informal_problem string | informal_answer string | header string | formal_answer string | formal_answer_type string | metainfo dict | independent_variables list | hypotheses list | conclusions list |
|---|---|---|---|---|---|---|---|---|
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))"
] |
Let $A$ be the set of all integers $n$ such that $1 \leq n \leq 2021$ and $\gcd(n,2021)=1$. For every nonnegative integer $j$, let $S(j)=\sum_{n \in A}n^j$. Determine all values of $j$ such that $S(j)$ is a multiple of $2021$. | the set of all j such that neither 42 nor 46 divides j | open Filter Topology | answer = ({j : ℕ | ¬(42 : ℕ) ∣ j ∧ ¬(46 : ℕ) ∣ j} : Set ℕ) | Set ℕ | {
"benchmark": "putnam_solving",
"index": 301
} | [
{
"t": "ℕ",
"v": null,
"name": "j",
"t_type": "Type"
}
] | [
{
"t": "Finset ℕ",
"v": null,
"name": "A",
"t_type": "Type"
},
{
"t": "ℕ → ℕ",
"v": null,
"name": "S",
"t_type": "Type"
},
{
"t": "A = {n | 1 ≤ n ∧ n ≤ 2021 ∧ Nat.gcd n 2021 = 1}",
"v": null,
"name": "hA",
"t_type": "Prop"
},
{
"t": "∀ j' : ℕ, S j' = ∑... | [
"((2021 ∣ S j) ↔ j ∈ answer)"
] |
Let $P(x)$ be a polynomial whose coefficients are all either $0$ or $1$. Suppose that $P(x)$ can be written as a product of two nonconstant polynomials with integer coefficients. Does it follow that $P(2)$ is a composite integer? | True | open Filter Topology | answer = True | Prop | {
"benchmark": "putnam_solving",
"index": 302
} | [] | [] | [
"((∀ P, (∀ n : ℕ, P.coeff n = 0 ∨ P.coeff n = 1) →\n (∃ Q R : Polynomial ℤ, Q.degree > 0 ∧ R.degree > 0 ∧ P = Q * R) →\n (P.eval 2 ≠ 0 ∧ P.eval 2 ≠ 1 ∧ ¬Prime (P.eval 2)))\n ↔ answer)"
] |
Determine the maximum value of the sum $S = \sum_{n=1}^\infty \frac{n}{2^n}(a_1a_2 \dots a_n)^{1/n}$ over all sequences $a_1,a_2,a_3,\dots$ of nonnegative real numbers satisfying $\sum_{k=1}^\infty a_k=1$. | 2 / 3 | open Filter Topology | answer = (2 / 3 : ℝ) | ℝ | {
"benchmark": "putnam_solving",
"index": 303
} | [] | [] | [
"(IsGreatest\n {S | ∃ a : ℕ+ → ℝ, (∑' k, a k = 1) ∧ (∀ k, 0 ≤ a k) ∧\n S = ∑' n : ℕ+, n / 2 ^ (n : ℕ) * (∏ k in Finset.Icc 1 n, a k) ^ (1 / n : ℝ)}\n answer)"
] |
Let $h(x,y)$ be a real-valued function that is twice continuously differentiable throughout $\mathbb{R}^2$, and define $\rho(x,y)=yh_x-xh_y$. Prove or disprove: For any positive constants $d$ and $r$ with $d>r$, there is a circle $\mathcal{S}$ of radius $r$ whose center is a distance $d$ away from the origin such that ... | True | open Filter Topology Metric | answer = True | Prop | {
"benchmark": "putnam_solving",
"index": 304
} | [] | [
{
"t": "((EuclideanSpace ℝ (Fin 2)) → ℝ) → (EuclideanSpace ℝ (Fin 2)) → ℝ",
"v": null,
"name": "rho",
