)"
] |
Evaluate the sum \begin{gather*} \sum_{k=0}^\infty \left( 3 \cdot \frac{\ln(4k+2)}{4k+2} - \frac{\ln(4k+3)}{4k+3} - \frac{\ln(4k+4)}{4k+4} - \frac{\ln(4k+5)}{4k+5} \right) \ = 3 \cdot \frac{\ln 2}{2} - \frac{\ln 3}{3} - \frac{\ln 4}{4} - \frac{\ln 5}{5} + 3 \cdot \frac{\ln 6}{6} - \frac{\ln 7}{7} \ - \frac{\ln 8}{8} - ... | (log 2) ^ 2 | open Topology Filter Real | answer = ((Real.log (2 : ℝ) : ℝ) ^ (2 : ℕ) : ℝ) | ℝ | {
"benchmark": "putnam_solving",
"index": 274
} | [] | [] | [
"((∑' k : ℕ, (3 * Real.log (4 * k + 2) / (4 * k + 2) - Real.log (4 * k + 3) / (4 * k + 3) - Real.log (4 * k + 4) / (4 * k + 4) - Real.log (4 * k + 5) / (4 * k + 5)) = answer))"
] |
Find the number of ordered $64$-tuples $(x_0,x_1,\dots,x_{63})$ such that $x_0,x_1,\dots,x_{63}$ are distinct elements of $\{1,2,\dots,2017\}$ and
\[
x_0 + x_1 + 2x_2 + 3x_3 + \cdots + 63 x_{63}
\]
is divisible by 2017. | 2016! / 1953! - 63! * 2016 | open Topology Filter Real Function Nat | answer = ((((2016 : ℕ)! : ℕ) / ((1953 : ℕ)! : ℕ) : ℕ) - (((63 : ℕ)! : ℕ) * (2016 : ℕ) : ℕ) : ℕ) | ℕ | {
"benchmark": "putnam_solving",
"index": 275
} | [] | [
{
"t": "Finset (Finset.range 64 → Finset.Icc 1 2017)",
"v": null,
"name": "S",
"t_type": "Type"
},
{
"t": "∀ x, x ∈ S ↔ (Injective x ∧ (2017 ∣ (∑ i : Finset.range 64, if i ≤ (⟨1, by norm_num⟩ : Finset.range 64) then (x i : ℤ) else i * (x i : ℤ))))",
"v": null,
"name": "hs",
"t_ty... | [
"(S.card = answer)"
] |
Find all ordered pairs $(a,b)$ of positive integers for which $\frac{1}{a} + \frac{1}{b} = \frac{3}{2018}$. | {⟨673, 1358114⟩, ⟨674, 340033⟩, ⟨1009, 2018⟩, ⟨2018, 1009⟩, ⟨340033, 674⟩, ⟨1358114, 673⟩} | answer =
(Insert.insert (((673 : ℤ), (1358114 : ℤ)) : ℤ × ℤ)
(Insert.insert (((674 : ℤ), (340033 : ℤ)) : ℤ × ℤ)
(Insert.insert (((1009 : ℤ), (2018 : ℤ)) : ℤ × ℤ)
(Insert.insert (((2018 : ℤ), (1009 : ℤ)) : ℤ × ℤ)
(Insert.insert (((340033 : ℤ), (674 : ℤ)) : ℤ × ℤ)
... | Set (ℤ × ℤ) | {
"benchmark": "putnam_solving",
"index": 276
} | [] | [
{
"t": "ℤ",
"v": null,
"name": "a",
"t_type": "Type"
},
{
"t": "ℤ",
"v": null,
"name": "b",
"t_type": "Type"
},
{
"t": "0 < a ∧ 0 < b",
"v": null,
"name": "h",
"t_type": "Prop"
}
] | [
"(((1 : ℚ) / a + (1 : ℚ) / b = (3 : ℚ) / 2018) ↔ (⟨a, b⟩ ∈ answer))"
] | |
