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 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 | open Filter Topology | answer =
({p : Polynomial ℝ |
(p.degree : WithBot ℕ) = (2 : WithBot ℕ) ∧
∃ (r1 : ℝ) (r2 : ℝ), r1 ≠ r2 ∧ (Polynomial.eval r1 p : ℝ) = (0 : ℝ) ∧ (Polynomial.eval r2 p : ℝ) = (0 : ℝ)} :
Set (Polynomial ℝ)) | Set (Polynomial ℝ) | {
"benchmark": "putnam_solving",
"index": 139
} | [
{
"t": "Polynomial ℝ",
"v": null,
"name": "p",
"t_type": "Type"
}
] | [
{
"t": "ℕ",
"v": null,
"name": "n",
"t_type": "Type"
},
{
"t": "n = p.degree",
"v": null,
"name": "hn",
"t_type": "Prop"
},
{
"t": "n ≥ 2",
"v": null,
"name": "hge",
"t_type": "Prop"
}
] | [
"(p ∈ answer ↔\n (∃ r : ℕ → ℝ, (∀ i : Fin (n - 1), r i < r (i + 1)) ∧\n (∀ i : Fin n, p.eval (r i) = 0) ∧\n (∀ i : Fin (n - 1), (Polynomial.derivative p).eval ((r i + r (i + 1)) / 2) = 0)))"
] |
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 ... | True | open Filter Metric Topology | answer = True | Prop | {
"benchmark": "putnam_solving",
"index": 140
} | [] | [] | [
"((∃ (c : ℕ → EuclideanSpace ℝ (Fin 2)) (r : ℕ → ℝ),\n (¬ ∃ p, MapClusterPt p atTop c) ∧\n (Summable <| fun i ↦ r i ^ 2) ∧\n (∀ L : AffineSubspace ℝ (EuclideanSpace ℝ (Fin 2)),\n Module.finrank ℝ L.direction = 1 → ∃ i, (↑L ∩ closedBall (c i) (r i)).Nonempty)) ↔\n answer)"
] |
Find the maximum value of $\int_0^y \sqrt{x^4+(y-y^2)^2}\,dx$ for $0 \leq y \leq 1$. | $\frac{1}{3}$ | open Filter Topology | answer = (1 / 3 : ℝ) | ℝ | {
"benchmark": "putnam_solving",
"index": 141
} | [] | [
{
"t": "Set.Icc (0 : ℝ) 1 → ℝ",
"v": null,
"name": "f",
"t_type": "Type"
},
{
"t": "∀ y : Set.Icc 0 1, f y = ∫ x in Set.Ioo 0 y, Real.sqrt (x ^ 4 + (y - y ^ 2) ^ 2)",
"v": null,
"name": "hf",
"t_type": "Prop"
}
] | [
"(IsGreatest (f '' (Set.Icc 0 1)) answer)"
] |
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 | open Filter Topology | answer = ({A : ℤ | ∃ x > (0 : ℤ), A = (x ^ (2 : ℕ) : ℤ)} : Set ℤ) | Set ℤ | {
"benchmark": "putnam_solving",
"index": 142
} | [
{
"t": "ℤ",
"v": null,
"name": "A",
"t_type": "Type"
}
] | [
{
"t": "ℤ → ℤ",
"v": null,
"name": "m",
"t_type": "Type"
},
{
"t": "ℤ → ℤ",
"v": null,
"name": "S",
"t_type": "Type"
},
{
"t": "ℕ → ℤ",
"v": null,
"name": "a",
"t_type": "Type"
},
{
"t": "∀ n, 0 ≤ n → (m n) ^ 2 ≤ n ∧ (∀ m' : ℤ, m' ^ 2 ≤ n → m' ≤ m n)",... | [
"((∃ (K : ℕ) (c : ℤ), ∀ k ≥ K, a k = c) ↔ A ∈ answer)"
] |
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 $\{x^2:x \in \mathbb{Z}_p\} \cap \{y^2+1:y \in \mathbb{Z}_p\}$? | $\lceil p / 4 \rceil$ | open Filter Topology | answer = fun (p : ℕ) => (⌈((↑p : ℝ) / (4 : ℝ) : ℝ)⌉₊ : ℕ) | ℕ → ℕ | {
"benchmark": "putnam_solving",
"index": 143
} | [
{
"t": "ℕ",
"v": null,
"name": "p",
"t_type": "Type"
}
] | [
{
"t": "Odd p",
"v": null,
"name": "podd",
"t_type": "Prop"
},
{
"t": "Prime p",
"v": null,
"name": "pprime",
"t_type": "Prop"
}
] | [
"(({z : ZMod p | ∃ x : ZMod p, z = x ^ 2} ∩ {z : ZMod p | ∃ y : ZMod p, z = y ^ 2 + 1}).encard = answer p)"
] |
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 : ℝ => |Real.log (a / b)|) | open Filter Topology | answer = fun (a b : ℝ) => (|(Real.log (a / b : ℝ) : ℝ)| : ℝ) | ℝ → ℝ → ℝ | {
"benchmark": "putnam_solving",
"index": 144
} | [
{
"t": "ℝ",
"v": null,
"name": "a",
"t_type": "Type"
},
{
"t": "ℝ",
"v": null,
"name": "b",
"t_type": "Type"
}
] | [
{
"t": "a > 0 ∧ b > 0",
"v": null,
"name": "abpos",
"t_type": "Prop"
}
] | [
"(IsGreatest {c | ∀ u, (0 < |u| ∧ |u| ≤ c) → (∀ x ∈ Set.Ioo 0 1, a ^ x * b ^ (1 - x) ≤ a * (Real.sinh (u * x) / Real.sinh u) + b * (Real.sinh (u * (1 - x)) / Real.sinh u))} (answer a b))"
] |
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 | open Topology Filter | answer = (1992 : ℝ) | ℝ | {
"benchmark": "putnam_solving",
"index": 145
} | [] | [
{
"t": "ℝ → ℝ",
"v": null,
"name": "C",
"t_type": "Type"
},
{
"t": "C = fun α ↦ taylorCoeffWithin (fun x ↦ (1 + x) ^ α) 1992 Set.univ 0",
"v": null,
"name": "hC",
"t_type": "Prop"
}
] | [
"(∫ y in (0)..1, C (-y - 1) * ∑ k in Finset.Icc (1 : ℕ) 1992, 1 / (y + k) = answer)"
] |
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 $\{(m + 1, 2^{m/2}, 2^{m/2})\}$ | open Topology Filter Nat | answer = fun (m : ℕ) =>