"t_type": "Type"
},
{
"t": "∀ (h : (EuclideanSpace ℝ (Fin 2)) → ℝ) (p : EuclideanSpace ℝ (Fin 2)),\n rho h p = (p 1) *\n deriv (fun x' : ℝ => h ((WithLp.equiv 2 (EuclideanSpace ℝ (Fi... | [
"((∀ h : (EuclideanSpace ℝ (Fin 2)) → ℝ, ContDiff ℝ 2 h →\n (∀ d > 0, ∀ r > 0, d > r → (∃ c : EuclideanSpace ℝ (Fin 2), dist c 0 = d ∧ (∫ p in (ball c r), rho h p) = 0))) ↔ answer)"
] |
Determine all ordered pairs of real numbers $(a,b)$ such that the line $y = ax+b$ intersects the curve $y = \ln(1+x^2)$ in exactly one point. | {(a, b) | (a = 0 ∧ b = 0) ∨ 1 ≤ |a| ∨ (0 < |a| ∧ |a| < 1 ∧ letI rm := (1 - √(1 - a ^ 2)) / a; letI rp := (1 + √(1 - a ^ 2)) / a; (b < Real.log (1 + rm ^ 2) - a * rm ∨ b > Real.log (1 + rp ^ 2) - a * rp))} | open Polynomial | answer =
({((a, b) : ℝ × ℝ) : ℝ × ℝ |
a = (0 : ℝ) ∧ b = (0 : ℝ) ∨
(1 : ℝ) ≤ (|a| : ℝ) ∨
(0 : ℝ) < (|a| : ℝ) ∧
(|a| : ℝ) < (1 : ℝ) ∧
(b <
((Real.log
((1 : ℝ) +
((((1 : ℝ) - (√((1 : ℝ) - (a ^ (2 :... | Set (ℝ × ℝ) | {
"benchmark": "putnam_solving",
"index": 305
} | [] | [] | [
"({a : (ℝ × ℝ) | ∃! x : ℝ, a.1 * x + a.2 = Real.log (1 + x^2)} = answer)"
] |
Let $n$ be an integer with $n \geq 2$. Over all real polynomials $p(x)$ of degree $n$, what is the largest possible number of negative coefficients of $p(x)^2$? | $2n - 2$ | open Polynomial | answer = fun (n : ℕ) => (((2 : ℕ) * n : ℕ) - (2 : ℕ) : ℕ) | ℕ → ℕ | {
"benchmark": "putnam_solving",
"index": 306
} | [
{
"t": "ℕ",
"v": null,
"name": "n",
"t_type": "Type"
}
] | [
{
"t": "n ≥ 2",
"v": null,
"name": "hn",
"t_type": "Prop"
},
{
"t": "Set ℝ[X]",
"v": null,
"name": "S",
"t_type": "Type"
},
{
"t": "S = {P | natDegree P = n}",
"v": null,
"name": "hS",
"t_type": "Prop"
},
{
"t": "ℝ[X] → ℕ",
"v": null,
"name": "... | [
"(sSup {negs (P^2) | P ∈ S} = answer n)"
] |
Alice and Bob play a game on a board consisting of one row of 2022 consecutive squares. They take turns placing tiles that cover two adjacent squares, with Alice going first. By rule, a tile must not cover a square that is already covered by another tile. The game ends when no tile can be placed according to this rule.... | 290 | answer = (290 : ℕ) | ℕ | {
"benchmark": "putnam_solving",
"index": 307
} | [] | [
{
"t": "Set (Fin 2022) → Set (Fin 2022) → Prop",
"v": null,
"name": "IsValidMove",
"t_type": "Type"
},
{
"t": "∀ x y, IsValidMove x y ↔\n (x = y ∧ ∀ i < 2021, i ∉ x → i + 1 ∈ x) ∨\n ∃ i < 2021, i ∉ x ∧ i + 1 ∉ x ∧ y = x ∪ {i, i + 1}",
"v": null,
"name": "IsValidMove_def",
... | [
"(IsGreatest\n {n | ∃ s, (∀ x, IsValidMove x (s x)) ∧ ∀ g,\n IsValidGame g → ConformsToStrategy g s → ∃ gh x, g = gh ++ [x] ∧ n ≤ xᶜ.ncard}\n answer)"
] | |