Let \( S_1, S_2, \ldots, S_{2^n-1} \) be the nonempty subsets of \( \{1, 2, \ldots, n\} \) in some order, and let \( M \) be the \( (2^n - 1) \times (2^n - 1) \) matrix whose \((i, j)\) entry is $m_{ij} = \begin{cases} 0 & \text{if } S_i \cap S_j = \emptyset; \\ 1 & \text{otherwise}. \end{cases} $ Calculate the determi... | 1 if n = 1, otherwise -1 | answer = fun (n : ℕ) => (if n = (1 : ℕ) then (1 : ℝ) else (-1 : ℝ) : ℝ) | ℕ → ℝ | {
"benchmark": "putnam_solving",
"index": 277
} | [
{
"t": "ℕ",
"v": null,
"name": "n",
"t_type": "Type"
}
] | [
{
"t": "Fin (2 ^ n - 1) → Set ℕ",
"v": null,
"name": "S",
"t_type": "Type"
},
{
"t": "Matrix (Fin (2 ^ n - 1)) (Fin (2 ^ n - 1)) ℝ",
"v": null,
"name": "M",
"t_type": "Type"
},
{
"t": "n ≥ 1",
"v": null,
"name": "npos",
"t_type": "Prop"
},
{
"t": "Set.... | [
"(M.det = answer n)"
] | |
Determine the greatest possible value of $\sum_{i=1}^{10} \cos(3x_i)$ for real numbers $x_1, x_2, \ldots, x_{10}$ satisfying $\sum_{i=1}^{10} \cos(x_i) = 0$. | 480/49 | answer = (480 / 49 : ℝ) | ℝ | {
"benchmark": "putnam_solving",
"index": 278
} | [] | [] | [
"(IsGreatest\n {∑ i, Real.cos (3 * x i) | (x : Fin 10 → ℝ) (hx : ∑ i, Real.cos (x i) = 0)}\n answer)"
] | |
Let $\mathcal{P}$ be the set of vectors defined by $\mathcal{P}=\left\{\left.\begin{pmatrix} a \\ b \end{pmatrix}\right| 0 \leq a \leq 2, 0 \leq b \leq 100,\text{ and }a,b \in \mathbb{Z}\right\}$. Find all $\mathbf{v} \in \mathcal{P}$ such that the set $\mathcal{P} \setminus \{\mathbf{v}\}$ obtained by omitting vector ... | {v : Mathlib.Vector ℤ 2 | ∃ b : ℤ, 0 ≤ b ∧ b ≤ 100 ∧ Even b ∧ v.toList = [1, b]} | answer =
({v : Fin (2 : ℕ) → ℤ |
∃ (b : ℤ),
(0 : ℤ) ≤ b ∧
b ≤ (100 : ℤ) ∧
Even b ∧
v =
(Matrix.vecCons (1 : ℤ) (Matrix.vecCons b (![] : Fin (0 : ℕ) → ℤ) : Fin (Nat.succ (0 : ℕ) : ℕ) → ℤ) :
Fin ((Nat.succ (0 : ℕ) : ℕ).succ : ℕ) → ℤ)} :... | Set (Fin 2 → ℤ) | {
"benchmark": "putnam_solving",
"index": 279
} | [
{
"t": "Fin 2 → ℤ",
"v": null,
"name": "v",
"t_type": "Type"
}
] | [
{
"t": "Finset (Fin 2 → ℤ)",
"v": null,
"name": "P",
"t_type": "Type"
},
{
"t": "Finset (Fin 2 → ℤ)",
"v": null,
"name": "Pvdiff",
"t_type": "Type"
},
{
"t": "P = {v' : Fin 2 → ℤ | 0 ≤ v' 0 ∧ v' 0 ≤ 2 ∧ 0 ≤ v' 1 ∧ v' 1 ≤ 100}",
"v": null,
"name": "hP",
"t_type... | [
"((v ∈ P ∧ (∃ Q R : Finset (Fin 2 → ℤ),\n (Q ∪ R = Pvdiff) ∧ (Q ∩ R = ∅) ∧ (Q.card = R.card) ∧\n (∑ q in Q, q 0 = ∑ r in R, r 0) ∧ (∑ q in Q, q 1 = ∑ r in R, r 1)))\n ↔ v ∈ answer)"