(if Odd m then (∅ : Set (ℕ × ℕ × ℕ))
else
({(((m + (1 : ℕ) : ℕ), ((((2 : ℕ) ^ (m / (2 : ℕ) : ℕ) : ℕ), ((2 : ℕ) ^ (m / (2 : ℕ) : ℕ) : ℕ)) : ℕ × ℕ)) :
ℕ × ℕ × ℕ)} :
Set (ℕ × ℕ × ℕ)) :
Set (ℕ × ℕ × ℕ)) | ℕ → Set (ℕ × ℕ × ℕ) | {
"benchmark": "putnam_solving",
"index": 146
} | [
{
"t": "ℕ",
"v": null,
"name": "m",
"t_type": "Type"
}
] | [
{
"t": "m > 0",
"v": null,
"name": "mpos",
"t_type": "Prop"
},
{
"t": "Set (ℕ × ℕ × ℕ)",
"v": null,
"name": "S",
"t_type": "Type"
},
{
"t": "∀ n x y : ℕ, (n, x, y) ∈ S ↔ n > 0 ∧ x > 0 ∧ y > 0 ∧ Coprime n m ∧ (x ^ 2 + y ^ 2) ^ m = (x * y) ^ n",
"v": null,
"name": "... | [
"(S = answer m)"
] |
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 ↦ ite (Even k) ((-1) ^ (k / 2) * factorial k) 0 | open Topology Filter Nat Function | answer = fun (k : ℕ) => (if Even k then (((-1 : ℝ) ^ (k / (2 : ℕ) : ℕ) : ℝ) * (↑(k ! : ℕ) : ℝ) : ℝ) else (0 : ℝ) : ℝ) | ℕ → ℝ | {
"benchmark": "putnam_solving",
"index": 147
} | [] | [
{
"t": "ℝ → ℝ",
"v": null,
"name": "f",
"t_type": "Type"
},
{
"t": "ContDiff ℝ ⊤ f",
"v": null,
"name": "hfdiff",
"t_type": "Prop"
},
{
"t": "∀ n : ℕ, n > 0 → f (1 / n) = n ^ 2 / (n ^ 2 + 1)",
"v": null,
"name": "hf",
"t_type": "Prop"
}
] | [
"(∀ k : ℕ, k > 0 → iteratedDeriv k f 0 = answer k)"
] |
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 ↦ 2 * n - 3 | open Topology Filter Nat Function | answer = fun (n : ℕ) => (((2 : ℤ) * (↑n : ℤ) : ℤ) - (3 : ℤ) : ℤ) | ℕ → ℤ | {
"benchmark": "putnam_solving",
"index": 148
} | [
{
"t": "ℕ",
"v": null,
"name": "n",
"t_type": "Type"
}
] | [
{
"t": "n ≥ 2",
"v": null,
"name": "hn",
"t_type": "Prop"
},
{
"t": "Finset ℝ → Set ℝ",
"v": null,
"name": "A",
"t_type": "Type"
},
{
"t": "A = fun S ↦ {x | ∃ a ∈ S, ∃ b ∈ S, a ≠ b ∧ (a + b) / 2 = x}",
"v": null,
"name": "hA",
"t_type": "Prop"
}
] | [
"(IsLeast {k : ℤ | ∃ S : Finset ℝ, S.card = n ∧ k = (A S).ncard} (answer n))"
] |
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 + π | open Topology Filter Nat Function Polynomial | answer = ((4 : ℝ) + (Real.pi : ℝ) : ℝ) | ℝ | {
"benchmark": "putnam_solving",
"index": 149
} | [] | [
{
"t": "(Fin 2 → ℝ) → (ℕ → ℝ)",
"v": null,
"name": "a",
"t_type": "Type"
},
{
"t": "∀ p, (a p) 0 = p 0 ∧ (∀ n, (a p) (n + 1) = (((a p) n) ^ 2 + (p 1) ^ 2) / 2)",
"v": null,
"name": "ha",
"t_type": "Prop"
}
] | [
"(answer = (MeasureTheory.volume {p | ∃ L, Tendsto (a p) atTop (𝓝 L)}).toReal)"
] |
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 | open Topology Filter Nat Function Polynomial | answer = (3984 : ℕ) | ℕ | {
"benchmark": "putnam_solving",
"index": 150
} | [] | [
{
"t": "Polynomial ℝ → Prop",
"v": null,
"name": "IsValid",
"t_type": "Type"
},
{
"t": "Polynomial ℝ → Polynomial ℝ → Prop",
"v": null,
"name": "pair",
"t_type": "Type"
},
{
"t": "∀ p, IsValid p ↔ p ≠ 0 ∧ p.degree < 1992 ∧ IsCoprime p (X ^ 3 - X)",
"v": null,
"nam... | [
"(IsLeast {k : ℕ | ∃ p f, IsValid p ∧ pair p f ∧ k = f.degree} answer)"
] |
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}
\righ... | False | open Topology Filter Nat Function Polynomial | answer = False | Prop | {
"benchmark": "putnam_solving",
"index": 151
} | [] | [
{
"t": "ℕ → ℚ",
"v": null,
"name": "D",
"t_type": "Type"
},
{
"t": "∀ n, D n = Matrix.det (fun i j : Fin (n - 1) ↦ ite (i = j) ((i : ℕ) + 3 : ℚ) 1)",
"v": null,
"name": "hD",
"t_type": "Prop"
}
] | [
"(answer ↔ (Bornology.IsBounded {x | ∃ n ≥ 2, D n / factorial n = x}))"
] |
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 $y=c$ betw... | 4/9 | answer = (4 / 9 : ℝ) | ℝ | {
"benchmark": "putnam_solving",
"index": 152
} | [] | [] | [
"(0 < answer ∧ answer < (4 * Real.sqrt 2) / 9 ∧\n (∫ x in Set.Ioo 0 ((Real.sqrt 2) / 3), max (answer - (2 * x - 3 * x ^ 3)) 0) =\n (∫ x in Set.Ioo 0 ((Real.sqrt 6) / 3), max ((2 * x - 3 * x ^ 3) - answer) 0))"
] | |
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 | answer = (3987 : ℕ) | ℕ | {
"benchmark": "putnam_solving",
"index": 153
} | [] | [] | [
"(IsLeast\n {n : ℕ | 0 < n ∧\n ∀ m ∈ Set.Ioo (0 : ℤ) (1993), ∃ k : ℤ,\n ((m : ℝ) / 1993 < (k : ℝ) / n) ∧ ((k : ℝ) / n < ((m : ℝ) + 1) / 1994) }\n answer)"
] | |
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) | answer = (((5 / 4 : ℚ), (-1 / 4 : ℚ)) : ℚ × ℚ) | ℚ × ℚ | {