Let $n$ be a positive integer. Determine, in terms of $n$, the largest integer $m$ with the following property: There exist real numbers $x_1,\dots,x_{2n}$ with $-1<x_1<x_2<\cdots<x_{2n}<1$ such that the sum of the lengths of the $n$ intervals $[x_1^{2k-1},x_2^{2k-1}],[x_3^{2k-1},x_4^{2k-1}],\dots,[x_{2n-1}^{2k-1},x_{2... | (fun n : ℕ => n) | open Set | answer = fun (n : ℕ) => n | ℕ → ℕ | {
"benchmark": "putnam_solving",
"index": 308
} | [
{
"t": "ℕ",
"v": null,
"name": "n",
"t_type": "Type"
}
] | [
{
"t": "0 < n",
"v": null,
"name": "hn",
"t_type": "Prop"
}
] | [
"(IsGreatest\n {m : ℕ | ∃ x : ℕ → ℝ,\n StrictMono x ∧ -1 < x 1 ∧ x (2 * n) < 1 ∧\n ∀ k ∈ Icc 1 m, ∑ i in Icc 1 n, ((x (2 * i) : ℝ) ^ (2 * k - 1) - (x (2 * i - 1)) ^ (2 * k - 1)) = 1}\n (answer n))"
] |
Let $\times$ represent the cross product in $\mathbb{R}^3$. For what positive integers $n$ does there exist a set $S \subset \mathbb{R}^3$ with exactly $n$ elements such that $S=\{v \times w:v,w \in S\}$? | {1, 7} | open Polynomial | answer = (Insert.insert (1 : ℕ) ({(7 : ℕ)} : Set ℕ) : Set ℕ) | Set ℕ | {
"benchmark": "putnam_solving",
"index": 309
} | [] | [
{
"t": "ℕ",
"v": null,
"name": "n",
"t_type": "Type"
},
{
"t": "Finset (Fin 3 → ℝ) → Prop",
"v": null,
"name": "P",
"t_type": "Type"
},
{
"t": "∀ S : Finset (Fin 3 → ℝ), P S ↔ (S = {u : Fin 3 → ℝ | ∃ v w : S, u = crossProduct v w})",
"v": null,
"name": "P_def",
... | [
"((0 < n ∧ ∃ S : Finset (Fin 3 → ℝ), S.card = n ∧ P S) ↔ n ∈ answer)"
] |
Assign to each positive real number a color, either red or blue. Let $D$ be the set of all distances $d>0$ such that there are two points of the same color at distance $d$ apart. Recolor the positive reals so that the numbers in $D$ are red and the numbers not in $D$ are blue. If we iterate this recoloring process, wil... | True | open Polynomial | answer = True | Prop | {
"benchmark": "putnam_solving",
"index": 310
} | [] | [
{
"t": "(ℝ → Prop) → (ℝ → Prop)",
"v": null,
"name": "recolor",
"t_type": "Type"
},
{
"t": "∀ isred d, recolor isred d ↔ ∃ p q : ℝ, p > 0 ∧ q > 0 ∧ p < q ∧ isred p = isred q ∧ q - p = d",
"v": null,
"name": "hrecolor",
"t_type": "Prop"
}
] | [
"((∀ isred, (∃ k, ∀ p > 0, (recolor^[k] isred) p)) ↔ answer)"
] |
Find all integers $n$ with $n \geq 4$ for which there exists a sequence of distinct real numbers $x_1,\dots,x_n$ such that each of the sets $\{x_1,x_2,x_3\},\{x_2,x_3,x_4\},\dots,\{x_{n-2},x_{n-1},x_n\},\{x_{n-1},x_n,x_1\}$, and $\{x_n,x_1,x_2\}$ forms a $3$-term arithmetic progression when arranged in increasing order... | {n : ℕ | 3 ∣ n ∧ n ≥ 9} | open Polynomial | answer = ({n : ℕ | (3 : ℕ) ∣ n ∧ n ≥ (9 : ℕ)} : Set ℕ) | Set ℕ | {
"benchmark": "putnam_solving",