] | |
Find all positive integers $n < 10^{100}$ for which simultaneously $n$ divides $2^n$, $n-1$ divides $2^n-1$, and $n-2$ divides $2^n - 2$. | {2^2, 2^4, 2^16, 2^256} | answer =
(Insert.insert ((2 : ℕ) ^ (2 : ℕ) : ℕ)
(Insert.insert ((2 : ℕ) ^ (4 : ℕ) : ℕ)
(Insert.insert ((2 : ℕ) ^ (16 : ℕ) : ℕ) ({((2 : ℕ) ^ (256 : ℕ) : ℕ)} : Set ℕ) : Set ℕ) :
Set ℕ) :
Set ℕ) | Set ℕ | {
"benchmark": "putnam_solving",
"index": 280
} | [] | [
{
"t": "ℕ",
"v": null,
"name": "n",
"t_type": "Type"
},
{
"t": "0 < n",
"v": null,
"name": "hn",
"t_type": "Prop"
}
] | [
"((n < 10^100 ∧ ((n : ℤ) ∣ (2^n : ℤ) ∧ (n - 1 : ℤ) ∣ (2^n - 1 : ℤ) ∧ (n - 2 : ℤ) ∣ (2^n - 2 : ℤ))) ↔ n ∈ answer)"
] | |
Determine all possible values of the expression
\[
A^3+B^3+C^3-3ABC
\]
where $A, B$, and $C$ are nonnegative integers. | the set of nonnegative integers not congruent to 3 or 6 modulo 9 | open Topology Filter | answer = ({n : ℤ | n ≥ (0 : ℤ) ∧ ¬n ≡ (3 : ℤ) [ZMOD (9 : ℤ)] ∧ ¬n ≡ (6 : ℤ) [ZMOD (9 : ℤ)]} : Set ℤ) | Set ℤ | {
"benchmark": "putnam_solving",
"index": 281
} | [] | [] | [
"({n : ℤ | ∃ A B C : ℤ, A ≥ 0 ∧ B ≥ 0 ∧ C ≥ 0 ∧ A^3 + B^3 + C^3 - 3*A*B*C = n} = answer)"
] |
Given real numbers $b_0, b_1, \dots, b_{2019}$ with $b_{2019} \neq 0$, let $z_1,z_2,\dots,z_{2019}$ be
the roots in the complex plane of the polynomial
\[
P(z) = \sum_{k=0}^{2019} b_k z^k.
\]
Let $\mu = (|z_1| + \cdots + |z_{2019}|)/2019$ be the average of the distances from $z_1,z_2,\dots,z_{2019}$ to the origin. Dete... | 2019^(-(1:ℝ)/2019) | open Topology Filter | answer = ((2019 : ℝ) ^ (-1 / 2019 : ℝ) : ℝ) | ℝ | {
"benchmark": "putnam_solving",
"index": 282
} | [] | [
{
"t": "Polynomial ℂ → Prop",
"v": null,
"name": "v",
"t_type": "Type"
},
{
"t": "v = fun b => b.degree = 2019 ∧ 1 ≤ (b.coeff 0).re ∧ (b.coeff 2019).re ≤ 2019 ∧\n (∀ i : Fin 2020, (b.coeff i).im = 0) ∧ (∀ i : Fin 2019, (b.coeff i).re < (b.coeff (i + 1)).re)",
"v": null,
"name": "hv"... | [
"(IsGreatest {M : ℝ | ∀ b, v b → μ b ≥ M} answer)"
] |
Let $f$ be a continuous real-valued function on $\mathbb{R}^3$. Suppose that for every sphere $S$ of radius $1$, the integral of $f(x,y,z)$ over the surface of $S$ equals $0$. Must $f(x,y,z)$ be identically 0? | False | open MeasureTheory Metric Topology Filter | answer = False | Prop | {
