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Find every real-valued function $f$ whose domain is an interval $I$ (finite or infinite) having 0 as a left-hand endpoint, such that for every positive member $x$ of $I$ the average of $f$ over the closed interval $[0, x]$ is equal to the geometric mean of the numbers $f(0)$ and $f(x)$.
the set of functions $f(x) = \frac{a}{(1 - c x)^2}$ where $a \geq 0$
open MeasureTheory Set
answer = ({f : ℝ β†’ ℝ | βˆƒ (a : ℝ) (c : ℝ), a β‰₯ (0 : ℝ) ∧ f = fun (x : ℝ) => (a / (((1 : ℝ) - (c * x : ℝ) : ℝ) ^ (2 : β„•) : ℝ) : ℝ)} : Set (ℝ β†’ ℝ))
Set (ℝ β†’ ℝ)
{ "benchmark": "putnam_solving", "index": 1 }
[]
[ { "t": "Set ℝ β†’ (ℝ β†’ ℝ) β†’ Prop", "v": null, "name": "P", "t_type": "Type" }, { "t": "βˆ€ s f, P s f ↔ 0 ≀ f ∧ βˆ€ x ∈ s, ⨍ t in Ico 0 x, f t = √(f 0 * f x)", "v": null, "name": "P_def", "t_type": "Prop" } ]
[ "((βˆ€ f,\n (P (Ioi 0) f β†’ βˆƒ g ∈ answer, EqOn f g (Ici 0)) ∧\n (βˆ€ e > 0, P (Ioo 0 e) f β†’ βˆƒ g ∈ answer, EqOn f g (Ico 0 e))) ∧\n βˆ€ f ∈ answer, P (Ioi 0) f ∨ (βˆƒ e > 0, P (Ioo 0 e) f))" ]
Evaluate in closed form \[ \sum_{k=1}^n {n \choose k} k^2. \]
$n(n+1)2^{n-2}$
answer = fun (n : β„•) => ((n * (n + (1 : β„•) : β„•) : β„•) * ((2 : β„•) ^ (n - (2 : β„•) : β„•) : β„•) : β„•)
β„• β†’ β„•
{ "benchmark": "putnam_solving", "index": 2 }
[ { "t": "β„•", "v": null, "name": "n", "t_type": "Type" } ]
[ { "t": "n β‰₯ 2", "v": null, "name": "hn", "t_type": "Prop" } ]
[ "(answer n = βˆ‘ k in Finset.Icc 1 n, Nat.choose n k * k^2)" ]
Find an integral formula (i.e., a function $z$ such that $y(x) = \int_{1}^{x} z(t) dt$) for the solution of the differential equation $$\delta (\delta - 1) (\delta - 2) \cdots (\delta - n + 1) y = f(x)$$ with the initial conditions $y(1) = y'(1) = \cdots = y^{(n-1)}(1) = 0$, where $n \in \mathbb{N}$, $f$ is continuous ...
$(x - t)^{n-1} \cdot f(t) / ((n-1)! \cdot t^n)$
open Nat Set Topology Filter
answer = fun (f : ℝ β†’ ℝ) (n : β„•) (x t : ℝ) => ((((x - t : ℝ) ^ (n - (1 : β„•) : β„•) : ℝ) * (f t : ℝ) : ℝ) / ((↑((n - (1 : β„•) : β„•)! : β„•) : ℝ) * (t ^ n : ℝ) : ℝ) : ℝ)
(ℝ β†’ ℝ) β†’ β„• β†’ ℝ β†’ ℝ β†’ ℝ
{ "benchmark": "putnam_solving", "index": 3 }
[ { "t": "β„•", "v": null, "name": "n", "t_type": "Type" }, { "t": "ℝ β†’ ℝ", "v": null, "name": "f", "t_type": "Type" } ]
[ { "t": "β„• β†’ (ℝ β†’ ℝ) β†’ (ℝ β†’ ℝ)", "v": null, "name": "P", "t_type": "Type" }, { "t": "P 0 = id ∧ βˆ€ i y, P (i + 1) y = P i (fun x ↦ x * deriv y x - i * y x)", "v": null, "name": "hP", "t_type": "Prop" }, { "t": "0 < n", "v": null, "name": "hn", "t_type": "Prop" ...
[ "((βˆ€ i < n, deriv^[i] y 1 = 0) ∧ (Ici 1).EqOn (P n y) f ↔\n βˆ€ x β‰₯ 1, y x = ∫ t in (1 : ℝ)..x, answer f n x t)" ]
For what integer $a$ does $x^2-x+a$ divide $x^{13}+x+90$?
2
open Topology Filter Polynomial
answer = (2 : β„€)
β„€
{ "benchmark": "putnam_solving", "index": 4 }
[]
[ { "t": "β„€", "v": null, "name": "a", "t_type": "Type" }, { "t": "Polynomial.X^2 - Polynomial.X + (Polynomial.C a) ∣ (Polynomial.X ^ 13 + Polynomial.X + (Polynomial.C 90))", "v": null, "name": "h_div", "t_type": "Prop" } ]
[ "(answer = a)" ]
Let $S$ be the set of all numbers of the form $2^m3^n$, where $m$ and $n$ are integers, and let $P$ be the set of all positive real numbers. Is $S$ dense in $P$?
True
open Topology Filter Polynomial
answer = True
Prop
{ "benchmark": "putnam_solving", "index": 5 }
[]
[ { "t": "Set ℝ", "v": null, "name": "S", "t_type": "Type" }, { "t": "S = {2 ^ m * 3 ^ n | (m : β„€) (n : β„€)}", "v": null, "name": "hS", "t_type": "Prop" } ]
[ "(answer ↔ (closure S βŠ‡ Set.Ioi (0 : ℝ)))" ]
Find every twice-differentiable real-valued function $f$ with domain the set of all real numbers and satisfying the functional equation $(f(x))^2-(f(y))^2=f(x+y)f(x-y)$ for all real numbers $x$ and $y$.
the set of functions of the form $A \sinh(k u)$, $A u$, or $A \sin(k u)$
open Topology Filter Polynomial
answer = ((({x : ℝ β†’ ℝ | βˆƒ (A : ℝ) (k : ℝ), (fun (u : ℝ) => (A * (Real.sinh (k * u : ℝ) : ℝ) : ℝ)) = x} : Set (ℝ β†’ ℝ)) βˆͺ ({x : ℝ β†’ ℝ | βˆƒ (A : ℝ), (fun (u : ℝ) => (A * u : ℝ)) = x} : Set (ℝ β†’ ℝ)) : Set (ℝ β†’ ℝ)) βˆͺ ({x : ℝ β†’ ℝ | βˆƒ (A : ℝ) (k : ℝ), (fun (u : ℝ) => (A * (Real.sin (k * u : ℝ) : ℝ) :...
Set (ℝ β†’ ℝ)
{ "benchmark": "putnam_solving", "index": 6 }
[]
[ { "t": "ℝ β†’ ℝ", "v": null, "name": "f", "t_type": "Type" } ]
[ "(f ∈ answer ↔\n (ContDiff ℝ 1 f ∧ Differentiable ℝ (deriv f) ∧\n βˆ€ x y : ℝ, (f x) ^ 2 - (f y) ^ 2 = f (x + y) * f (x - y)))" ]
Let $\alpha$ be a real number. Find all continuous real-valued functions $f : [0, 1] \to (0, \infty)$ such that \begin{align*} \int_0^1 f(x) dx &= 1, \\ \int_0^1 x f(x) dx &= \alpha, \\ \int_0^1 x^2 f(x) dx &= \alpha^2. \\ \end{align*}
the empty set
open Set
answer = fun (x : ℝ) => (βˆ… : Set (ℝ β†’ ℝ))
ℝ β†’ Set (ℝ β†’ ℝ)
{ "benchmark": "putnam_solving", "index": 7 }
[]
[ { "t": "ℝ", "v": null, "name": "Ξ±", "t_type": "Type" } ]
[ "(answer Ξ± = {f : ℝ β†’ ℝ | (βˆ€ x ∈ Icc 0 1, f x > 0) ∧ ContinuousOn f (Icc 0 1) ∧ ∫ x in (0)..1, f x = 1 ∧ ∫ x in (0)..1, x * f x = Ξ± ∧ ∫ x in (0)..1, x^2 * f x = Ξ±^2})" ]
Let $\triangle ABC$ satisfy $\angle CAB < \angle BCA < \frac{\pi}{2} < \angle ABC$. If the bisector of the external angle at $A$ meets line $BC$ at $P$, the bisector of the external angle at $B$ meets line $CA$ at $Q$, and $AP = BQ = AB$, find $\angle CAB$.