"index": 311
} | [
{
"t": "ℕ",
"v": null,
"name": "n",
"t_type": "Type"
}
] | [
{
"t": "n ≥ 4",
"v": null,
"name": "hn",
"t_type": "Prop"
},
{
"t": "ℝ → ℝ → ℝ → Prop",
"v": null,
"name": "ap3",
"t_type": "Type"
},
{
"t": "(ℕ → ℝ) → Prop",
"v": null,
"name": "xprog",
"t_type": "Type"
},
{
"t": "∀ x0 x1 x2, ap3 x0 x1 x2 ↔ ∀ o0 o1 o2... | [
"((∃ x, xprog x) ↔ n ∈ answer)"
] |
For $0 \leq p \leq 1/2$, let $X_1,X_2,\dots$ be independent random variables such that
$X_i=\begin{cases}
1 & \text{with probability $p$,} \\
-1 & \text{with probability $p$,} \\
0 & \text{with probability $1-2p$,}
\end{cases}$
for all $i \geq 1$. Given a positive integer $n$ and integers $b,a_1,\dots,a_n$, let $P(b,a_... | Set.Icc 0 (1 / 4) | open Polynomial | answer = (Set.Icc (0 : ℝ) (1 / 4 : ℝ) : Set ℝ) | Set ℝ | {
"benchmark": "putnam_solving",
"index": 312
} | [] | [
{
"t": "ℝ",
"v": null,
"name": "p",
"t_type": "Type"
},
{
"t": "(n : ℕ) → Set (Fin n → ℤ)",
"v": null,
"name": "Xset",
"t_type": "Type"
},
{
"t": "(n : ℕ) → (Fin n → ℤ) → ℝ",
"v": null,
"name": "Xprob",
"t_type": "Type"
},
{
"t": "(n : ℕ) → ℤ → (Fin n ... | [
"((0 ≤ p ∧ p ≤ 1 / 2 ∧ (∀ n > 0, ∀ (b : ℤ) (a : Fin n → ℤ), P n 0 a ≥ P n b a)) ↔ p ∈ answer)"
] |
Find all continuous functions $f:\mathbb{R}^+ \to \mathbb{R}^+$ such that $f(xf(y))+f(yf(x))=1+f(x+y)$ for all $x,y>0$. | the set of functions of the form f(x) = 1 / (1 + c * x) where c ≥ 0 | open Polynomial | answer =
({f : ↑(Set.Ioi (0 : ℝ) : Set ℝ) → ↑(Set.Ioi (0 : ℝ) : Set ℝ) |
∃ c ≥ (0 : ℝ),
∀ (x : ↑(Set.Ioi (0 : ℝ) : Set ℝ)),
(↑(f x : ↑(Set.Ioi (0 : ℝ) : Set ℝ)) : ℝ) = ((1 : ℝ) / ((1 : ℝ) + (c * (↑x : ℝ) : ℝ) : ℝ) : ℝ)} :
Set (↑(Set.Ioi (0 : ℝ) : Set ℝ) → ↑(Set.Ioi (0 : ℝ) : Set ℝ))) | Set (Set.Ioi (0 : ℝ) → Set.Ioi (0 : ℝ)) | {
"benchmark": "putnam_solving",
"index": 313
} | [
{
"t": "Set.Ioi (0 : ℝ) → Set.Ioi (0 : ℝ)",
"v": null,
"name": "f",
"t_type": "Type"
}
] | [
{
"t": "Continuous f",
"v": null,
"name": "hf",
"t_type": "Prop"
}
] | [
"((∃ fr : ℝ → ℝ, (∀ x : Set.Ioi (0 : ℝ), fr x = f x) ∧ (∀ x y : Set.Ioi (0 : ℝ), fr (x * fr y) + fr (y * fr x) = 1 + fr (x + y))) ↔ f ∈ answer)"
] |
For a positive integer $n$, let $f_n(x) = \cos(x) \cos(2x) \cos(3x) \cdots \cos(nx)$. Find the smallest $n$ such that $|f_n''(0)| > 2023$. | 18 | open Nat | answer = (18 : ℕ) | ℕ | {
"benchmark": "putnam_solving",
"index": 314
} | [] | [
{
"t": "ℕ → ℝ → ℝ",
"v": null,
"name": "f",
"t_type": "Type"
},
{
"t": "∀ n > 0, f n = fun x : ℝ => ∏ i in Finset.Icc 1 n, Real.cos (i * x)",
"v": null,
"name": "hf",
"t_type": "Prop"
}
] | [
"(IsLeast {n | 0 < n ∧ |iteratedDeriv 2 (f n) 0| > 2023} answer)"
] |