"benchmark": "putnam_solving",
"index": 283
} | [] | [
{
"t": "(EuclideanSpace ℝ (Fin 3) → ℝ) → Prop",
"v": null,
"name": "P",
"t_type": "Type"
},
{
"t": "∀ f, P f ↔ ∀ C, ∫ x in sphere C 1, f x ∂μH[2] = 0",
"v": null,
"name": "P_def",
"t_type": "Prop"
}
] | [
"((∀ f, Continuous f → P f → f = 0) ↔ answer)"
] |
Let $p$ be an odd prime number, and let $\mathbb{F}_p$ denote the field of integers modulo $p$. Let $\mathbb{F}_p[x]$ be the ring of polynomials over $\mathbb{F}_p$, and let $q(x) \in \mathbb{F}_p[x]$ be given by $q(x)=\sum_{k=1}^{p-1} a_kx^k$, where $a_k=k^{(p-1)/2}\mod{p}$. Find the greatest nonnegative integer $n$ s... | (p - 1) / 2 | open Topology Filter | answer = fun (p : ℕ) => ((p - (1 : ℕ) : ℕ) / (2 : ℕ) : ℕ) | ℕ → ℕ | {
"benchmark": "putnam_solving",
"index": 284
} | [
{
"t": "ℕ",
"v": null,
"name": "p",
"t_type": "Type"
}
] | [
{
"t": "Polynomial (ZMod p)",
"v": null,
"name": "q",
"t_type": "Type"
},
{
"t": "ℕ → ZMod p",
"v": null,
"name": "a",
"t_type": "Type"
},
{
"t": "ℕ → Polynomial (ZMod p)",
"v": null,
"name": "npoly",
"t_type": "Type"
},
{
"t": "Odd p",
"v": null,
... | [
"(IsGreatest {n | npoly n ∣ q} (answer p))"
] |
Denote by $\mathbb{Z}^2$ the set of all points $(x,y)$ in the plane with integer coordinates. For each integer $n \geq 0$, let $P_n$ be the subset of $\mathbb{Z}^2$ consisting of the point $(0,0)$ together with all points $(x,y)$ such that $x^2+y^2=2^k$ for some integer $k \leq n$. Determine, as a function of $n$, the ... | 5n + 1 | open Topology Filter | answer = fun (n : ℕ) => (((5 : ℕ) * n : ℕ) + (1 : ℕ) : ℕ) | ℕ → ℕ | {
"benchmark": "putnam_solving",
"index": 285
} | [
{
"t": "ℕ",
"v": null,
"name": "n",
"t_type": "Type"
}
] | [
{
"t": "Set (Fin 2 → ℤ)",
"v": null,
"name": "Pn",
"t_type": "Type"
},
{
"t": "(Fin 2 → ℤ) → EuclideanSpace ℝ (Fin 2)",
"v": null,
"name": "pZtoR",
"t_type": "Type"
},
{
"t": "Finset (Fin 2 → ℤ) → Prop",
"v": null,
"name": "sPnsquare",
"t_type": "Type"
},
... | [
"({sPn : Finset (Fin 2 → ℤ) | (sPn : Set (Fin 2 → ℤ)) ⊆ Pn ∧ sPnsquare sPn}.encard = answer n)"
] |
For all $n \geq 1$, let
\[
a_n = \sum_{k=1}^{n-1} \frac{\sin \left( \frac{(2k-1)\pi}{2n} \right)}{\cos^2 \left( \frac{(k-1)\pi}{2n} \right) \cos^2 \left( \frac{k\pi}{2n} \right)}.
\]
Determine
\[
\lim_{n \to \infty} \frac{a_n}{n^3}.