"benchmark": "putnam_solving",
"index": 154
} | [] | [] | [
"((MeasureTheory.volume\n {p : Fin 2 → ℝ | 0 < p 0 ∧ p 0 < 1 ∧ 0 < p 1 ∧ p 1 < 1 ∧ Even (round (p 0 / p 1))}\n ).toReal = answer.1 + answer.2 * Real.pi)"
] | |
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 $332 \leq n \leq 350$ | open Filter Topology | answer = ({n : ℤ | (315 : ℤ) ≤ n ∧ n ≤ (325 : ℤ) ∨ (332 : ℤ) ≤ n ∧ n ≤ (350 : ℤ)} : Set ℤ) | Set ℤ | {
"benchmark": "putnam_solving",
"index": 155
} | [
{
"t": "ℤ",
"v": null,
"name": "n",
"t_type": "Type"
}
] | [] | [
"(n ∈ answer ↔ (0 < n ∧ {m : ℕ | |n - m ^ 2| ≤ 250}.encard = 15))"
] |
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 : ℝ | c < 243 / 8} | open Filter Topology | answer = ({c : ℝ | c < (243 / 8 : ℝ)} : Set ℝ) | Set ℝ | {
"benchmark": "putnam_solving",
"index": 156
} | [
{
"t": "ℝ",
"v": null,
"name": "c",
"t_type": "Type"
}
] | [] | [
"(answer = {c : ℝ | (∃ m b : ℝ,\n {x : ℝ | m * x + b = x ^ 4 + 9 * x ^ 3 + c * x ^ 2 + 9 * x + 4}.encard = 4)})"
] |
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 | open Filter Topology | answer = (Set.Iio (1 : ℝ) : Set ℝ) | Set ℝ | {
"benchmark": "putnam_solving",
"index": 157
} | [] | [] | [
"({k | ∀ f (hf : (∀ x, 0 < f x ∧ f x < deriv f x) ∧ Differentiable ℝ f),\n ∃ N, ∀ x > N, Real.exp (k * x) < f x} = answer)"
] |
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 ⟨a,b⟩ := x; a = b} | open Filter Topology Real | answer = ({((a, b) : ℝ × ℝ) : ℝ × ℝ | a = b} : Set (ℝ × ℝ)) | Set (ℝ × ℝ) | {
"benchmark": "putnam_solving",
"index": 158
} | [] | [
{
"t": "(ℝ × ℝ) → Prop",
"v": null,
"name": "habconv",
"t_type": "Type"
},
{
"t": "habconv = fun ⟨a,b⟩ => ∃ limit : ℝ, Tendsto (fun t : ℝ => ∫ x in (Set.Icc b t), (sqrt (sqrt (x + a) - sqrt x) - sqrt (sqrt x - sqrt (x - b)))) atTop (𝓝 limit)",
"v": null,
"name": "habconv_def",
"... | [
"(∀ ab : ℝ × ℝ, ab.1 > 0 ∧ ab.2 > 0 → (habconv ab ↔ ab ∈ answer))"
] |
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... | True | open Filter Topology Real | answer = True | Prop | {
"benchmark": "putnam_solving",
"index": 159
} | [] | [
{
"t": "ℕ",
"v": null,
"name": "n",
"t_type": "Type"
},
{
"t": "Fin n → (ℝ → ℝ)",
"v": null,
"name": "x",
"t_type": "Type"
},
{
"t": "Fin n → Fin n → ℝ",
"v": null,
"name": "a",
"t_type": "Type"
},
{
"t": "0 < n",
"v": null,
"name": "npos",
... | [
"(answer ↔ ¬(∀ b : Fin n → ℝ, (∀ t : ℝ, ∑ i : Fin n, (b i) * ((x i) t) = 0) → (∀ i, b i = 0)))"
] |
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 o... | 45 if n = 1, 10 * 45² if n = 2, and 0 otherwise | open Filter Topology Real Nat | answer = fun (n : ℕ) =>
(if n = (1 : ℕ) then (45 : ℤ) else (if n = (2 : ℕ) then ((10 : ℤ) * ((45 : ℤ) ^ (2 : ℕ) : ℤ) : ℤ) else (0 : ℤ) : ℤ) :
ℤ) | ℕ → ℤ | {
"benchmark": "putnam_solving",
"index": 160
} | [
{
"t": "ℕ",
"v": null,
"name": "n",
"t_type": "Type"
}
] | [
{
"t": "n > 0",
"v": null,
"name": "hn",
"t_type": "Prop"
},
{
"t": "Set (ℕ → ℕ)",
"v": null,
"name": "digits_set",
"t_type": "Type"
},
{
"t": "digits_set = {f | f 0 ≠ 0 ∧ (∀ i : Fin (n ^ 2), f i ≤ 9) ∧ (∀ i ≥ n ^ 2, f i = 0)}",
"v": null,
"name": "hdigits_set",
... | [
"(∑' f : digits_set, (digits_to_matrix f).det = answer n)"
] |
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. | ⟨3,1,5,2⟩ | open Filter Topology Real Nat | answer = (((3 : ℤ), (((1 : ℤ), (((5 : ℤ), (2 : ℤ)) : ℤ × ℤ)) : ℤ × ℤ × ℤ)) : ℤ × ℤ × ℤ × ℤ) | ℤ × ℤ × ℤ × ℤ | {
"benchmark": "putnam_solving",
"index": 161
} | [] | [
{
"t": "ℝ",
"v": null,
"name": "contfrac",
"t_type": "Type"
},
{
"t": "contfrac = 2207 - 1 / contfrac",
"v": null,
"name": "hcontfrac",
"t_type": "Prop"
},
{
"t": "1 < contfrac",
"v": null,
"name": "hcontfrac'",
"t_type": "Prop"
}
] | [
"(let ⟨a, b, c, d⟩ := answer;\n contfrac ^ ((1 : ℝ) / 8) = (a + b * sqrt c) / d)"
] |
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 | open Metric | answer = fun (O1 O2 : EuclideanSpace ℝ (Fin (2 : ℕ))) =>
({p : EuclideanSpace ℝ (Fin (2 : ℕ)) |
(Dist.dist p (midpoint ℝ O1 O2 : EuclideanSpace ℝ (Fin (2 : ℕ))) : ℝ) ≥ (1 : ℝ) ∧
(Dist.dist p (midpoint ℝ O1 O2 : EuclideanSpace ℝ (Fin (2 : ℕ))) : ℝ) ≤ (2 : ℝ)} :