Let $n$ be an even positive integer. Let $p$ be a monic, real polynomial of degree $2n$; that is to say, $p(x) = x^{2n} + a_{2n-1} x^{2n-1} + \cdots + a_1 x + a_0$ for some real coefficients $a_0, \dots, a_{2n-1}$. Suppose that $p(1/k) = k^2$ for all integers $k$ such that $1 \leq |k| \leq n$. Find all other real numbe... | {1/n!, -1/n!} | open Nat | answer = fun (n : ℕ) =>
(Insert.insert ((1 : ℝ) / (↑(n ! : ℕ) : ℝ) : ℝ) ({((-1 : ℝ) / (↑(n ! : ℕ) : ℝ) : ℝ)} : Set ℝ) : Set ℝ) | ℕ → Set ℝ | {
"benchmark": "putnam_solving",
"index": 315
} | [
{
"t": "ℕ",
"v": null,
"name": "n",
"t_type": "Type"
}
] | [
{
"t": "n > 0 ∧ Even n",
"v": null,
"name": "hn",
"t_type": "Prop"
},
{
"t": "Polynomial ℝ",
"v": null,
"name": "p",
"t_type": "Type"
},
{
"t": "Polynomial.Monic p ∧ p.degree = 2*n",
"v": null,
"name": "hp",
"t_type": "Prop"
},
{
"t": "Set ℝ",
"v":... | [
"({x : ℝ | x ≠ 0 ∧ p.eval (1/x) = x^2} \\ S = answer n)"
] |
Determine the smallest positive real number $r$ such that there exist differentiable functions $f\colon \mathbb{R} \to \mathbb{R}$ and $g\colon \mathbb{R} \to \mathbb{R}$ satisfying
\begin{enumerate}
\item[(a)] $f(0) > 0$,
\item[(b)] $g(0) = 0$,
\item[(c)] $|f'(x)| \leq |g(x)|$ for all $x$,
\item[(d)] $|g'(x)| \leq... | π/2 | open Nat | answer = ((Real.pi : ℝ) / (2 : ℝ) : ℝ) | ℝ | {
"benchmark": "putnam_solving",
"index": 316
} | [] | [] | [
"(answer = sInf {r > 0 | ∃ f g : ℝ → ℝ,\n Differentiable ℝ f ∧ Differentiable ℝ g ∧\n (∀ x : ℝ, |deriv f x| ≤ |g x| ∧ |deriv g x| ≤ |f x|) ∧\n f 0 > 0 ∧ g 0 = 0 ∧ f r = 0})"
] |
For a nonnegative integer $k$, let $f(k)$ be the number of ones in the base 3 representation of $k$. Find all complex numbers $z$ such that \[ \sum_{k=0}^{3^{1010}-1} (-2)^{f(k)} (z+k)^{2023} = 0. \] | {-(3^1010 - 1)/2, -(3^1010 - 1)/2 + Real.sqrt (9^1010 - 1) * Complex.I/4, -(3^1010 - 1)/2 - Real.sqrt (9^1010 - 1) * Complex.I/4} | open Nat | answer =
(Insert.insert ((-(((3 : ℂ) ^ (1010 : ℕ) : ℂ) - (1 : ℂ) : ℂ) : ℂ) / (2 : ℂ) : ℂ)
(Insert.insert
(((-(((3 : ℂ) ^ (1010 : ℕ) : ℂ) - (1 : ℂ) : ℂ) : ℂ) / (2 : ℂ) : ℂ) +
(((↑(√(((9 : ℝ) ^ (1010 : ℕ) : ℝ) - (1 : ℝ) : ℝ) : ℝ) : ℂ) * (Complex.I : ℂ) : ℂ) / (4 : ℂ) : ℂ) :
ℂ)
... | Set ℂ | {
"benchmark": "putnam_solving",
"index": 317
} | [] | [] | [
"({z : ℂ | ∑ k in Finset.Icc 0 (3^1010 - 1),\n (-2)^(((digits 3 k).filter (fun x => x = 1)).length) * (z + k)^2023 = 0} = answer)"
] |
Alice and Bob play a game in which they take turns choosing integers from $1$ to $n$. Before any integers are chosen, Bob selects a goal of 'odd' or 'even'. On the first turn, Alice chooses one of the $n$ integers. On the second turn, Bob chooses one of the remaining integers. They continue alternately choosing one of ... | the set of all positive integers | open Finset | answer = ({n : ℕ | (0 : ℕ) < n} : Set ℕ) | Set ℕ | {