\] | 8/π^3 | open Topology Filter Set | answer = ((8 : ℝ) / ((Real.pi : ℝ) ^ (3 : ℕ) : ℝ) : ℝ) | ℝ | {
"benchmark": "putnam_solving",
"index": 286
} | [] | [
{
"t": "ℕ → ℝ",
"v": null,
"name": "a",
"t_type": "Type"
},
{
"t": "a = fun n : ℕ => ∑ k : Icc (1 : ℤ) (n - 1),\nReal.sin ((2*k - 1)*Real.pi/(2*n))/((Real.cos ((k - 1)*Real.pi/(2*n))^2)*(Real.cos (k*Real.pi/(2*n))^2))",
"v": null,
"name": "ha",
"t_type": "Prop"
}
] | [
"(Tendsto (fun n : ℕ => (a n)/n^3) atTop (𝓝 answer))"
] |
Let $\mathcal{F}$ be the set of functions $f(x,y)$ that are twice continuously differentiable for $x \geq 1,y \geq 1$ and that satisfy the following two equations (where subscripts denote partial derivatives):
\begin{gather*}
xf_x+yf_y=xy\ln(xy), \\
x^2f_{xx}+y^2f_{yy}=xy.
\end{gather*}
For each $f \in \mathcal{F}$, le... | $2 \ln 2 - \frac{1}{2}$ | open Topology Filter Set Matrix | answer = (((2 : ℝ) * (Real.log (2 : ℝ) : ℝ) : ℝ) - (1 / 2 : ℝ) : ℝ) | ℝ | {
"benchmark": "putnam_solving",
"index": 287
} | [] | [
{
"t": "(Fin 2 → ℝ) → ℝ",
"v": null,
"name": "f",
"t_type": "Type"
},
{
"t": "ℝ → ℝ → (Fin 2 → ℝ)",
"v": null,
"name": "vec",
"t_type": "Type"
},
{
"t": "ContDiff ℝ 2 f",
"v": null,
"name": "fdiff",
"t_type": "Prop"
},
{
"t": "∀ x y : ℝ, (vec x y) 0 = ... | [
"(sInf {f (vec (s + 1) (s + 1)) - f (vec (s + 1) s) - f (vec s (s + 1)) + f (vec s s) | s ≥ 1} = answer)"
] |
Let $F_m$ be the $m$th Fibonacci number, defined by $F_1 = F_2 = 1$ and $F_m = F_{m-1} + F_{m-2}$ for all $m \geq 3$. Let $p(x)$ be the polynomial of degree $1008$ such that $p(2n + 1) = F_{2n+1}$ for $n = 0,1,2,\ldots,1008$. Find integers $j$ and $k$ such that $p(2019) = F_j - F_k$. | ⟨2019, 1010⟩ | open Topology Filter Set Matrix | answer = (((2019 : ℕ), (1010 : ℕ)) : ℕ × ℕ) | ℕ × ℕ | {
"benchmark": "putnam_solving",
"index": 288
} | [] | [
{
"t": "ℕ → ℤ",
"v": null,
"name": "F",
"t_type": "Type"
},
{
"t": "Polynomial ℝ",
"v": null,
"name": "P",
"t_type": "Type"
},
{
"t": "∀ x, x ≥ 1 → F (x + 2) = F (x + 1) + F x",
"v": null,
"name": "hF",
"t_type": "Prop"
},
{
"t": "F 1 = 1 ∧ F 2 = 1",
... | [
"(∀ j k : ℕ, (P.eval 2019 = F j - F k) ↔ ⟨j, k⟩ = answer)"
] |
Let \( \mathbb{Z}^n \) be the integer lattice in \( \mathbb{R}^n \). Two points in \( \mathbb{Z}^n \) are called neighbors if they differ by exactly 1 in one coordinate and are equal in all other coordinates. For which integers \( n \geq 1 \) does there exist a set of points \( S \subset \mathbb{Z}^n \) satisfying the ... | the set of all integers n ≥ 1 | open Topology Filter Set Matrix | answer = (Set.Ici (1 : ℕ) : Set ℕ) | Set ℕ | {
"benchmark": "putnam_solving",
"index": 289
} | [] | [