Ο€ / 15
open EuclideanGeometry Real
answer = ((Ο€ : ℝ) / (15 : ℝ) : ℝ)
ℝ
{ "benchmark": "putnam_solving", "index": 8 }
[]
[ { "t": "EuclideanSpace ℝ (Fin 2)", "v": null, "name": "A", "t_type": "Type" }, { "t": "EuclideanSpace ℝ (Fin 2)", "v": null, "name": "B", "t_type": "Type" }, { "t": "EuclideanSpace ℝ (Fin 2)", "v": null, "name": "C", "t_type": "Type" }, { "t": "Euclide...
[ "(∠ C A B = answer)" ]
How many orderings of the integers from $1$ to $n$ satisfy the condition that, for every integer $i$ except the first, there exists some earlier integer in the ordering which differs from $i$ by $1$?
$2^{n-1}$
open EuclideanGeometry Topology Filter Complex
answer = fun (n : β„•) => ((2 : β„•) ^ (n - (1 : β„•) : β„•) : β„•)
β„• β†’ β„•
{ "benchmark": "putnam_solving", "index": 9 }
[ { "t": "β„•", "v": null, "name": "n", "t_type": "Type" } ]
[ { "t": "n > 0", "v": null, "name": "npos", "t_type": "Prop" } ]
[ "({p ∈ permsOfFinset (Finset.Icc 1 n) | βˆ€ m ∈ Finset.Icc 2 n, βˆƒ k ∈ Finset.Ico 1 m, p m = p k + 1 ∨ p m = p k - 1}.card = answer n)" ]
Find $$\lim_{n \to \infty} \int_{0}^{1} \int_{0}^{1} \cdots \int_{0}^{1} \cos^2\left(\frac{\pi}{2n}(x_1 + x_2 + \cdots + x_n)\right) dx_1 dx_2 \cdots dx_n.$$
1 / 2
open EuclideanGeometry Topology Filter Complex
answer = (1 / 2 : ℝ)
ℝ
{ "benchmark": "putnam_solving", "index": 10 }
[]
[]
[ "(Tendsto (fun n : β„• ↦ ∫ x in {x : Fin (n+1) β†’ ℝ | βˆ€ k : Fin (n+1), x k ∈ Set.Icc 0 1}, (Real.cos (Real.pi/(2 * (n+1)) * βˆ‘ k : Fin (n+1), x k))^2) atTop (𝓝 answer))" ]
Consider polynomial forms $ax^2-bx+c$ with integer coefficients which have two distinct zeros in the open interval $0<x<1$. Exhibit with a proof the least positive integer value of $a$ for which such a polynomial exists.
5
open Polynomial
answer = (5 : β„•)
β„•
{ "benchmark": "putnam_solving", "index": 11 }
[]
[ { "t": "Set β„€", "v": null, "name": "S", "t_type": "Type" }, { "t": "S = {a | βˆƒ P : Polynomial β„€, P.degree = 2 ∧ (βˆƒ z1 z2 : Set.Ioo (0 : ℝ) 1, z1 β‰  z2 ∧ aeval (z1 : ℝ) P = 0 ∧ aeval (z2 : ℝ) P = 0) ∧P.coeff 2 = a ∧ a > 0}", "v": null, "name": "hS", "t_type": "Prop" } ]
[ "(IsLeast S answer)" ]
Given real numbers $\{a_i\}$ and $\{b_i\}$, ($i=1,2,3,4$), such that $a_1b_2-a_2b_1 \neq 0$. Consider the set of all solutions $(x_1,x_2,x_3,x_4)$ of the simultaneous equations $a_1x_1+a_2x_2+a_3x_3+a_4x_4=0$ and $b_1x_1+b_2x_2+b_3x_3+b_4x_4=0$, for which no $x_i$ ($i=1,2,3,4$) is zero. Each such solution generates a $...
8
open Nat Topology Filter
answer = (8 : β„•)
β„•
{ "benchmark": "putnam_solving", "index": 12 }
[]
[ { "t": "(Fin 4 β†’ ℝ) β†’ (Fin 4 β†’ ℝ) β†’ Prop", "v": null, "name": "abneq0", "t_type": "Type" }, { "t": "abneq0 = (fun a b : Fin 4 β†’ ℝ => a 0 * b 1 - a 1 * b 0 β‰  0)", "v": null, "name": "habneq0", "t_type": "Prop" }, { "t": "(Fin 4 β†’ ℝ) β†’ (Fin 4 β†’ ℝ) β†’ β„•", "v": null, "...
[ "((βˆƒ a b : Fin 4 β†’ ℝ, abneq0 a b ∧ numtuples a b = answer) ∧ (βˆ€ a b : Fin 4 β†’ ℝ, abneq0 a b β†’ numtuples a b ≀ answer))" ]
Let $V$ be the set of all quadratic polynomials with real coefficients such that $|P(x)| \le 1$ for all $x \in [0, 1]$. Find the supremum of $|P'(0)|$ across all $P \in V$.
8
open Finset Polynomial
answer = (8 : ℝ)
ℝ
{ "benchmark": "putnam_solving", "index": 13 }
[]
[ { "t": "Set ℝ[X]", "v": null, "name": "V", "t_type": "Type" }, { "t": "V = {P : ℝ[X] | P.degree = 2 ∧ βˆ€ x ∈ Set.Icc 0 1, |P.eval x| ≀ 1}", "v": null, "name": "V_def", "t_type": "Prop" } ]
[ "(sSup {|(derivative P).eval 0| | P ∈ V} = answer)" ]
Find all polynomials of the form $\sum_{0}^{n} a_{i} x^{n-i}$ with $n \ge 1$ and $a_i = \pm 1$ for all $0 \le i \le n$ whose roots are all real.
{X - 1, -(X - 1), X + 1, -(X + 1), X^2 + X - 1, -(X^2 + X - 1), X^2 - X - 1, -(X^2 - X - 1), X^3 + X^2 - X - 1, -(X^3 + X^2 - X - 1), X^3 - X^2 - X + 1, -(X^3 - X^2 - X + 1)}
open Finset Polynomial
answer = (Insert.insert ((Polynomial.X : β„‚[X]) - (1 : β„‚[X]) : β„‚[X]) (Insert.insert (-((Polynomial.X : β„‚[X]) - (1 : β„‚[X]) : β„‚[X]) : β„‚[X]) (Insert.insert ((Polynomial.X : β„‚[X]) + (1 : β„‚[X]) : β„‚[X]) (Insert.insert (-((Polynomial.X : β„‚[X]) + (1 : β„‚[X]) : β„‚[X]) : β„‚[X]) (Inse...
Set β„‚[X]
{ "benchmark": "putnam_solving", "index": 14 }
[]
[]
[ "({P : β„‚[X] | P.natDegree β‰₯ 1 ∧ (βˆ€ k ∈ Set.Icc 0 P.natDegree, P.coeff k = 1 ∨ P.coeff k = -1) ∧\n βˆ€ z : β„‚, P.eval z = 0 β†’ βˆƒ r : ℝ, r = z} = answer)" ]
Let $p$ be a prime number. Find the number of distinct $2 \times 2$ matrices $$\begin{pmatrix} a & b \\ c & d \end{pmatrix}$$ such that $a, b, c, d \in \{0, 1, ..., p - 1\}$, $a + d \equiv 1 \pmod p$, and $ad - bc \equiv 0 \pmod p$.
$p^2 + p$
open Finset Polynomial Topology Filter Metric
answer = fun (p : β„•) => ((p ^ (2 : β„•) : β„•) + p : β„•)
β„• β†’ β„•
{ "benchmark": "putnam_solving", "index": 15 }
[ { "t": "β„•", "v": null, "name": "p", "t_type": "Type" } ]
[ { "t": "Nat.Prime p", "v": null, "name": "hp", "t_type": "Prop" } ]
[ "({M : Matrix (Fin 2) (Fin 2) (ZMod p) | M 0 0 + M 1 1 = 1 ∧ M 0 0 * M 1 1 - M 0 1 * M 1 0 = 0}.ncard = answer p)" ]
What are the possible ranges (across all real inputs $x$ and $y$) of a polynomial $f(x, y)$ with real coefficients?
{{x} | x : ℝ} βˆͺ {Set.Ici x | x : ℝ} βˆͺ {Set.Iic x | x : ℝ} βˆͺ {Set.Iio x | x : ℝ} βˆͺ {Set.Ioi x | x : ℝ} βˆͺ {Set.univ}
open Matrix Filter Topology Set Nat
answer = (((((({x : Set ℝ | βˆƒ (x_1 : ℝ), ({x_1} : Set ℝ) = x} : Set (Set ℝ)) βˆͺ ({x : Set ℝ | βˆƒ (x_1 : ℝ), (Set.Ici x_1 : Set ℝ) = x} : Set (Set ℝ)) : Set (Set ℝ)) βˆͺ ({x : Set ℝ | βˆƒ (x_1 : ℝ), (Set.Iic x_1 : Set ℝ) = x} : Set (Set ℝ)) : Set (S...