Set (EuclideanSpace ℝ (Fin (2 : ℕ)))) | (EuclideanSpace ℝ (Fin 2)) → (EuclideanSpace ℝ (Fin 2)) → Set (EuclideanSpace ℝ (Fin 2)) | {
"benchmark": "putnam_solving",
"index": 162
} | [
{
"t": "EuclideanSpace ℝ (Fin 2)",
"v": null,
"name": "O1",
"t_type": "Type"
},
{
"t": "EuclideanSpace ℝ (Fin 2)",
"v": null,
"name": "O2",
"t_type": "Type"
}
] | [
{
"t": "Set (EuclideanSpace ℝ (Fin 2))",
"v": null,
"name": "C1",
"t_type": "Type"
},
{
"t": "Set (EuclideanSpace ℝ (Fin 2))",
"v": null,
"name": "C2",
"t_type": "Type"
},
{
"t": "C1 = Metric.sphere O1 1",
"v": null,
"name": "hC1",
"t_type": "Prop"
},
{
... | [
"({M : EuclideanSpace ℝ (Fin 2) | ∃ X Y, X ∈ C1 ∧ Y ∈ C2 ∧ M = midpoint ℝ X Y} = answer O1 O2)"
] |
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 | answer = False | Prop | {
"benchmark": "putnam_solving",
"index": 163
} | [] | [
{
"t": "Fin 20 → Set (Fin 6)",
"v": null,
"name": "choices",
"t_type": "Type"
}
] | [
"(answer ↔ (∃ (students : Finset (Fin 20)) (courses : Finset (Fin 6)),\n students.card = 5 ∧\n courses.card = 2 ∧\n (↑courses ⊆ ⋂ s ∈ students, choices s ∨ ↑courses ⊆ ⋂ s ∈ students, (choices s)ᶜ)))"
] | |
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 | open Function | answer = fun (c : ℝ) =>
(if c ≤ (1 / 4 : ℝ) then ({f : ℝ → ℝ | ∃ (d : ℝ), ∀ (x : ℝ), (f x : ℝ) = d} : Set (ℝ → ℝ))
else
({f : ℝ → ℝ |
ContinuousOn f (Set.Icc (0 : ℝ) c : Set ℝ) ∧
(f (0 : ℝ) : ℝ) = (f c : ℝ) ∧
(∀ (x : ℝ), x > (0 : ℝ) → (f x : ℝ) = (f ((x ^ (2 : ℕ) : ℝ) + c... | ℝ → Set (ℝ → ℝ) | {
"benchmark": "putnam_solving",
"index": 164
} | [
{
"t": "ℝ",
"v": null,
"name": "c",
"t_type": "Type"
},
{
"t": "ℝ → ℝ",
"v": null,
"name": "f",
"t_type": "Type"
}
] | [
{
"t": "c > 0",
"v": null,
"name": "cgt0",
"t_type": "Prop"
}
] | [
"((Continuous f ∧ ∀ x : ℝ, f x = f (x ^ 2 + c)) ↔ f ∈ answer c)"
] |
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 | open Function | answer = (Nat.fib : ℕ → ℕ) | ℕ → ℕ | {
"benchmark": "putnam_solving",
"index": 165
} | [] | [
{
"t": "Finset ℕ → Prop",
"v": null,
"name": "selfish",
"t_type": "Type"
},
{
"t": "ℕ",
"v": null,
"name": "n",
"t_type": "Type"
},
{
"t": "∀ s : Finset ℕ, selfish s = (s.card ∈ s)",
"v": null,
"name": "hselfish",
"t_type": "Prop"
},
{
"t": "n ≥ 1",
... | [
"({s : Finset ℕ | (s : Set ℕ) ⊆ Set.Icc 1 n ∧ selfish s ∧ (∀ ss : Finset ℕ, ss ⊂ s → ¬selfish ss)}.encard = answer n)"
] |
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 $x_1x_2+x_2x_3+\cdots+x_{n-1}x_n+x_nx_1$. | $(2n^3 + 3n^2 - 11n + 18) / 6$ | open Function | answer = fun (n : ℕ) =>
((((((2 : ℕ) * (n ^ (3 : ℕ) : ℕ) : ℕ) + ((3 : ℕ) * (n ^ (2 : ℕ) : ℕ) : ℕ) : ℕ) - ((11 : ℕ) * n : ℕ) : ℕ) + (18 : ℕ) :
ℕ) /
(6 : ℕ) :
ℕ) | ℕ → ℕ | {
"benchmark": "putnam_solving",
"index": 166
} | [
{
"t": "ℕ",
"v": null,
"name": "n",
"t_type": "Type"
}
] | [
{
"t": "n ≥ 2",
"v": null,
"name": "hn",
"t_type": "Prop"
}
] | [
"(IsGreatest\n {k | ∃ x : ℕ → ℤ,\n (x '' (Finset.range n) = Set.Icc (1 : ℤ) n) ∧\n ∑ i : Fin n, x i * x ((i + 1) % n) = k}\n (answer n))"
] |
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 | open Function Nat | answer = False | Prop | {
"benchmark": "putnam_solving",
"index": 167
} | [] | [
{
"t": "Matrix (Fin 2) (Fin 2) ℝ → Matrix (Fin 2) (Fin 2) ℝ",
"v": null,
"name": "matsin",
"t_type": "Type"
},
{
"t": "Matrix (Fin 2) (Fin 2) ℝ",
"v": null,
"name": "mat1996",
"t_type": "Type"
},
{
"t": "∀ A, matsin A = ∑' n : ℕ, ((-(1 : ℝ)) ^ n / (2 * n + 1)!) • A ^ (2 *... | [
"((∃ A, matsin A = mat1996) ↔ answer)"
] |
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,... | $2^{\lfloor (n + 2) / 2 \rfloor} + 2^{\lfloor (n + 1) / 2 \rfloor} - 2$ | open Function Nat | answer = fun (n : ℕ) =>
((((2 : ℕ) ^ (⌊((n + (2 : ℕ) : ℕ) / (2 : ℕ) : ℕ)⌋₊ : ℕ) : ℕ) +
((2 : ℕ) ^ (⌊((n + (1 : ℕ) : ℕ) / (2 : ℕ) : ℕ)⌋₊ : ℕ) : ℕ) :
ℕ) -
(2 : ℕ) :
ℕ) | ℕ → ℕ | {
"benchmark": "putnam_solving",
"index": 168
} | [
{
"t": "ℕ",
"v": null,
"name": "n",
"t_type": "Type"
}
] | [
{
"t": "(Fin n → ℤˣ) → Fin n → Fin n → ℤ",
"v": null,
"name": "Δ",
"t_type": "Type"
},
{
"t": "(Fin n → ℤˣ) → Prop",
"v": null,
"name": "balanced",
"t_type": "Type"
},
{
"t": "∀ S, ∀ a b, a ≤ b → Δ S a b = ∑ i in Finset.Icc a b, (S i : ℤ)",