{
"t": "ℕ",
"v": null,
"name": "n",
"t_type": "Type"
},
{
"t": "(Fin n → ℤ) → (Fin n → ℤ) → Prop",
"v": null,
"name": "neighbors",
"t_type": "Type"
},
{
"t": "∀ p q, neighbors p q ↔ (∃ i : Fin n, abs (p i - q i) = 1 ∧ ∀ j ≠ i, p j = q j)",
"v": null,
"name": "neig... | [
"((1 ≤ n ∧ ∃ S : Set (Fin n → ℤ),\n (∀ p ∈ S, ∀ q, neighbors p q → q ∉ S) ∧ (∀ p ∉ S, {q ∈ S | neighbors p q}.encard = 1)) ↔ n ∈ answer)"
] |
Find the number of positive integers $N$ satisfying: (i) $N$ is divisible by $2020$, (ii) $N$ has at most $2020$ decimal digits, (iii) The decimal digits of $N$ are a string of consecutive ones followed by a string of consecutive zeros. | 508536 | answer = (508536 : ℕ) | ℕ | {
"benchmark": "putnam_solving",
"index": 290
} | [] | [] | [
"(Set.ncard {x : ℕ | (2020 ∣ x) ∧ (Nat.log 10 x) + 1 ≤ 2020 ∧ (∃ k l, k ≥ l ∧ x = ∑ i in Finset.range (k - l + 1), 10 ^ (i + l))} = answer)"
] | |
Let $k$ be a nonnegative integer. Evaluate
\[
\sum_{j=0}^k 2^{k-j} \binom{k+j}{j}.
\] | 4^k | answer = fun (k : ℕ) => ((4 : ℕ) ^ k : ℕ) | ℕ → ℕ | {
"benchmark": "putnam_solving",
"index": 291
} | [
{
"t": "ℕ",
"v": null,
"name": "k",
"t_type": "Type"
}
] | [] | [
"((∑ j in Finset.Icc 0 k, 2 ^ (k - j) * Nat.choose (k + j) j) = answer k)"
] | |
Let $a_0 = \pi/2$, and let $a_n = \sin(a_{n-1})$ for $n \geq 1$. Determine whether
\[
\sum_{n=1}^\infty a_n^2
\]
converges. | False | open Filter Topology Set | answer = False | Prop | {
"benchmark": "putnam_solving",
"index": 292
} | [] | [
{
"t": "ℕ → ℝ",
"v": null,
"name": "a",
"t_type": "Type"
},
{
"t": "a 0 = Real.pi / 2",
"v": null,
"name": "ha0",
"t_type": "Prop"
},
{
"t": "∀ n, a (n+1) = Real.sin (a n)",
"v": null,
"name": "ha",
"t_type": "Prop"
}
] | [
"((∃ L, Tendsto (fun m : ℕ => ∑ n in Finset.Icc 1 m, (a n)^2) atTop (𝓝 L)) ↔ answer)"
] |
Let $a_n$ be the number of sets $S$ of positive integers for which
\[
\sum_{k \in S} F_k = n,
\]
where the Fibonacci sequence $(F_k)_{k \geq 1}$ satisfies $F_{k+2} = F_{k+1} + F_k$ and begins $F_1 = 1, F_2 = 1, F_3 = 2, F_4 = 3$. Find the largest integer $n$ such that $a_n = 2020$. | (Nat.fib 4040) - 1 | open Filter Topology Set | answer = ((↑(Nat.fib (4040 : ℕ) : ℕ) : ℤ) - (1 : ℤ) : ℤ) | ℤ | {
"benchmark": "putnam_solving",
"index": 293
} | [] | [
{
"t": "ℤ → ℕ",
"v": null,
"name": "a",
"t_type": "Type"
},
{
"t": "a = fun n : ℤ => {S : Finset ℕ | (∀ k ∈ S, k > 0) ∧ ∑ k : S, Nat.fib k = n}.ncard",
"v": null,
"name": "ha",
"t_type": "Prop"
}
] | [
"(IsGreatest {n | a n = 2020} answer)"
] |
For a positive integer $N$, let $f_N$ be the function defined by
\[
f_N(x) = \sum_{n=0}^N \frac{N+1/2-n}{(N+1)(2n+1)} \sin((2n+1)x).