Set (Set ℝ)
{ "benchmark": "putnam_solving", "index": 16 }
[]
[]
[ "(answer = {{z : ℝ | βˆƒ x : Fin 2 β†’ ℝ, MvPolynomial.eval x f = z} | f : MvPolynomial (Fin 2) ℝ})" ]
Show that a finite group can not be the union of two of its proper subgroups. Does the statement remain true if 'two' is replaced by 'three'?
False
open Matrix Filter Topology Set Nat
answer = False
Prop
{ "benchmark": "putnam_solving", "index": 17 }
[]
[ { "t": "β„• β†’ Prop", "v": null, "name": "P", "t_type": "Type" }, { "t": "βˆ€ n, P n ↔ βˆ€ (G : Type) [Group G] [Finite G],\n βˆ€ H : Fin n β†’ Subgroup G, (βˆ€ i, H i < ⊀) β†’ ⋃ i, (H i : Set G) < ⊀", "v": null, "name": "P_def", "t_type": "Prop" } ]
[ "(answer ↔ (P 3))" ]
Find the length of the longest possible sequence of equal nonzero digits (in base 10) in which a perfect square can terminate. Also, find the smallest square that attains this length.
(3, 1444)
open Metric Set EuclideanGeometry
answer = (((3 : β„•), (1444 : β„•)) : β„• Γ— β„•)
β„• Γ— β„•
{ "benchmark": "putnam_solving", "index": 18 }
[]
[ { "t": "β„• β†’ β„•", "v": null, "name": "L", "t_type": "Type" }, { "t": "βˆ€ n : β„•, L n ≀ (Nat.digits 10 n).length ∧\n(βˆ€ k : β„•, k < L n β†’ (Nat.digits 10 n)[k]! = (Nat.digits 10 n)[0]!) ∧\n(L n β‰  (Nat.digits 10 n).length β†’ (Nat.digits 10 n)[L n]! β‰  (Nat.digits 10 n)[0]!)", "v": null, "name":...
[ "((βˆƒ n : β„•, (Nat.digits 10 (n^2))[0]! β‰  0 ∧ L (n^2) = answer.1) ∧\n(βˆ€ n : β„•, (Nat.digits 10 (n^2))[0]! β‰  0 β†’ L (n^2) ≀ answer.1) ∧\n(βˆƒ m : β„•, m^2 = answer.2) ∧\nL (answer.2) = answer.1 ∧\n(Nat.digits 10 answer.2)[0]! β‰  0 ∧\nβˆ€ n : β„•, (Nat.digits 10 (n^2))[0]! β‰  0 ∧ L (n^2) = answer.1 β†’ n^2 β‰₯ answer.2)" ]
Evaluate the infinite product $\lim_{n \to \infty} \frac{1}{n^4} \prod_{i = 1}^{2n} (n^2 + i^2)^{1/n}$.
$e^{2 \ln 5 - 4 + 2 \arctan 2}$
open Metric Set EuclideanGeometry Filter Topology
answer = (Real.exp ((((2 : ℝ) * (Real.log (5 : ℝ) : ℝ) : ℝ) - (4 : ℝ) : ℝ) + ((2 : ℝ) * (Real.arctan (2 : ℝ) : ℝ) : ℝ) : ℝ) : ℝ)
ℝ
{ "benchmark": "putnam_solving", "index": 19 }
[]
[]
[ "(Tendsto (fun n => 1/(n^4) * ∏ i in Finset.Icc (1 : β„€) (2*n), ((n^2 + i^2) : ℝ)^((1 : ℝ)/n)) atTop (𝓝 answer))" ]
Determine all polynomials $P(x)$ such that $P(x^2 + 1) = (P(x))^2 + 1$ and $P(0) = 0$.
{Polynomial.X}
open Set
answer = ({(Polynomial.X : Polynomial ℝ)} : Set (Polynomial ℝ))
Set (Polynomial ℝ)
{ "benchmark": "putnam_solving", "index": 20 }
[]
[ { "t": "Polynomial ℝ", "v": null, "name": "P", "t_type": "Type" } ]
[ "(βˆ€ P : Polynomial ℝ, P ∈ answer ↔ (P.eval 0 = 0 ∧ (βˆ€ x : ℝ, P.eval (x^2 + 1) = (P.eval x)^2 + 1)))" ]
After each play of a certain game of solitaire, the player receives either $a$ or $b$ points, where $a$ and $b$ are positive integers with $a > b$; scores accumulate from play to play. If there are $35$ unattainable scores, one of which is $58$, find $a$ and $b$.
(11, 8)
open Set MvPolynomial
answer = (((11 : β„€), (8 : β„€)) : β„€ Γ— β„€)
β„€ Γ— β„€
{ "benchmark": "putnam_solving", "index": 21 }
[]
[ { "t": "β„€", "v": null, "name": "a", "t_type": "Type" }, { "t": "β„€", "v": null, "name": "b", "t_type": "Type" }, { "t": "a > 0 ∧ b > 0 ∧ a > b", "v": null, "name": "hab", "t_type": "Prop" }, { "t": "β„€ β†’ β„€ β†’ Prop", "v": null, "name": "pab", "...
[ "(pab a b ↔ a = answer.1 ∧ b = answer.2)" ]
Find all functions $F : \mathbb{R} \setminus \{0, 1\} \to \mathbb{R}$ that satisfy $F(x) + F\left(\frac{x - 1}{x}\right) = 1 + x$ for all $x \in \mathbb{R} \setminus \{0, 1\}$.
$\left\{x \mapsto \frac{x^3 - x^2 - 1}{2x(x - 1)}\right\}$
open Set MvPolynomial
answer = ({fun (x : ℝ) => ((((x ^ (3 : β„•) : ℝ) - (x ^ (2 : β„•) : ℝ) : ℝ) - (1 : ℝ) : ℝ) / (((2 : ℝ) * x : ℝ) * (x - (1 : ℝ) : ℝ) : ℝ) : ℝ)} : Set (ℝ β†’ ℝ))
Set (ℝ β†’ ℝ)
{ "benchmark": "putnam_solving", "index": 22 }
[]
[ { "t": "Set ℝ", "v": null, "name": "S", "t_type": "Type" }, { "t": "S = univ \\ {0, 1}", "v": null, "name": "hS", "t_type": "Prop" }, { "t": "(ℝ β†’ ℝ) β†’ Prop", "v": null, "name": "P", "t_type": "Type" }, { "t": "P = fun (F : ℝ β†’ ℝ) => βˆ€ x ∈ S, F x + F (...
[ "(βˆ€ F ∈ answer, P F ∧ βˆ€ f : ℝ β†’ ℝ, P f β†’ βˆƒ F ∈ answer, (βˆ€ x ∈ S, f x = F x))" ]
We call a function $f$ from $[0,1]$ to the reals to be supercontinuous on $[0,1]$ if the Cesaro-limit exists for the sequence $f(x_1), f(x_2), f(x_3), \dots$ whenever it does for the sequence $x_1, x_2, x_3 \dots$. Find all supercontinuous functions on $[0,1]$.
the set of all linear functions on [0,1]
open EuclideanGeometry Filter Topology Set
answer = ({f : ℝ β†’ ℝ | βˆƒ (A : ℝ) (B : ℝ), βˆ€ (x : ℝ), x ∈ (Set.Icc (0 : ℝ) (1 : ℝ) : Set ℝ) β†’ (f x : ℝ) = ((A * x : ℝ) + B : ℝ)} : Set (ℝ β†’ ℝ))
Set (ℝ β†’ ℝ)
{ "benchmark": "putnam_solving", "index": 23 }
[]
[ { "t": "(β„• β†’ ℝ) β†’ Prop", "v": null, "name": "climit_exists", "t_type": "Type" }, { "t": "(ℝ β†’ ℝ) β†’ Prop", "v": null, "name": "supercontinuous", "t_type": "Type" }, { "t": "βˆ€ x, climit_exists x ↔ βˆƒ C : ℝ, Tendsto (fun n => (βˆ‘ i in Finset.range n, (x i))/(n : ℝ)) atTop (𝓝 ...
[ "({f | supercontinuous f} = answer)" ]
Let $x : \mathbb{R} \to \mathbb{R}$ be a twice differentiable function whose second derivative is nonstrictly decreasing. If $x(t) - x(0) = s$, $x'(0) = 0$, and $x'(t) = v$ for some $t > 0$, find the maximum possible value of $t$ in terms of $s$ and $v$.
$2s / v$
open EuclideanGeometry Filter Topology Set MeasureTheory Metric
answer = fun (s v : ℝ) => (((2 : ℝ) * s : ℝ) / v : ℝ)
ℝ β†’ ℝ β†’ ℝ
{ "benchmark": "putnam_solving", "index": 24 }
[ { "t": "ℝ", "v": null, "name": "s", "t_type": "Type" }, { "t": "ℝ", "v": null, "name": "v", "t_type": "Type" } ]
[ { "t": "s > 0", "v": null, "name": "hs", "t_type": "Prop" }, { "t": "v > 0", "v": null, "name": "hv", "t_type": "Prop" }, { "t": "ℝ β†’ (ℝ β†’ ℝ) β†’ Prop", "v": null, "name": "valid", "t_type": "Type" }, { "t": "βˆ€ t x, valid t x ↔\n DifferentiableOn ℝ ...