"v": null,
"name": "hΔ"... | [
"({S : Fin n → ℤˣ | balanced S}.ncard = answer n)"
] |
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*} | \(\sqrt{e}\) | open Filter Topology | answer = (√(Real.exp (1 : ℝ) : ℝ) : ℝ) | ℝ | {
"benchmark": "putnam_solving",
"index": 169
} | [] | [
{
"t": "ℝ → ℝ",
"v": null,
"name": "series1",
"t_type": "Type"
},
{
"t": "ℝ → ℝ",
"v": null,
"name": "series2",
"t_type": "Type"
},
{
"t": "series1 = fun x => ∑' n : ℕ, (-1)^n * x^(2*n + 1)/(∏ i : Finset.range n, 2 * ((i : ℝ) + 1))",
"v": null,
"name": "hseries1",... | [
"(Tendsto (fun t => ∫ x in Set.Icc 0 t, series1 x * series2 x) atTop (𝓝 answer))"
] |
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 | open Filter Topology | answer = True | Prop | {
"benchmark": "putnam_solving",
"index": 170
} | [] | [
{
"t": "(n : ℕ+) → Set (Fin n → ℕ+)",
"v": null,
"name": "N",
"t_type": "Type"
},
{
"t": "N = fun (n : ℕ+) => {t : Fin n → ℕ+ | (∀ i j : Fin n, i < j → t i <= t j) ∧ (∑ i : Fin n, (1 : ℝ)/(t i) = 1) }",
"v": null,
"name": "hN",
"t_type": "Prop"
}
] | [
"(Odd (N 10).ncard ↔ answer)"
] |
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) | open Filter Topology | answer = fun (n k : ℤ) => (↑(((n.toNat : ℕ) - (1 : ℕ) : ℕ).choose ((k.toNat : ℕ) - (1 : ℕ) : ℕ) : ℕ) : ℝ) | ℤ → ℤ → ℝ | {
"benchmark": "putnam_solving",
"index": 171
} | [
{
"t": "ℤ",
"v": null,
"name": "n",
"t_type": "Type"
}
] | [
{
"t": "n > 0",
"v": null,
"name": "hn",
"t_type": "Prop"
},
{
"t": "ℝ → (ℤ → ℝ)",
"v": null,
"name": "x",
"t_type": "Type"
},
{
"t": "∀ c, x c 0 = 0",
"v": null,
"name": "hx0",
"t_type": "Prop"
},
{
"t": "∀ c, x c 1 = 1",
"v": null,
"name": "h... | [
"(∀ k : Set.Icc 1 n, x (sSup S) k = answer n k)"
] |
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 | open Filter Topology | answer = fun (n : ℕ) => (↑n : ℝ) | ℕ → ℝ | {
"benchmark": "putnam_solving",
"index": 172
} | [
{
"t": "ℝ",
"v": null,
"name": "r",
"t_type": "Type"
}
] | [
{
"t": "ℝ → ℝ",
"v": null,
"name": "dist_to_int",
"t_type": "Type"
},
{
"t": "dist_to_int r = |r - round r|",
"v": null,
"name": "h_dist_to_int",
"t_type": "Prop"
},
{
"t": "ℕ → ℝ",
"v": null,
"name": "F",
"t_type": "Type"
},
{
"t": "F = fun (n : ℕ) =>... | [
"(∀ n, n > 0 → F n = answer n)"
] |
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 ≤ n ∧ n ≤ 4) ∨ (20 ≤ n ∧ n ≤ 24) ∨ (100 ≤ n ∧ n ≤ 104) ∨ (120 ≤ n ∧ n ≤ 124)} | open Filter Topology Bornology Set | answer =
({n : ℕ |
(1 : ℕ) ≤ n ∧ n ≤ (4 : ℕ) ∨
(20 : ℕ) ≤ n ∧ n ≤ (24 : ℕ) ∨ (100 : ℕ) ≤ n ∧ n ≤ (104 : ℕ) ∨ (120 : ℕ) ≤ n ∧ n ≤ (124 : ℕ)} :
Set ℕ) | Set ℕ | {
"benchmark": "putnam_solving",
"index": 173
} | [
{
"t": "ℕ",
"v": null,
"name": "n",
"t_type": "Type"
}
] | [
{
"t": "n > 0",
"v": null,
"name": "hn",
"t_type": "Prop"
}
] | [
"(n ∈ answer ↔ ¬5 ∣ (∑ m in Finset.Icc 1 n, 1/m : ℚ).den)"
] |
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 $n \equiv 1 \pmod{6}$ | answer = ({n : ℕ | n ≡ (1 : ℕ) [MOD (6 : ℕ)]} : Set ℕ) | Set ℕ | {
"benchmark": "putnam_solving",
"index": 174
} | [] | [
{
"t": "ℕ → List ℕ",
"v": null,
"name": "A",
"t_type": "Type"
},
{
"t": "A 1 = [0]",
"v": null,
"name": "hA1",
"t_type": "Prop"
},
{
"t": "A 2 = [1]",
"v": null,
"name": "hA2",
"t_type": "Prop"
},
{
"t": "∀ n > 0, A (n + 2) = A (n + 1) ++ A n",
"v"... | [
"({n | 1 ≤ n ∧ 11 ∣ Nat.ofDigits 10 (A n).reverse} = answer)"
] | |
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 | open Set Function Metric | answer = (6 : ℝ) | ℝ | {
"benchmark": "putnam_solving",
"index": 175
} | [] | [] | [
"(sInf {((x + 1/x)^6 - (x^6 + 1/x^6) - 2)/((x + 1/x)^3 + (x^3 + 1/x^3)) | x > (0 : ℝ)} = answer)"
] |
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 $0$ | open Set Function Metric | answer = fun (a b : ℝ) =>
(if a > b then (√(((2 : ℝ) * (a ^ (2 : ℕ) : ℝ) : ℝ) + ((2 : ℝ) * (b ^ (2 : ℕ) : ℝ) : ℝ) : ℝ) : ℝ) else (0 : ℝ) : ℝ) | ℝ → ℝ → ℝ | {
"benchmark": "putnam_solving",
"index": 176
} | [
{
"t": "ℝ",
"v": null,
"name": "a",
"t_type": "Type"
},
{
"t": "ℝ",
"v": null,
"name": "b",
"t_type": "Type"
}
] | [
{
"t": "0 < b ∧ b < a",
"v": null,
"name": "hab",
"t_type": "Prop"
}
] | [