\]
Determine the smallest constant $M$ such that $f_N(x) \leq M$ for all $N$ and all real $x$. | π/4 | open Filter Topology Set | answer = ((Real.pi : ℝ) / (4 : ℝ) : ℝ) | ℝ | {
"benchmark": "putnam_solving",
"index": 294
} | [] | [
{
"t": "ℤ → (ℝ → ℝ)",
"v": null,
"name": "f",
"t_type": "Type"
},
{
"t": "f = fun N : ℤ => fun x : ℝ =>\n ∑ n in Finset.Icc 0 N, (N + 1/2 - n)/((N + 1)*(2*n + 1)) * Real.sin ((2*n + 1)*x)",
"v": null,
"name": "hf",
"t_type": "Prop"
}
] | [
"(answer = sSup {y | ∃ᵉ (N > 0) (x : ℝ), y = f N x})"
] |
For a positive integer $n$, define $d(n)$ to be the sum of the digits of $n$ when written in binary (for example, $d(13) = 1+1+0+1=3)$. Let
\[
S = \sum_{k=1}^{2020} (-1)^{d(k)} k^3.
\]
Determine $S$ modulo 2020. | 1990 | open Filter Topology Set | answer = (1990 : ℕ) | ℕ | {
"benchmark": "putnam_solving",
"index": 295
} | [] | [
{
"t": "ℕ → ℕ",
"v": null,
"name": "d",
"t_type": "Type"
},
{
"t": "ℤ",
"v": null,
"name": "S",
"t_type": "Type"
},
{
"t": "d = fun n : ℕ => ∑ i : Fin (Nat.digits 2 n).length, (Nat.digits 2 n)[i]!",
"v": null,
"name": "hd",
"t_type": "Prop"
},
{
"t": "... | [
"(S % 2020 = answer)"
] |
Let $n$ be a positive integer, and let $V_n$ be the set of integer $(2n+1)$-tuples $\mathbf{v} = (s_0, s_1, \cdots, s_{2n-1}, s_{2n})$ for which $s_0 = s_{2n} = 0$ and $|s_j - s_{j-1}| = 1$ for $j=1,2,\cdots,2n$. Define \[ q(\mathbf{v}) = 1 + \sum_{j=1}^{2n-1} 3^{s_j}, \] and let $M(n)$ be the average of $\frac{1}{q(\m... | 1 / 4040 | open Filter Topology Set | answer = (1 / 4040 : ℝ) | ℝ | {
"benchmark": "putnam_solving",
"index": 296
} | [] | [
{
"t": "ℕ → Set (ℕ → ℤ)",
"v": null,
"name": "V",
"t_type": "Type"
},
{
"t": "ℕ → (ℕ → ℤ) → ℝ",
"v": null,
"name": "q",
"t_type": "Type"
},
{
"t": "ℕ → ℝ",
"v": null,
"name": "M",
"t_type": "Type"
},
{
"t": "V = fun n ↦ ({s : ℕ → ℤ | s 0 = 0 ∧ (∀ j ≥ 2... | [
"(M 2020 = answer)"
] |
A grasshopper starts at the origin in the coordinate plane and makes a sequence of hops.
Each hop has length $5$, and after each hop the grasshopper is at a point whose coordinates are both integers; thus, there are $12$ possible locations for the grasshopper after the first hop.