[ "(IsGreatest {t | βˆƒ x : ℝ β†’ ℝ, valid t x} (answer s v))" ]
Consider an infinite series whose $n$th term is given by $\pm \frac{1}{n}$, where the actual values of the $\pm$ signs repeat in blocks of $8$ (so the $\frac{1}{9}$ term has the same sign as the $\frac{1}{1}$ term, and so on). Call such a sequence balanced if each block contains four $+$ and four $-$ signs. Prove that ...
True
open Nat Set MeasureTheory Topology Filter
answer = True
Prop
{ "benchmark": "putnam_solving", "index": 25 }
[]
[ { "t": "List ℝ", "v": null, "name": "L", "t_type": "Type" }, { "t": "L.length = 8 ∧ βˆ€ i : Fin L.length, L[i] = 1 ∨ L[i] = -1", "v": null, "name": "hL", "t_type": "Prop" }, { "t": "β„•", "v": null, "name": "pluses", "t_type": "Type" }, { "t": "pluses = {i...
[ "(answer ↔ ((βˆƒ l : ℝ, Tendsto S atTop (𝓝 l)) β†’ pluses = 4))" ]
How many zeros does the function $f(x) = 2^x - 1 - x^2$ have on the real line?
3
open Nat Set MeasureTheory Topology Filter
answer = (3 : β„•)
β„•
{ "benchmark": "putnam_solving", "index": 26 }
[]
[ { "t": "ℝ β†’ ℝ", "v": null, "name": "f", "t_type": "Type" }, { "t": "f = fun x => 2^x - 1 - x^2", "v": null, "name": "hf", "t_type": "Prop" } ]
[ "(answer = {x : ℝ | f x = 0}.ncard)" ]
Suppose $f$ is a function on $[0,1]$ with continuous derivative satisfying $0 < f'(x) \leq 1$ and $f 0 = 0$. Prove that $\left[\int_0^1 f(x) dx\right]]^2 \geq \int_0^1 (f(x))^3 dx$, and find an example where equality holds.
the identity function $f(x) = x$
open Nat Set MeasureTheory Topology Filter
answer = fun (x : ℝ) => x
ℝ β†’ ℝ
{ "benchmark": "putnam_solving", "index": 27 }
[]
[]
[ "(ContDiff ℝ 1 answer ∧ (βˆ€ x : ℝ, 0 < deriv answer x ∧ deriv answer x ≀ 1) ∧ answer 0 = 0 ∧ (∫ x in Icc 0 1, answer x)^2 = ∫ x in Icc 0 1, (answer x)^3)" ]
Call a set of positive integers 'conspiratorial' if no three of them are pairwise relatively prime. What is the largest number of elements in any conspiratorial subset of the integers 1 through 16?
11
open Set
answer = (11 : β„•)
β„•
{ "benchmark": "putnam_solving", "index": 28 }
[]
[ { "t": "Set β„€ β†’ Prop", "v": null, "name": "conspiratorial", "t_type": "Type" }, { "t": "βˆ€ S, conspiratorial S ↔ βˆ€ a ∈ S, βˆ€ b ∈ S, βˆ€ c ∈ S, (a > 0 ∧ b > 0 ∧ c > 0) ∧ ((a β‰  b ∧ b β‰  c ∧ a β‰  c) β†’ (Int.gcd a b > 1 ∨ Int.gcd b c > 1 ∨ Int.gcd a c > 1))", "v": null, "name": "hconspiratorial...
[ "(IsGreatest {k | βˆƒ S, S βŠ† Icc 1 16 ∧ conspiratorial S ∧ S.encard = k} answer)" ]
A well-known theorem asserts that a prime $p > 2$ can be written as the sum of two perfect squres if and only if $p \equiv 1 \bmod 4$. Find which primes $p > 2$ can be written in each of the following forms, using (not necessarily positive) integers $x$ and $y$: (a) $x^2 + 16y^2$, (b) $4x^2 + 4xy + 5y^2$.
({p : β„• | p.Prime ∧ p ≑ 1 [MOD 8]}, {p : β„• | p.Prime ∧ p ≑ 5 [MOD 8]})
open Set
answer = ((({p : β„• | Nat.Prime p ∧ p ≑ (1 : β„•) [MOD (8 : β„•)]} : Set β„•), ({p : β„• | Nat.Prime p ∧ p ≑ (5 : β„•) [MOD (8 : β„•)]} : Set β„•)) : Set β„• Γ— Set β„•)
(Set β„•) Γ— (Set β„•)
{ "benchmark": "putnam_solving", "index": 29 }
[ { "t": "β„•", "v": null, "name": "p", "t_type": "Type" } ]
[ { "t": "βˆ€ p : β„•, p.Prime ∧ p > 2 β†’ ((βˆƒ m n : β„€, p = m^2 + n^2) ↔ p ≑ 1 [MOD 4])", "v": null, "name": "h_assumption", "t_type": "Prop" } ]
[ "(βˆ€ p : β„•,\n ((p.Prime ∧ p > 2 ∧ (βˆƒ x y : β„€, p = x^2 + 16*y^2)) ↔ p ∈ answer.1) ∧\n ((p.Prime ∧ p > 2 ∧ (βˆƒ x y : β„€, p = 4*x^2 + 4*x*y + 5*y^2)) ↔ p ∈ answer.2))" ]
Evaluate in closed form: $\frac{1}{2^{n-1}} \sum_{k < n/2} (n-2k)*{n \choose k}$.
(fun n ↦ (1 : β„š) / ((2 : β„š) ^ ((n :β„•) - 1)) * (n * (n - 1).choose ⌊n / 2βŒ‹β‚Š))
open Set Nat
answer = fun (n : β„•) => (((1 : β„š) / ((2 : β„š) ^ (n - (1 : β„•) : β„•) : β„š) : β„š) * ((↑n : β„š) * (↑((n - (1 : β„•) : β„•).choose (⌊(n / (2 : β„•) : β„•)βŒ‹β‚Š : β„•) : β„•) : β„š) : β„š) : β„š)
β„• β†’ β„š
{ "benchmark": "putnam_solving", "index": 30 }
[ { "t": "β„•", "v": null, "name": "n", "t_type": "Type" } ]
[ { "t": "0 < n", "v": null, "name": "hn", "t_type": "Prop" } ]
[ "((1 : β„š) / (2 ^ (n - 1)) * βˆ‘ k in Finset.Icc 0 ⌊n / 2βŒ‹β‚Š, (n - 2 * k) * (n.choose k) = answer n)" ]
Given $n$, let $k(n)$ be the minimal degree of any monic integral polynomial $f$ such that the value of $f(x)$ is divisible by $n$ for every integer $x$. Find the value of $k(1000000)$.
25
open Set Nat Polynomial
answer = (25 : β„•)
β„•
{ "benchmark": "putnam_solving", "index": 31 }
[]
[ { "t": "Polynomial β„€ β†’ Prop", "v": null, "name": "hdivnallx", "t_type": "Type" }, { "t": "hdivnallx = fun f => Monic f ∧ (βˆ€ x : β„€, (10^6 : β„€) ∣ f.eval x)", "v": null, "name": "hdivnallx_def", "t_type": "Prop" } ]
[ "(sInf {d : β„• | βˆƒ f : Polynomial β„€, hdivnallx f ∧ d = f.natDegree} = answer)" ]
For a set with $1000$ elements, how many subsets are there whose candinality is respectively $\equiv 0 \bmod 3, \equiv 1 \bmod 3, \equiv 2 \bmod 3$?
((2^1000 - 1)/3, (2^1000 - 1)/3, 1 + (2^1000 - 1)/3)
open Set Nat Polynomial Filter Topology
answer = ((((((2 : β„•) ^ (1000 : β„•) : β„•) - (1 : β„•) : β„•) / (3 : β„•) : β„•), ((((((2 : β„•) ^ (1000 : β„•) : β„•) - (1 : β„•) : β„•) / (3 : β„•) : β„•), ((1 : β„•) + ((((2 : β„•) ^ (1000 : β„•) : β„•) - (1 : β„•) : β„•) / (3 : β„•) : β„•) : β„•)) : β„• Γ— β„•)) : β„• Γ— β„• Γ— β„•)
β„• Γ— β„• Γ— β„•
{ "benchmark": "putnam_solving", "index": 32 }
[]
[ { "t": "β„€", "v": null, "name": "n", "t_type": "Type" }, { "t": "n = 1000", "v": null, "name": "hn", "t_type": "Prop" }, { "t": "β„•", "v": null, "name": "count0", "t_type": "Type" }, { "t": "β„•", "v": null, "name": "count1", "t_type": "Type" ...