"(sInf {d : ℝ | ∃ (c : ℝ) (x : ℝ), 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) ∧\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) ∧\n Real.sqrt ((a - c)^2 + (b - 0)^2) + Real.sqrt ((a - x)^... |
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 ⟨n,m⟩ := nm; multiplicity 2 n ≠ multiplicity 2 m} | open Set Function Metric | answer = ({((n, m) : ℕ × ℕ) : ℕ × ℕ | (multiplicity (2 : ℕ) n : ℕ) ≠ (multiplicity (2 : ℕ) m : ℕ)} : Set (ℕ × ℕ)) | Set (ℕ × ℕ) | {
"benchmark": "putnam_solving",
"index": 177
} | [
{
"t": "ℕ",
"v": null,
"name": "n",
"t_type": "Type"
},
{
"t": "ℕ",
"v": null,
"name": "m",
"t_type": "Type"
}
] | [
{
"t": "ℕ → ℕ → ℤ",
"v": null,
"name": "quantity",
"t_type": "Type"
},
{
"t": "quantity = fun n m => ∑ i in Finset.range (m * n), (-1)^(i/m + i/n)",
"v": null,
"name": "hquantity",
"t_type": "Prop"
},
{
"t": "n > 0 ∧ m > 0",
"v": null,
"name": "hnm",
"t_type":... | [
"(quantity n m = 0 ↔ ⟨n, m⟩ ∈ answer)"
] |
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 | open Set Function Metric | answer = (1 : ℕ) | ℕ | {
"benchmark": "putnam_solving",
"index": 178
} | [] | [
{
"t": "ℕ",
"v": null,
"name": "N",
"t_type": "Type"
},
{
"t": "N = ∑ i in Finset.range 1998, 10^i",
"v": null,
"name": "hN",
"t_type": "Prop"
}
] | [
"(answer = (Nat.floor (10^1000 * Real.sqrt N)) % 10)"
] |
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 | answer = True | Prop | {
"benchmark": "putnam_solving",
"index": 179
} | [] | [] | [
"(answer ↔ ∃ f g h : Polynomial ℝ, ∀ x : ℝ, |f.eval x| - |g.eval x| + h.eval x = if x < -1 then -1 else (if (x ≤ 0) then 3 * x + 2 else -2 * x + 2))"
] | |
Sum the series \[\sum_{m=1}^\infty \sum_{n=1}^\infty \frac{m^2 n}{3^m(n3^m+m3^n)}.\] | 9/32 | open Filter Topology Metric | answer = (9 / 32 : ℝ) | ℝ | {
"benchmark": "putnam_solving",
"index": 180
} | [] | [] | [
"(Tendsto (fun i => ∑ m in Finset.range i, ∑' n : ℕ, (((m + 1)^2*(n+1))/(3^(m + 1) * ((n+1)*3^(m + 1) + (m + 1)*3^(n+1))) : ℝ)) atTop (𝓝 answer))"
] |
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 | open Filter Topology Metric | answer = (3 : ℝ) | ℝ | {
"benchmark": "putnam_solving",
"index": 181
} | [] | [
{
"t": "Set (ℝ × ℝ)",
"v": null,
"name": "A",
"t_type": "Type"
},
{
"t": "A = {xy | 0 ≤ xy.1 ∧ xy.1 < 1 ∧ 0 ≤ xy.2 ∧ xy.2 < 1}",
"v": null,
"name": "hA",
"t_type": "Prop"
},
{
"t": "ℝ → ℝ → ℝ",
"v": null,
"name": "S",
"t_type": "Type"
},
{
"t": "S = fu... | [
"(Tendsto (fun xy : (ℝ × ℝ) => (1 - xy.1 * xy.2^2) * (1 - xy.1^2 * xy.2) * (S xy.1 xy.2)) (𝓝[A] ⟨1,1⟩) (𝓝 answer))"
] |
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 | open Filter Topology Metric | answer = fun (n : ℕ) => ((1 : ℝ) - (((↑n : ℝ) ^ (2 : ℕ) : ℝ) / (4 : ℝ) : ℝ) : ℝ) | ℕ → ℝ | {
"benchmark": "putnam_solving",
"index": 182
} | [
{
"t": "ℕ",
"v": null,
"name": "n",
"t_type": "Type"
}
] | [
{
"t": "n ≥ 3",
"v": null,
"name": "hn",
"t_type": "Prop"
},
{
"t": "ℝ",
"v": null,
"name": "theta",
"t_type": "Type"
},
{
"t": "theta = 2 * Real.pi / n",
"v": null,
"name": "htheta",
"t_type": "Prop"
},
{
"t": "Matrix (Fin n) (Fin n) ℝ",
"v": null... | [
"((1 + A).det = answer n)"
] |
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 $(0, A^2)$ | open Topology Filter | answer = fun (A : ℝ) => (Set.Ioo (0 : ℝ) (A ^ (2 : ℕ) : ℝ) : Set ℝ) | ℝ → Set ℝ | {
"benchmark": "putnam_solving",
"index": 183
} | [
{
"t": "ℝ",
"v": null,
"name": "A",
"t_type": "Type"
}
] | [
{
"t": "A > 0",
"v": null,
"name": "Apos",
"t_type": "Prop"
}
] | [
"({S : ℝ |\n ∃ x : ℕ → ℝ,\n (∀ j : ℕ, x j > 0) ∧\n (∑' j : ℕ, x j) = A ∧\n (∑' j : ℕ, (x j) ^ 2) = S}\n = answer A)"
] |
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 : ℤ | ∃ k : ℤ, k^2 = m ∨ 2*k^2 = m} | open Topology Filter Polynomial Set | answer = ({m : ℤ | ∃ (k : ℤ), (k ^ (2 : ℕ) : ℤ) = m ∨ ((2 : ℤ) * (k ^ (2 : ℕ) : ℤ) : ℤ) = m} : Set ℤ) | Set ℤ | {
"benchmark": "putnam_solving",
"index": 184
} | [] | [
{
"t": "ℤ → Polynomial ℤ",
"v": null,
"name": "P",
"t_type": "Type"
},
{
"t": "P = fun m : ℤ => (Polynomial.X)^4 - (Polynomial.C (2*m + 4))*(Polynomial.X)^2 + Polynomial.C ((m - 2)^2)",
"v": null,
"name": "hP",
"t_type": "Prop"
}
] | [
"({m : ℤ | ∃ a b, P m = a * b ∧\n(∃ n ∈ Ici 1, a.coeff n ≠ 0) ∧ (∃ n ∈ Ici 1, b.coeff n ≠ 0)} = answer)"
] |