What is the smallest number of hops nee... | 578 | open Filter Topology | answer = (578 : ℕ) | ℕ | {
"benchmark": "putnam_solving",
"index": 297
} | [] | [
{
"t": "List (ℤ × ℤ) → Prop",
"v": null,
"name": "P",
"t_type": "Type"
},
{
"t": "∀ l, P l ↔ l.Chain' fun p q ↦ (p.1 - q.1) ^ 2 + (p.2 - q.2) ^ 2 = 25",
"v": null,
"name": "P_def",
"t_type": "Prop"
}
] | [
"(IsLeast\n {k | ∃ l, P ((0, 0) :: l) ∧ l.getLast! = (2021, 2021) ∧ l.length = k}\n answer)"
] |
For every positive real number $x$, let $g(x)=\lim_{r \to 0}((x+1)^{r+1}-x^{r+1})^\frac{1}{r}$. Find $\lim_{x \to \infty}\frac{g(x)}{x}$. | Real.exp 1 | open Filter Topology | answer = (Real.exp (1 : ℝ) : ℝ) | ℝ | {
"benchmark": "putnam_solving",
"index": 298
} | [] | [
{
"t": "ℝ → ℝ",
"v": null,
"name": "g",
"t_type": "Type"
},
{
"t": "∀ x > 0, Tendsto (fun r : ℝ => ((x + 1) ^ (r + 1) - x ^ (r + 1)) ^ (1 / r)) (𝓝[>] 0) (𝓝 (g x))",
"v": null,
"name": "hg",
"t_type": "Prop"
}
] | [
"(Tendsto (fun x : ℝ => g x / x) atTop (𝓝 answer))"
] |
Determine all positive integers $N$ for which the sphere $x^2+y^2+z^2=N$ has an inscribed regular tetrahedron whose vertices have integer coordinates. | {3 * m ^ 2 | m > 0} | open Filter Topology | answer = ({x : ℕ | ∃ m > (0 : ℕ), ((3 : ℕ) * (m ^ (2 : ℕ) : ℕ) : ℕ) = x} : Set ℕ) | Set ℕ | {
"benchmark": "putnam_solving",
"index": 299
} | [] | [
{
"t": "ℕ",
"v": null,
"name": "N",
"t_type": "Type"
},
{
"t": "Set (EuclideanSpace ℝ (Fin 3))",
"v": null,
"name": "Nsphere",
"t_type": "Type"
},
{
"t": "Nsphere = {p | (p 0) ^ 2 + (p 1) ^ 2 + (p 2) ^ 2 = (N : ℝ)}",
"v": null,
"name": "hNsphere",
"t_type": "P... | [
"((0 < N ∧ ∃ A B C D : EuclideanSpace ℝ (Fin 3),\n A ∈ Nsphere ∧ B ∈ Nsphere ∧ C ∈ Nsphere ∧ D ∈ Nsphere ∧\n intcoords A ∧ intcoords B ∧ intcoords C ∧ intcoords D ∧\n (∃ s > 0, dist A B = s ∧ dist A C = s ∧ dist A D = s ∧ dist B C = s ∧ dist B D = s ∧ dist C D = s))\n ↔ N ∈ answer)"
] |
Let
\[
I(R) = \iint_{x^2+y^2 \leq R^2} \left( \frac{1+2x^2}{1+x^4+6x^2y^2+y^4} - \frac{1+y^2}{2+x^4+y^4} \right)\,dx\,dy.
\]
Find
\[
\lim_{R \to \infty} I(R),
\]
or show that this limit does not exist. | $\frac{\sqrt{2}}{2} \pi \log 2$ | open Filter Topology Metric | answer = ((((√(2 : ℝ) : ℝ) / (2 : ℝ) : ℝ) * (Real.pi : ℝ) : ℝ) * (Real.log (2 : ℝ) : ℝ) : ℝ) | ℝ | {
"benchmark": "putnam_solving",
"index": 300
} | [] | [
{
"t": "ℝ → Set (EuclideanSpace ℝ (Fin 2))",
"v": null,
"name": "S",
"t_type": "Type"
},
{
"t": "S = fun R => ball (0 : EuclideanSpace ℝ (Fin 2)) R",
"v": null,
"name": "hS",
"t_type": "Prop"
},
{
"t": "ℝ → ℝ",
"v": null,
"name": "I",
"t_type": "Type"
},
{... | [
"(Tendsto I atTop (𝓝 answer))"
] |
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