[ "((count0, count1, count2) = answer)" ]
If an integer $n$ can be written as the sum of two triangular numbers (that is, $n = \frac{a^2 + a}{2} + \frac{b^2 + b}{2}$ for some integers $a$ and $b$), express $4n + 1$ as the sum of the squares of two integers $x$ and $y$, giving $x$ and $y$ in terms of $a$ and $b$. Also, show that if $4n + 1 = x^2 + y^2$ for some...
(fun (a, b) => a + b + 1, fun (a, b) => a - b)
open Polynomial
answer = ((fun (x : β„€ Γ— β„€) => match x with | ((a, b) : β„€ Γ— β„€) => ((a + b : β„€) + (1 : β„€) : β„€), fun (x : β„€ Γ— β„€) => match x with | ((a, b) : β„€ Γ— β„€) => (a - b : β„€)) : (β„€ Γ— β„€ β†’ β„€) Γ— (β„€ Γ— β„€ β†’ β„€))
((β„€ Γ— β„€) β†’ β„€) Γ— ((β„€ Γ— β„€) β†’ β„€)
{ "benchmark": "putnam_solving", "index": 33 }
[]
[ { "t": "(β„€ Γ— β„€ Γ— β„€) β†’ Prop", "v": null, "name": "nab", "t_type": "Type" }, { "t": "(β„€ Γ— β„€ Γ— β„€) β†’ Prop", "v": null, "name": "nxy", "t_type": "Type" }, { "t": "nab = fun (n, a, b) => n = (a^2 + (a : β„š))/2 + (b^2 + (b : β„š))/2", "v": null, "name": "hnab", "t_type"...
[ "((βˆ€ n a b : β„€, nab (n, a, b) β†’ nxy (n, answer.1 (a, b), answer.2 (a, b))) ∧ βˆ€ n : β„€, (βˆƒ x y : β„€, nxy (n, x, y)) β†’ βˆƒ a b : β„€, nab (n, a, b))" ]
For which ordered pairs $(b, c)$ of real numbers do both roots of $z^2 + bz + c$ lie strictly inside the unit disk (i.e., $\{|z| < 1\}$) in the complex plane?
$c < 1 \land c - b > -1 \land c + b > -1$
open Polynomial
βˆ€ (b : ℝ) (c : ℝ), answer ((b, c) : ℝ Γ— ℝ) = (c < (1 : ℝ) ∧ (c - b : ℝ) > (-1 : ℝ) ∧ (c + b : ℝ) > (-1 : ℝ))
(ℝ Γ— ℝ) β†’ Prop
{ "benchmark": "putnam_solving", "index": 34 }
[]
[]
[ "(βˆ€ (b : ℝ) (c : ℝ), (βˆ€ z : β„‚, (Polynomial.X^2 + (Polynomial.C (b : β„‚)) * Polynomial.X + (Polynomial.C (c : β„‚)) : Polynomial β„‚).eval z = 0 β†’ β€–zβ€– < 1) ↔ answer (b, c))" ]
If $a$, $b$, and $c$ are real numbers satisfying $0 < a < b < c$, at what points in the set $$\{(x, y, z) \in \mathbb{R}^3 : x^b + y^b + z^b = 1, x \ge 0, y \ge 0, z \ge 0\}$$ does $f(x, y, z) = x^a + y^b + z^c$ attain its maximum and minimum?
(fun (a, b, c) => ((a/b)^(1/(b - a)), (1 - ((a/b)^(1/(b - a)))^b)^(1/b), 0), fun (a, b, c) => (0, (1 - ((b/c)^(1/(c - b)))^b)^(1/b), (b/c)^(1/(c - b))))
open Polynomial
answer = ((fun (x : ℝ Γ— ℝ Γ— ℝ) => match x with | ((a, ((b, c) : ℝ Γ— ℝ)) : ℝ Γ— ℝ Γ— ℝ) => ((((a / b : ℝ) ^ ((1 : ℝ) / (b - a : ℝ) : ℝ) : ℝ), (((((1 : ℝ) - (((a / b : ℝ) ^ ((1 : ℝ) / (b - a : ℝ) : ℝ) : ℝ) ^ b : ℝ) : ℝ) ^ ((1 : ℝ) / b : ℝ) : ℝ), (0 : ℝ)) : ℝ Γ—...
((ℝ Γ— ℝ Γ— ℝ) β†’ (ℝ Γ— ℝ Γ— ℝ)) Γ— ((ℝ Γ— ℝ Γ— ℝ) β†’ (ℝ Γ— ℝ Γ— ℝ))
{ "benchmark": "putnam_solving", "index": 35 }
[ { "t": "ℝ", "v": null, "name": "a", "t_type": "Type" }, { "t": "ℝ", "v": null, "name": "b", "t_type": "Type" }, { "t": "ℝ", "v": null, "name": "c", "t_type": "Type" } ]
[ { "t": "0 < a ∧ a < b ∧ b < c", "v": null, "name": "hi", "t_type": "Prop" }, { "t": "(ℝ Γ— ℝ Γ— ℝ) β†’ Prop", "v": null, "name": "P", "t_type": "Type" }, { "t": "(ℝ Γ— ℝ Γ— ℝ) β†’ ℝ", "v": null, "name": "f", "t_type": "Type" }, { "t": "P = fun (x, y, z) => x β‰₯...
[ "((P (answer.1 (a, b, c)) ∧ βˆ€ x y z : ℝ, P (x, y, z) β†’\nf (x, y, z) ≀ f (answer.1 (a, b, c))) ∧\n(P (answer.2 (a, b, c)) ∧ βˆ€ x y z : ℝ, P (x, y, z) β†’\nf (x, y, z) β‰₯ f (answer.2 (a, b, c))))" ]
Let $n = 2m$, where $m$ is an odd integer greater than 1. Let $\theta = e^{2\pi i/n}$. Expression $(1 - \theta)^{-1}$ explicitly as a polynomial in $\theta$ \[ a_k \theta^k + a_{k-1}\theta^{k-1} + \dots + a_1\theta + a_0\], with integer coefficients $a_i$.
$\sum_{j=0}^{(m-1)/2} \theta^{2j+1}$
open Polynomial Real Complex
answer = fun (m : β„•) => βˆ‘ j ∈ (Finset.range ((m - (1 : β„•) : β„•) / (2 : β„•) : β„•) : Finset β„•), ((Polynomial.X : β„€[X]) ^ (((2 : β„•) * j : β„•) + (1 : β„•) : β„•) : β„€[X])
β„• β†’ Polynomial β„€
{ "benchmark": "putnam_solving", "index": 36 }
[ { "t": "β„•", "v": null, "name": "m", "t_type": "Type" } ]
[ { "t": "Odd m ∧ m > 1", "v": null, "name": "hm", "t_type": "Prop" }, { "t": "β„‚", "v": null, "name": "ΞΈ", "t_type": "Type" }, { "t": "ΞΈ = cexp (2 * Real.pi * I / (2 * m))", "v": null, "name": "hΞΈ", "t_type": "Prop" } ]
[ "(1/(1 - ΞΈ) = Polynomial.aeval ΞΈ (answer m))" ]
Let $H$ be a subgroup of the additive group of ordered pairs of integers under componentwise addition. If $H$ is generated by the elements $(3, 8)$, $(4, -1)$, and $(5, 4)$, then $H$ is also generated by two elements $(1, b)$ and $(0, a)$ for some integer $b$ and positive integer $a$. Find $a$.
7
open Polynomial Real Complex
answer = (7 : β„€)
β„€
{ "benchmark": "putnam_solving", "index": 37 }
[]
[ { "t": "Set (β„€ Γ— β„€)", "v": null, "name": "H", "t_type": "Type" }, { "t": "H = {h : (β„€ Γ— β„€) | βˆƒ u v w : β„€, h = (u*3 + v*4 + w*5, u*8 + v*(-1) + w*4)}", "v": null, "name": "hH", "t_type": "Prop" } ]
[ "((βˆƒ b : β„€, H = {h : (β„€ Γ— β„€) | βˆƒ u v : β„€, h = (u, u*b + v*answer)}) ∧ answer > 0)" ]
Let $s_k (a_1, a_2, \dots, a_n)$ denote the $k$-th elementary symmetric function; that is, the sum of all $k$-fold products of the $a_i$. For example, $s_1 (a_1, \dots, a_n) = \sum_{i=1}^{n} a_i$, and $s_2 (a_1, a_2, a_3) = a_1a_2 + a_2a_3 + a_1a_3$. Find the supremum $M_k$ (which is never attained) of $$\frac{s_k (a_1...
fun k : β„• => (1: ℝ)/(Nat.factorial k)
open Polynomial Real Complex Matrix Filter Topology Multiset
answer = fun (k : β„•) => ((1 : ℝ) / (↑(k.factorial : β„•) : ℝ) : ℝ)
β„• β†’ ℝ
{ "benchmark": "putnam_solving", "index": 38 }
[]
[]
[ "(βˆ€ k : β„•, k > 0 β†’ (βˆ€ a : Multiset ℝ, (βˆ€ i ∈ a, i > 0) ∧ card a β‰₯ k β†’\n(esymm a k)/(esymm a 1)^k ≀ answer k) ∧\nβˆ€ M : ℝ, M < answer k β†’ (βˆƒ a : Multiset ℝ, (βˆ€ i ∈ a, i > 0) ∧ card a β‰₯ k ∧\n(esymm a k)/(esymm a 1)^k > M))" ]
Let $C = \{(x, y) \in \mathbb{R}^2 : x^2 + y^2 = 1\}$ denote the unit circle. Does there exist $B \subseteq C$ for which $B$ is topologically closed and contains exactly one point from each pair of diametrically opposite points in $C$?