"benchmark": "putnam_solving",
"index": 318
} | [] | [
{
"t": "List ℕ → Prop",
"v": null,
"name": "IsValidGame",
"t_type": "Type"
},
{
"t": "∀ g, IsValidGame g ↔ g.Nodup ∧ (∀ i ∈ g, i ∈ Icc 1 g.length)",
"v": null,
"name": "IsValidGame_def",
"t_type": "Prop"
},
{
"t": "List ℕ → ZMod 2",
"v": null,
"name": "parityOf",
... | [
"({n : ℕ | 0 < n ∧ ∃ s, IsWinningFor n s} = answer)"
] |
Consider an $m$-by-$n$ grid of unit squares, indexed by $(i,j)$ with $1 \leq i \leq m$ and $1 \leq j \leq n$. There are $(m-1)(n-1)$ coins, which are initially placed in the squares $(i,j)$ with $1 \leq i \leq m-1$ and $1 \leq j \leq n-1$. If a coin occupies the square $(i,j)$ with $i \leq m-1$ and $j \leq n-1$ and the... | $\binom{m + n - 2}{m - 1}$ | open Nat | answer = fun (m n : ℕ) => (((m + n : ℕ) - (2 : ℕ) : ℕ).choose (m - (1 : ℕ) : ℕ) : ℕ) | ℕ → ℕ → ℕ | {
"benchmark": "putnam_solving",
"index": 319
} | [
{
"t": "ℕ",
"v": null,
"name": "m",
"t_type": "Type"
},
{
"t": "ℕ",
"v": null,
"name": "n",
"t_type": "Type"
}
] | [
{
"t": "ℕ → ℕ → Bool",
"v": null,
"name": "initcoins",
"t_type": "Type"
},
{
"t": "initcoins = (fun i j : ℕ => (i ≤ m - 2 ∧ j ≤ n - 2 : Bool))",
"v": null,
"name": "hinitcoins",
"t_type": "Prop"
},
{
"t": "(ℕ → ℕ → Bool) → (ℕ → ℕ → Bool) → Prop",
"v": null,
"name"... | [
"({config : ℕ → ℕ → Bool | ∃ seq : List (ℕ → ℕ → Bool), IsLegalSeq seq ∧ config = seq.getLast!}.encard = answer m n)"
] |
For each positive integer $n$, let $k(n)$ be the number of ones in the binary representation of $2023 * n$. What is the minimum value of $k(n)$? | 3 | open Nat | answer = (3 : ℕ) | ℕ | {
"benchmark": "putnam_solving",
"index": 320
} | [] | [] | [
"(sInf {(digits 2 (2023 * n)).sum | n > 0} = answer)"
] |
A sequence $y_1, y_2, \ldots, y_k$ of real numbers is called zigzag if $k = 1$, or if $y_2 - y_1, y_3 - y_2, \ldots, y_k - y_{k-1}$ are nonzero and alternate in sign. Let $X_1, X_2,\ldots, X_n$ be chosen independently from the uniform distribution on $[0,1]$. Let $a(X_1, X_2, \ldots, X_n)$ be the largest value of $k$ f... | (2n + 2) / 3 | open ProbabilityTheory Set | answer = fun (n : ℕ) => ((((2 : ℝ) * (↑n : ℝ) : ℝ) + (2 : ℝ) : ℝ) / (3 : ℝ) : ℝ) | ℕ → ℝ | {
"benchmark": "putnam_solving",
"index": 321
} | [
{
"t": "ℕ",
"v": null,
"name": "n",
"t_type": "Type"
}
] | [
{
"t": "{k : ℕ} → (Fin k → ℝ) → Prop",
"v": null,
"name": "IsZigZag",
"t_type": "Type"
},
{
"t": "∀ (k : ℕ) [NeZero k] (y : Fin k → ℝ),\n IsZigZag y ↔ k = 1 ∨ ((∀ i, i + 1 < k → y (i + 1) ≠ y i)) ∧\n (∀ i, i + 2 < k → (y (i + 2) < y (i + 1) ↔ y i < y (i + 1)))",