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 $\left(\frac{3^{1/5} + 1}{2}, \frac{3^{1/5} - 1}{2}\right)$ | open Topology Filter Polynomial Set | answer =
({((((((3 : ℝ) ^ (1 / 5 : ℝ) : ℝ) + (1 : ℝ) : ℝ) / (2 : ℝ) : ℝ),
((((3 : ℝ) ^ (1 / 5 : ℝ) : ℝ) - (1 : ℝ) : ℝ) / (2 : ℝ) : ℝ)) :
ℝ × ℝ)} :
Set (ℝ × ℝ)) | Set (ℝ × ℝ) | {
"benchmark": "putnam_solving",
"index": 185
} | [] | [
{
"t": "ℝ",
"v": null,
"name": "x",
"t_type": "Type"
},
{
"t": "ℝ",
"v": null,
"name": "y",
"t_type": "Type"
},
{
"t": "x ≠ 0",
"v": null,
"name": "hx",
"t_type": "Prop"
},
{
"t": "y ≠ 0",
"v": null,
"name": "hy",
"t_type": "Prop"
},
{
... | [
"(eq1 ∧ eq2 ↔ (x, y) ∈ answer)"
] |
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 | open Topology Filter Polynomial Set | answer = (3 : ℝ) | ℝ | {
"benchmark": "putnam_solving",
"index": 186
} | [] | [] | [
"(∑' n : Set.Ici 1, ((2 : ℝ) ^ (round (Real.sqrt n)) + (2 : ℝ) ^ (-round (Real.sqrt n))) / 2 ^ (n : ℝ) = answer)"
] |
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 | open Topology Filter Polynomial Set | answer = True | Prop | {
"benchmark": "putnam_solving",
"index": 187
} | [] | [
{
"t": "ℤ → ℝ",
"v": null,
"name": "a",
"t_type": "Type"
},
{
"t": "∀ n ≥ 1, a n > 0 ∧ a n < a (n + 1)",
"v": null,
"name": "h_pos_inc",
"t_type": "Prop"
},
{
"t": "Tendsto (fun n : ℤ => a (n + 1) / (n + 1)) atTop (𝓝 0)",
"v": null,
"name": "h_limit",
"t_type... | [
"({n : ℤ | n > 0 ∧ (∀ i ∈ Set.Icc 1 (n - 1), a (n - i) + a (n + i) < 2 * a n)}.Infinite ↔ answer)"
] |
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! | open Nat | answer = fun (k n : ℕ) => (((-(↑k : ℝ) : ℝ) ^ n : ℝ) * (↑(n ! : ℕ) : ℝ) : ℝ) | ℕ → ℕ → ℝ | {
"benchmark": "putnam_solving",
"index": 188
} | [
{
"t": "ℕ",
"v": null,
"name": "k",
"t_type": "Type"
}
] | [
{
"t": "ℕ → Polynomial ℝ",
"v": null,
"name": "P",
"t_type": "Type"
},
{
"t": "k > 0",
"v": null,
"name": "kpos",
"t_type": "Prop"
},
{
"t": "∀ n x, iteratedDeriv n (fun x' : ℝ => 1 / (x' ^ k - 1)) x = ((P n).eval x) / ((x ^ k - 1) ^ (n + 1))",
"v": null,
"name": ... | [
"(∀ n, (P n).eval 1 = answer k n)"
] |
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} | open Nat Set Topology Filter | answer = ({(2 : ℕ)} : Set ℕ) | Set ℕ | {
"benchmark": "putnam_solving",
"index": 189
} | [] | [
{
"t": "ℕ → ℕ → ℝ",
"v": null,
"name": "f",
"t_type": "Type"
},
{
"t": "∀ b : ℕ, f b 1 = 1 ∧ f b 2 = 2 ∧ ∀ n ∈ Ici 3, f b n = n * f b (Nat.digits b n).length",
"v": null,
"name": "hf",
"t_type": "Prop"
}
] | [
"({b ∈ Ici 2 | ∃ L : ℝ, Tendsto (fun m : ℕ => ∑ n in Finset.Icc 1 m, 1/(f b n)) atTop (𝓝 L)} = answer)"
] |
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: $4, 2 + 2, 1 + 1 + 2, 1 + 1 + 1 + 1$ | $n$ | open MvPolynomial | answer = fun (n : ℕ) => n | ℕ → ℕ | {
"benchmark": "putnam_solving",
"index": 190
} | [
{
"t": "ℕ",
"v": null,
"name": "n",
"t_type": "Type"
}
] | [
{
"t": "n > 0",
"v": null,
"name": "hn",
"t_type": "Prop"
}
] | [
"(Set.encard {a : ℕ → ℤ |\n ∃ k > 0, (∑ i : Fin k, a i = n) ∧\n (∀ i : Fin k, a i > 0) ∧\n (∀ i : Fin (k - 1), a i ≤ a (i + 1)) ∧\n a (k - 1) ≤ a 0 + 1 ∧ (∀ i ≥ k, a i = 0)} = answer n)"
] |
Find the minimum value of $|\sin x+\cos x+\tan x+\cot x+\sec x+\csc x|$ for real numbers $x$. | $2\sqrt{2} - 1$ | open Set | answer = (((2 : ℝ) * (√(2 : ℝ) : ℝ) : ℝ) - (1 : ℝ) : ℝ) | ℝ | {
"benchmark": "putnam_solving",
"index": 191
} | [] | [
{
"t": "ℝ → ℝ",
"v": null,
"name": "f",
"t_type": "Type"
},
{
"t": "∀ x : ℝ, 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_type": "Prop"
}
] | [
"(IsLeast (Set.range f) answer)"
] |
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 | open MvPolynomial Set | answer = True | Prop | {
"benchmark": "putnam_solving",
"index": 192
} | [] | [
{
"t": "Set ℕ → ℕ → ℕ",
"v": null,
"name": "r",
"t_type": "Type"
},
{
"t": "∀ S n, r S n = ∑' s1 : S, ∑' s2 : S, if (s1 ≠ s2 ∧ s1 + s2 = n) then 1 else 0",
"v": null,
"name": "hr",
"t_type": "Prop"
}
] | [
"((∃ A B : Set ℕ, A ∪ B = ℕ ∧ A ∩ B = ∅ ∧ (∀ n : ℕ, r A n = r B n)) ↔ answer)"
] |
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 | open MvPolynomial Set | answer = False | Prop | {
"benchmark": "putnam_solving",
"index": 193
} | [] | [] | [