False
open Polynomial Real Complex Matrix Filter Topology Multiset
answer = False
Prop
{ "benchmark": "putnam_solving", "index": 39 }
[]
[ { "t": "ℝ Γ— ℝ β†’ Prop", "v": null, "name": "P", "t_type": "Type" }, { "t": "P = fun (x, y) => x^2 + y^2 = 1", "v": null, "name": "hP", "t_type": "Prop" } ]
[ "((βˆƒ B βŠ† setOf P, IsClosed B ∧ βˆ€ x y : ℝ, P (x, y) β†’ Xor' ((x, y) ∈ B) ((-x, -y) ∈ B)) ↔ answer)" ]
Find all integer solutions $(p, r, q, s)$ of the equation $|p^r - q^s| = 1$, where $p$ and $q$ are prime and $r$ and $s$ are greater than $1$.
{(3, 2, 2, 3), (2, 3, 3, 2)}
answer = (Insert.insert (((3 : β„•), (((2 : β„•), (((2 : β„•), (3 : β„•)) : β„• Γ— β„•)) : β„• Γ— β„• Γ— β„•)) : β„• Γ— β„• Γ— β„• Γ— β„•) ({(((2 : β„•), (((3 : β„•), (((3 : β„•), (2 : β„•)) : β„• Γ— β„•)) : β„• Γ— β„• Γ— β„•)) : β„• Γ— β„• Γ— β„• Γ— β„•)} : Set (β„• Γ— β„• Γ— β„• Γ— β„•)) : Set (β„• Γ— β„• Γ— β„• Γ— β„•))
Set (β„• Γ— β„• Γ— β„• Γ— β„•)
{ "benchmark": "putnam_solving", "index": 40 }
[]
[]
[ "({a : β„• Γ— β„• Γ— β„• Γ— β„• | Nat.Prime a.1 ∧ Nat.Prime a.2.2.1 ∧ a.2.1 > 1 ∧ a.2.2.2 > 1 ∧ |(a.1^a.2.1 : β„€) - a.2.2.1^a.2.2.2| = 1} = answer)" ]
Let $r$ be a real root of $P(x) = x^3 + ax^2 + bx - 1$, where $a$ and $b$ are integers and $P$ is irreducible over the rationals. Suppose that $r + 1$ is a root of $x^3 + cx^2 + dx + 1$, where $c$ and $d$ are also integers. Express another root $s$ of $P$ as a function of $r$ that does not depend on the values of $a$, ...
$\left(-\frac{1}{r + 1}, -\frac{r + 1}{r}\right)$
open Polynomial
answer = ((fun (r : ℝ) => ((-1 : ℝ) / (r + (1 : ℝ) : ℝ) : ℝ), fun (r : ℝ) => ((-(r + (1 : ℝ) : ℝ) : ℝ) / r : ℝ)) : (ℝ β†’ ℝ) Γ— (ℝ β†’ ℝ))
(ℝ β†’ ℝ) Γ— (ℝ β†’ ℝ)
{ "benchmark": "putnam_solving", "index": 41 }
[ { "t": "ℝ", "v": null, "name": "r", "t_type": "Type" } ]
[ { "t": "β„€", "v": null, "name": "a", "t_type": "Type" }, { "t": "β„€", "v": null, "name": "b", "t_type": "Type" }, { "t": "β„€", "v": null, "name": "c", "t_type": "Type" }, { "t": "β„€", "v": null, "name": "d", "t_type": "Type" }, { "t": "...
[ "(βˆƒ s : ℝ, Polynomial.aeval s P = 0 ∧ (s = answer.1 r ∨ s = answer.2 r))" ]
Find $$\lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^{n}\left(\left\lfloor \frac{2n}{k} \right\rfloor - 2\left\lfloor \frac{n}{k} \right\rfloor\right).$$ Your answer should be in the form $\ln(a) - b$, where $a$ and $b$ are positive integers.
ln(4) - 1
open Polynomial Filter Topology
answer = (((4 : β„•), (1 : β„•)) : β„• Γ— β„•)
β„• Γ— β„•
{ "benchmark": "putnam_solving", "index": 42 }
[]
[]
[ "(Tendsto (fun n : β„• => ((1 : ℝ)/n)*βˆ‘ k in Finset.Icc (1 : β„€) n, (Int.floor ((2*n)/k) - 2*Int.floor (n/k))) atTop\n (𝓝 (Real.log answer.1 - answer.2)))" ]
Let $G$ be a group generated by two elements $A$ and $B$; i.e., every element of $G$ can be expressed as a finite word $A^{n_1}B^{n_2} \cdots A^{n_{k-1}}B^{n_k}$, where the $n_i$ can assume any integer values and $A^0 = B^0 = 1$. Further assume that $A^4 = B^7 = ABA^{-1}B = 1$, but $A^2 \ne 1$ and $B \ne 1$. Find the n...
(8, {[(0, 0)], [(2, 0)], [(0, 1)], [(0, 2)], [(0, 3)], [(0, 4)], [(0, 5)], [(0, 6)]})
open Polynomial Filter Topology
answer = (((8 : β„•), (Insert.insert ((((0 : β„€), (0 : β„€)) : β„€ Γ— β„€) :: ([] : List (β„€ Γ— β„€)) : List (β„€ Γ— β„€)) (Insert.insert ((((2 : β„€), (0 : β„€)) : β„€ Γ— β„€) :: ([] : List (β„€ Γ— β„€)) : List (β„€ Γ— β„€)) (Insert.insert ((((0 : β„€), (1 : β„€)) : β„€ Γ— β„€) :: ([] : List (β„€ Γ— β„€)) : List (β„€ Γ— β„€)) ...
β„• Γ— Set (List (β„€ Γ— β„€))
{ "benchmark": "putnam_solving", "index": 43 }
[]
[ { "t": "Type*", "v": null, "name": "G", "t_type": "Type" }, { "t": "Group G", "v": null, "name": null, "t_type": "Type" }, { "t": "G", "v": null, "name": "A", "t_type": "Type" }, { "t": "G", "v": null, "name": "B", "t_type": "Type" }, {...
[ "(S.ncard = answer.1 ∧ S = {word w | w ∈ answer.2})" ]
Find $$\sum_{k=0}^{n} (-1)^k {n \choose k} (x - k)^n.$$
fun n => C (Nat.factorial n)
open Polynomial Filter Topology ProbabilityTheory MeasureTheory
answer = fun (n : β„•) => (((Polynomial.C : β„€ β†’+* β„€[X]) : β„€ β†’ β„€[X]) (↑(n.factorial : β„•) : β„€) : β„€[X])
β„• β†’ Polynomial β„€
{ "benchmark": "putnam_solving", "index": 44 }
[]
[]
[ "(βˆ€ n : β„•, βˆ‘ k in Finset.range (n + 1), C ((-(1 : β„€))^k * Nat.choose n k) * (X - (C (k : β„€)))^n = answer n)" ]
Show that if four distinct points of the curve $y = 2x^4 + 7x^3 + 3x - 5$ are collinear, then their average $x$-coordinate is some constant $k$. Find $k$.
$-7/8$
answer = (-7 / 8 : ℝ)
ℝ
{ "benchmark": "putnam_solving", "index": 45 }
[]
[ { "t": "ℝ β†’ ℝ", "v": null, "name": "y", "t_type": "Type" }, { "t": "y = fun x ↦ 2 * x ^ 4 + 7 * x ^ 3 + 3 * x - 5", "v": null, "name": "hy", "t_type": "Prop" }, { "t": "Finset ℝ", "v": null, "name": "S", "t_type": "Type" }, { "t": "S.card = 4", "v"...
[ "((βˆ‘ x in S, x) / 4 = answer)" ]
Find all real solutions $(a, b, c, d)$ to the equations $a + b + c = d$, $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{d}$.