"v": null,
... | [
"(𝔼[(↑) ∘ a] = answer n)"
] |
For a nonnegative integer $n$ and a strictly increasing sequence of real numbers $t_0,t_1,\dots,t_n$, let $f(t)$ be the corresponding real-valued function defined for $t \geq t_0$ by the following properties:
\begin{enumerate}
\item[(a)] $f(t)$ is continuous for $t \geq t_0$, and is twice differentiable for all $t>t_0$... | 29 | open Nat Topology Filter | answer = (29 : ℝ) | ℝ | {
"benchmark": "putnam_solving",
"index": 322
} | [] | [
{
"t": "ℕ → (ℕ → ℝ) → Set ℝ",
"v": null,
"name": "tne",
"t_type": "Type"
},
{
"t": "∀ n ts, tne n ts = {t | t > ts 0 ∧ ∀ i : Fin n, t ≠ ts (i.1 + 1)}",
"v": null,
"name": "htne",
"t_type": "Prop"
}
] | [
"(IsLeast\n {(T : ℝ) | 0 ≤ T ∧ ∃ (n : ℕ) (ts : ℕ → ℝ) (f : ℝ → ℝ),\n ∀ k : Fin n, ts (k.1 + 1) ≥ ts k.1 + 1 ∧\n ContinuousOn f (Set.Ici (ts 0)) ∧\n ContDiffOn ℝ 1 f (tne n ts) ∧\n DifferentiableOn ℝ (derivWithin f (tne n ts)) (tne n ts) ∧\n f (ts 0) = 0.5 ∧\n (∀ k : Fin (n + 1),\n ... |
Determine which positive integers $n$ have the following property: For all integers $m$ that are relatively prime to $n$, there exists a permutation $\pi:\{1,2,\dots,n\} \to \{1,2,\dots,n\}$ such that $\pi(\pi(k)) \equiv mk \pmod{n}$ for all $k \in \{1,2,\dots,n\}$. | {n : ℕ | n = 1 ∨ n ≡ 2 [MOD 4]} | open Nat Topology Filter | answer = ({n : ℕ | n = (1 : ℕ) ∨ n ≡ (2 : ℕ) [MOD (4 : ℕ)]} : Set ℕ) | Set ℕ | {
"benchmark": "putnam_solving",
"index": 323
} | [] | [] | [
"answer = {n : ℕ | 0 < n ∧ (∀ m : ℤ, IsRelPrime m n → ∃ p : Equiv.Perm (Fin n),\n ∀ k : Fin n, (p (p k)).1 + 1 ≡ m * (k.1 + 1) [ZMOD n])}"
] |
Let $n$ be a positive integer. For $i$ and $j$ in $\{1,2,\dots,n\}$, let $s(i,j)$ be the number of pairs $(a,b)$ of nonnegative integers satisfying $ai+bj=n$. Let $S$ be the $n$-by-$n$ matrix whose $(i,j)$ entry is $s(i,j)$. For example, when $n=5$, we have $S = \begin{bmatrix}
6 & 3 & 2 & 2 & 2 \\
3 & 0 & 1 & 0 & 1 \\... | $(-1)^{\lceil n/2 \rceil-1}2\lceil\frac{n}{2}\rceil$ | open Nat Topology Filter | answer = fun (n : ℕ) =>
((((-1 : ℤ) ^ ((⌈((↑n : ℚ) / (2 : ℚ) : ℚ)⌉₊ : ℕ) + (1 : ℕ) : ℕ) : ℤ) * (2 : ℤ) : ℤ) *
(↑(⌈((↑n : ℚ) / (2 : ℚ) : ℚ)⌉₊ : ℕ) : ℤ) :
ℤ) | ℕ → ℤ | {
"benchmark": "putnam_solving",
"index": 324
} | [
{
"t": "ℕ",
"v": null,
"name": "n",
"t_type": "Type"
}
] | [
{
"t": "Matrix (Fin n) (Fin n) ℤ",
"v": null,
"name": "S",
"t_type": "Type"
},
{
"t": "n > 0",
"v": null,
"name": "npos",
"t_type": "Prop"
},
{
"t": "∀ i j : Fin n, S i j = ∑' a : ℕ, ∑' b : ℕ, if a * (i.1 + 1) + b * (j.1 + 1) = (n : ℤ) then 1 else 0",
"v": null,
"... | [
"(S.det = answer n)"
] |
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