informal_problem string | informal_answer string | header string | formal_answer string | formal_answer_type string | metainfo dict | independent_variables list | hypotheses list | conclusions list |
|---|---|---|---|---|---|---|---|---|
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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