$d = a \land b = -c \lor d = b \land a = -c \lor d = c \land a = -b$
answer = fun (a b c d : ℝ) => d = a ∧ b = (-c : ℝ) ∨ d = b ∧ a = (-c : ℝ) ∨ d = c ∧ a = (-b : ℝ)
ℝ β†’ ℝ β†’ ℝ β†’ ℝ β†’ Prop
{ "benchmark": "putnam_solving", "index": 46 }
[ { "t": "ℝ", "v": null, "name": "a", "t_type": "Type" }, { "t": "ℝ", "v": null, "name": "b", "t_type": "Type" }, { "t": "ℝ", "v": null, "name": "c", "t_type": "Type" }, { "t": "ℝ", "v": null, "name": "d", "t_type": "Type" } ]
[]
[ "(answer a b c d ↔\n a β‰  0 β†’ b β‰  0 β†’ c β‰  0 β†’ d β‰  0 β†’ (a + b + c = d ∧ 1 / a + 1 / b + 1 / c = 1 / d))" ]
Let $f, g, h$ be functions $\mathbb{R} \to \mathbb{R}$. Find an expression for $h(x)$ in terms of $f$ and $g$ such that $f(x) = \frac{h(x + 1) + h(x - 1)}{2}$ and $g(x) = \frac{h(x + 4) + h(x - 4)}{2}$.
$h(x) = g(x) - f(x - 3) + f(x - 1) + f(x + 1) - f(x + 3)$
answer = fun (f g : ℝ β†’ ℝ) (x : ℝ) => (((((g x : ℝ) - (f (x - (3 : ℝ) : ℝ) : ℝ) : ℝ) + (f (x - (1 : ℝ) : ℝ) : ℝ) : ℝ) + (f (x + (1 : ℝ) : ℝ) : ℝ) : ℝ) - (f (x + (3 : ℝ) : ℝ) : ℝ) : ℝ)
(ℝ β†’ ℝ) β†’ (ℝ β†’ ℝ) β†’ (ℝ β†’ ℝ)
{ "benchmark": "putnam_solving", "index": 47 }
[ { "t": "ℝ β†’ ℝ", "v": null, "name": "f", "t_type": "Type" }, { "t": "ℝ β†’ ℝ", "v": null, "name": "g", "t_type": "Type" } ]
[ { "t": "ℝ β†’ ℝ", "v": null, "name": "h", "t_type": "Type" }, { "t": "βˆ€ x, f x = (h (x + 1) + h (x - 1)) / 2", "v": null, "name": "hf", "t_type": "Prop" }, { "t": "βˆ€ x, g x = (h (x + 4) + h (x - 4)) / 2", "v": null, "name": "hg", "t_type": "Prop" } ]
[ "(h = answer f g)" ]
Find $\sum_{n=0}^{\infty} \frac{x^{2^n}}{1 - x^{2^{n+1}}}$ as a rational function of $x$ for $x \in (0, 1)$.
$\frac{x}{1 - x}$
open RingHom Set
answer = ((RatFunc.X : RatFunc ℝ) / ((1 : RatFunc ℝ) - (RatFunc.X : RatFunc ℝ) : RatFunc ℝ) : RatFunc ℝ)
RatFunc ℝ
{ "benchmark": "putnam_solving", "index": 48 }
[]
[ { "t": "ℝ", "v": null, "name": "x", "t_type": "Type" }, { "t": "x ∈ Ioo 0 1", "v": null, "name": "hx", "t_type": "Prop" } ]
[ "(answer.eval (id ℝ) x = βˆ‘' n : β„•, x ^ 2 ^ n / (1 - x ^ 2 ^ (n + 1)))" ]
Let $X$ be the square $[0, 1] \times [0, 1]$, and let $f : X \to \mathbb{R}$ be continuous. If $\int_Y f(x, y) \, dx \, dy = 0$ for all squares $Y$ such that \begin{itemize} \item[(1)] $Y \subseteq X$, \item[(2)] $Y$ has sides parallel to those of $X$, \item[(3)] at least one of $Y$'s sides is contained in the boundary...
True
open RingHom Set Nat
answer = True
Prop
{ "benchmark": "putnam_solving", "index": 49 }
[]
[ { "t": "Set (ℝ Γ— ℝ)", "v": null, "name": "X", "t_type": "Type" }, { "t": "X = Set.prod (Icc 0 1) (Icc 0 1)", "v": null, "name": "hX", "t_type": "Prop" }, { "t": "(ℝ Γ— ℝ) β†’ ℝ", "v": null, "name": "room", "t_type": "Type" }, { "t": "room = fun (a,b) ↦ mi...
[ "((βˆ€ f : (ℝ Γ— ℝ) β†’ ℝ, Continuous f β†’ (βˆ€ P ∈ X, ∫ x in (P.1 - room P)..(P.1 + room P), ∫ y in (P.2 - room P)..(P.2 + room P), f (x, y) = 0) β†’ (βˆ€ P ∈ X, f P = 0)) ↔ answer)" ]
Find $\prod_{n=2}^{\infty} \frac{(n^3 - 1)}{(n^3 + 1)}$.
2/3
open RingHom Set Nat Filter Topology
answer = (2 / 3 : ℝ)
ℝ
{ "benchmark": "putnam_solving", "index": 50 }
[]
[]
[ "(Tendsto (fun N ↦ ∏ n in Finset.Icc (2 : β„€) N, ((n : ℝ) ^ 3 - 1) / (n ^ 3 + 1)) atTop (𝓝 answer))" ]
An ordered triple $(a, b, c)$ of positive irrational numbers with $a + b + c = 1$ is considered $\textit{balanced}$ if all three elements are less than $\frac{1}{2}$. If a triple is not balanced, we can perform a ``balancing act'' $B$ defined by $B(a, b, c) = (f(a), f(b), f(c))$, where $f(x) = 2x - 1$ if $x > 1/2$ and ...
False
open RingHom Set Nat Filter Topology
answer = False
Prop
{ "benchmark": "putnam_solving", "index": 51 }
[]
[ { "t": "ℝ Γ— ℝ Γ— ℝ β†’ Prop", "v": null, "name": "P", "t_type": "Type" }, { "t": "ℝ Γ— ℝ Γ— ℝ β†’ Prop", "v": null, "name": "balanced", "t_type": "Type" }, { "t": "ℝ Γ— ℝ Γ— ℝ β†’ ℝ Γ— ℝ Γ— ℝ", "v": null, "name": "B", "t_type": "Type" }, { "t": "P = fun (a, b, c) =...
[ "((βˆ€ t : ℝ Γ— ℝ Γ— ℝ, P t β†’ βˆƒ n : β„•, balanced (B^[n] t)) ↔ answer)" ]
Let $p(x) = 2(x^6 + 1) + 4(x^5 + x) + 3(x^4 + x^2) + 5x^3$. For $k$ with $0 < k < 5$, let \[ I_k = \int_0^{\infty} \frac{x^k}{p(x)} \, dx. \] For which $k$ is $I_k$ smallest?
2
open Set Polynomial
answer = (2 : β„•)
β„•
{ "benchmark": "putnam_solving", "index": 52 }
[]
[ { "t": "Polynomial ℝ", "v": null, "name": "p", "t_type": "Type" }, { "t": "p = 2 * (Polynomial.X ^ 6 + 1) + 4 * (Polynomial.X ^ 5 + Polynomial.X) + 3 * (Polynomial.X ^ 4 + Polynomial.X ^ 2) + 5 * Polynomial.X ^ 3", "v": null, "name": "hp", "t_type": "Prop" }, { "t": "β„• β†’ ...
[ "(IsLeast {y | βˆƒ k ∈ Ioo 0 5, I k = y} answer)" ]
Find \[ \sum_{i=1}^{\infty} \sum_{j=1}^{\infty} \frac{1}{i^2j + 2ij + ij^2}. \]
7 / 4
open Set Real
answer = (7 / 4 : β„š)
β„š
{ "benchmark": "putnam_solving", "index": 53 }
[]
[]
[ "((βˆ‘' i : β„•+, βˆ‘' j : β„•+, (1 : β„š) / (i ^ 2 * j + 2 * i * j + i * j ^ 2)) = answer)" ]
Find the real polynomial $p(x)$ of degree $4$ with largest possible coefficient of $x^4$ such that $p([-1, 1]) \subseteq [0, 1]$.