"(answer = (∃ a b c d : Polynomial ℝ, (∀ x y : ℝ, 1 + x * y + x ^ 2 * y ^ 2 = a.eval x * c.eval y + b.eval x * d.eval y)))"
] |
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... | True | open Nat Topology Filter | answer = True | Prop | {
"benchmark": "putnam_solving",
"index": 194
} | [] | [
{
"t": "(ℕ → Fin 2) → ℕ → ℝ",
"v": null,
"name": "S",
"t_type": "Type"
},
{
"t": "∀ attempts, ∀ N ≥ 1, S attempts N = (∑ i : Fin N, (attempts i).1) / N",
"v": null,
"name": "hS",
"t_type": "Prop"
}
] | [
"(answer ↔ (∀ attempts a b,\n (1 ≤ a ∧ a < b ∧ S attempts a < 0.8 ∧ S attempts b > 0.8) →\n (∃ c : ℕ, a < c ∧ c < b ∧ S attempts c = 0.8)))"
] |
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... | fun n z ↦ z + n | open Nat Topology Filter | answer = fun (n : ℕ) (z : ℂ) => (z + (↑n : ℂ) : ℂ) | ℕ → ℂ → ℂ | {
"benchmark": "putnam_solving",
"index": 195
} | [
{
"t": "ℕ",
"v": null,
"name": "n",
"t_type": "Type"
}
] | [
{
"t": "n ≥ 2",
"v": null,
"name": "nge2",
"t_type": "Prop"
},
{
"t": "ℕ → ℂ → ℂ",
"v": null,
"name": "R",
"t_type": "Type"
},
{
"t": "ℕ → ℂ → ℂ",
"v": null,
"name": "Rk",
"t_type": "Type"
},
{
"t": "R 0 = id ∧ ∀ k : ℕ, R (k + 1) = Rk (k + 1) ∘ R k",
... | [
"(R n = answer n)"
] |
Evaluate $\lim_{x \to 1^-} \prod_{n=0}^\infty \left(\frac{1+x^{n+1}}{1+x^n}\right)^{x^n}$. | 2 / e | open Nat Topology Filter | answer = ((2 : ℝ) / (Real.exp (1 : ℝ) : ℝ) : ℝ) | ℝ | {
"benchmark": "putnam_solving",
"index": 196
} | [] | [
{
"t": "ℝ → ℝ",
"v": null,
"name": "xprod",
"t_type": "Type"
},
{
"t": "∀ x ∈ Set.Ioo 0 1,\n Tendsto (fun N ↦ ∏ n in Finset.range N, ((1 + x ^ (n + 1)) / (1 + x ^ n)) ^ (x ^ n))\n atTop (𝓝 (xprod x))",
"v": null,
"name": "hxprod",
"t_type": "Prop"
}
] | [
"(Tendsto xprod (𝓝[<] 1) (𝓝 answer))"
] |
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... | fun n ↦ if n = 1 then 0 else 2 ^ (n - 2) | open Nat Set | answer = fun (n : ℕ) => (if n = (1 : ℕ) then (0 : ℕ) else ((2 : ℕ) ^ (n - (2 : ℕ) : ℕ) : ℕ) : ℕ) | ℕ → ℕ | {
"benchmark": "putnam_solving",
"index": 197
} | [
{
"t": "ℕ",
"v": null,
"name": "n",
"t_type": "Type"
}
] | [
{
"t": "n > 0",
"v": null,
"name": "npos",
"t_type": "Prop"
},
{
"t": "Set (ℤ × ℤ)",
"v": null,
"name": "S",
"t_type": "Type"
},
{
"t": "ℤ × ℤ → ℤ × ℤ → Prop",
"v": null,
"name": "unit",
"t_type": "Type"
},
{
"t": "(ℕ → ℤ × ℤ) → Prop",
"v": null,
... | [
"({p : ℕ → ℤ × ℤ | rooktour p ∧ p 1 = (1, 1) ∧ p (3 * n) = ((n : ℤ), 1)}.encard = answer n)"
] |
Evaluate $\int_0^1 \frac{\ln(x+1)}{x^2+1}\,dx$. | $\frac{\pi \ln 2}{8}$ | open Nat Set | answer = (((Real.pi : ℝ) * (Real.log (2 : ℝ) : ℝ) : ℝ) / (8 : ℝ) : ℝ) | ℝ | {
"benchmark": "putnam_solving",
"index": 198
} | [] | [] | [
"(∫ x in (0:ℝ)..1, (Real.log (x+1))/(x^2 + 1) = answer)"
] |
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) | open Nat Set | answer =
(((MvPolynomial.X (1 : Fin (2 : ℕ)) : MvPolynomial (Fin (2 : ℕ)) ℝ) -
((2 : MvPolynomial (Fin (2 : ℕ)) ℝ) * (MvPolynomial.X (0 : Fin (2 : ℕ)) : MvPolynomial (Fin (2 : ℕ)) ℝ) :
MvPolynomial (Fin (2 : ℕ)) ℝ) :
MvPolynomial (Fin (2 : ℕ)) ℝ) *
(((MvPolynomial.X (1 : Fin (2 : ℕ... | MvPolynomial (Fin 2) ℝ | {
"benchmark": "putnam_solving",
"index": 199
} | [] | [] | [
"(answer ≠ 0)",
"(∀ a : ℝ, MvPolynomial.eval (fun n : Fin 2 => if (n = 0) then (Int.floor a : ℝ) else (Int.floor (2 * a))) answer = 0)"
] |
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) : ℕ × (ℕ → ℤ) | (n = 1 ∧ k 0 = 1) ∨ (n = 3 ∧ (k '' {0, 1, 2} = {2, 3, 6})) ∨ (n = 4 ∧ (∀ i : Fin 4, k i = 4))} | open Nat Set | answer =
({((n, k) : ℕ × (ℕ → ℤ)) : ℕ × (ℕ → ℤ) |
n = (1 : ℕ) ∧ (k (0 : ℕ) : ℤ) = (1 : ℤ) ∨
n = (3 : ℕ) ∧
(k '' (Insert.insert (0 : ℕ) (Insert.insert (1 : ℕ) ({(2 : ℕ)} : Set ℕ) : Set ℕ) : Set ℕ) : Set ℤ) =
(Insert.insert (2 : ℤ) (Insert.insert (3 : ℤ) ({(6 : ℤ)} : Set ℤ) : Set... | Set (ℕ × (ℕ → ℤ)) | {
"benchmark": "putnam_solving",
"index": 200
} | [] | [] | [
"({((n : ℕ), (k : ℕ → ℤ)) | (n > 0) ∧ (∀ i ∈ Finset.range n, k i > 0) ∧ (∑ i in Finset.range n, k i = 5 * n - 4) ∧ (∑ i : Finset.range n, (1 : ℝ) / (k i) = 1)} = answer)"
] |
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))"
] |
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