$4x^4 - 4x^2 + 1$
open Set Real Filter Topology Polynomial
answer = ((((4 : ℝ[X]) * ((Polynomial.X : ℝ[X]) ^ (4 : β„•) : ℝ[X]) : ℝ[X]) - ((4 : ℝ[X]) * ((Polynomial.X : ℝ[X]) ^ (2 : β„•) : ℝ[X]) : ℝ[X]) : ℝ[X]) + (1 : ℝ[X]) : ℝ[X])
Polynomial ℝ
{ "benchmark": "putnam_solving", "index": 54 }
[]
[ { "t": "Set (Polynomial ℝ)", "v": null, "name": "S", "t_type": "Type" }, { "t": "S = {p : Polynomial ℝ | p.degree = 4 ∧ βˆ€ x ∈ Icc (-1 : ℝ) 1, p.eval x ∈ Icc 0 1}", "v": null, "name": "hS", "t_type": "Prop" } ]
[ "(answer ∈ S ∧ (βˆ€ p ∈ S, p.coeff 4 ≀ answer.coeff 4))" ]
For which positive integers $n$ and $a_1, a_2, \dots, a_n$ with $\sum_{i = 1}^{n} a_i = 1979$ does $\prod_{i = 1}^{n} a_i$ attain the greatest value?
Multiset.replicate 659 3 + {2}
answer = ((Multiset.replicate (659 : β„•) (3 : β„•) : Multiset β„•) + ({(2 : β„•)} : Multiset β„•) : Multiset β„•)
Multiset β„•
{ "benchmark": "putnam_solving", "index": 55 }
[]
[ { "t": "Multiset β„• β†’ Prop", "v": null, "name": "P", "t_type": "Type" }, { "t": "βˆ€ a, P a ↔ Multiset.card a > 0 ∧ (βˆ€ i ∈ a, i > 0) ∧ a.sum = 1979", "v": null, "name": "hP", "t_type": "Prop" } ]
[ "(P answer ∧ βˆ€ a : Multiset β„•, P a β†’ answer.prod β‰₯ a.prod)" ]
For which real numbers $k$ does there exist a continuous function $f : \mathbb{R} \to \mathbb{R}$ such that $f(f(x)) = kx^9$ for all real $x$?
$k \geq 0$
answer = fun (k : ℝ) => k β‰₯ (0 : ℝ)
ℝ β†’ Prop
{ "benchmark": "putnam_solving", "index": 56 }
[ { "t": "ℝ", "v": null, "name": "k", "t_type": "Type" } ]
[]
[ "(answer k ↔ βˆƒ f : ℝ β†’ ℝ, Continuous f ∧ βˆ€ x : ℝ, f (f x) = k * x^9)" ]
Let $x_1, x_2, x_3, \dots$ be a sequence of nonzero real numbers such that $$x_n = \frac{x_{n-2}x_{n-1}}{2x_{n-2}-x_{n-1}}$$ for all $n \ge 3$. For which real values of $x_1$ and $x_2$ does $x_n$ attain integer values for infinitely many $n$?
fun (a, b) => βˆƒ m : β„€, a = m ∧ b = m
answer = fun (x : ℝ Γ— ℝ) => match x with | ((a, b) : ℝ Γ— ℝ) => βˆƒ (m : β„€), a = (↑m : ℝ) ∧ b = (↑m : ℝ)
(ℝ Γ— ℝ) β†’ Prop
{ "benchmark": "putnam_solving", "index": 57 }
[ { "t": "β„• β†’ ℝ", "v": null, "name": "x", "t_type": "Type" } ]
[ { "t": "βˆ€ n : β„•, x n β‰  0 ∧ (n β‰₯ 3 β†’ x n = (x (n - 2))*(x (n - 1))/(2*(x (n - 2)) - (x (n - 1))))", "v": null, "name": "hx", "t_type": "Prop" } ]
[ "((βˆ€ m : β„•, βˆƒ n : β„•, n > m ∧ βˆƒ a : β„€, a = x n) ↔ answer (x 1, x 2))" ]
Let $A$ be a set of $2n$ points in the plane, $n$ colored red and $n$ colored blue, such that no three points in $A$ are collinear. Must there exist $n$ closed straight line segments, each connecting one red and one blue point in $A$, such that no two of the $n$ line segments intersect?
True
open Set
answer = True
Prop
{ "benchmark": "putnam_solving", "index": 58 }
[]
[ { "t": "Finset (Fin 2 β†’ ℝ) Γ— Finset (Fin 2 β†’ ℝ) β†’ Prop", "v": null, "name": "A", "t_type": "Type" }, { "t": "A = fun (R, B) => R.card = B.card ∧ R ∩ B = βˆ… ∧\n βˆ€ u : Finset (Fin 2 β†’ ℝ), u βŠ† R βˆͺ B β†’ u.card = 3 β†’ Β¬Collinear ℝ (u : Set (Fin 2 β†’ ℝ))", "v": null, "name": "hA", "t_ty...
[ "(answer ↔\n (βˆ€ R B, A (R, B) β†’\n βˆƒ v : Finset ((Fin 2 β†’ ℝ) Γ— (Fin 2 β†’ ℝ)),\n (βˆ€ L ∈ v, βˆ€ M ∈ v, L β‰  M β†’ βˆ€ x ∈ Icc 0 1, βˆ€ y ∈ Icc 0 1,\n Real.sqrt ((w (L.1, L.2) x 0 - w (M.1, M.2) y 0)^2 + (w (L.1, L.2) x 1 - w (M.1, M.2) y 1)^2) β‰  0) ∧\n v.card = R.card ∧ βˆ€ L ∈ v, L.1 ∈ R ∧ L....
If $0 < a < b$, find $$\lim_{t \to 0} \left( \int_{0}^{1}(bx + a(1-x))^t dx \right)^{\frac{1}{t}}$$ in terms of $a$ and $b$.
fun (a, b) => (Real.exp (-1))*(b^b/a^a)^(1/(b-a))
open Set Topology Filter
answer = fun (x : ℝ Γ— ℝ) => match x with | ((a, b) : ℝ Γ— ℝ) => ((Real.exp (-1 : ℝ) : ℝ) * (((b ^ b : ℝ) / (a ^ a : ℝ) : ℝ) ^ ((1 : ℝ) / (b - a : ℝ) : ℝ) : ℝ) : ℝ)
ℝ Γ— ℝ β†’ ℝ
{ "benchmark": "putnam_solving", "index": 59 }
[]
[]
[ "(βˆ€ a b : ℝ, 0 < a ∧ a < b β†’ Tendsto (fun t : ℝ => (∫ x in Icc 0 1, (b*x + a*(1 - x))^t)^(1/t)) (𝓝[β‰ ] 0) (𝓝 (answer (a, b))))" ]
Let $F$ be a finite field with $n$ elements, and assume $n$ is odd. Suppose $x^2 + bx + c$ is an irreducible polynomial over $F$. For how many elements $d \in F$ is $x^2 + bx + c + d$ irreducible?
fun n : β„• ↦ (n - (1 : β„€)) / 2
open Set Topology Filter Polynomial
answer = fun (n : β„•) => (((↑n : β„€) - (1 : β„€) : β„€) / (2 : β„€) : β„€)
β„• β†’ β„€
{ "benchmark": "putnam_solving", "index": 60 }
[ { "t": "β„•", "v": null, "name": "n", "t_type": "Type" } ]
[ { "t": "Type*", "v": null, "name": "F", "t_type": "Type" }, { "t": "Field F", "v": null, "name": null, "t_type": "Type" }, { "t": "Fintype F", "v": null, "name": null, "t_type": "Type" }, { "t": "n = Fintype.card F", "v": null, "name": "hn", ...
[ "({d : F | Irreducible (p + (C d))}.ncard = answer n)" ]
Let $r$ and $s$ be positive integers. Derive a formula for the number of ordered quadruples $(a,b,c,d)$ of positive integers such that $3^r \cdot 7^s=\text{lcm}[a,b,c]=\text{lcm}[a,b,d]=\text{lcm}[a,c,d]=\text{lcm}[b,c,d]$. The answer should be a function of $r$ and $s$. (Note that $\text{lcm}[x,y,z]$ denotes the least...
(fun r s : β„• => (1 + 4 * r + 6 * r ^ 2) * (1 + 4 * s + 6 * s ^ 2))
answer = fun (r s : β„•) => ((((1 : β„•) + ((4 : β„•) * r : β„•) : β„•) + ((6 : β„•) * (r ^ (2 : β„•) : β„•) : β„•) : β„•) * (((1 : β„•) + ((4 : β„•) * s : β„•) : β„•) + ((6 : β„•) * (s ^ (2 : β„•) : β„•) : β„•) : β„•) : β„•)
β„• β†’ β„• β†’ β„•
{ "benchmark": "putnam_solving", "index": 61 }
[ { "t": "β„•", "v": null, "name": "r", "t_type": "Type" }, { "t": "β„•", "v": null, "name": "s", "t_type": "Type" } ]
[ { "t": "β„• β†’ β„• β†’ β„• β†’ β„• β†’ Prop", "v": null, "name": "abcdlcm", "t_type": "Type" }, { "t": "r > 0 ∧ s > 0", "v": null, "name": "rspos", "t_type": "Prop" }, { "t": "βˆ€ a b c d : β„•, abcdlcm a b c d ↔\n (a > 0 ∧ b > 0 ∧ c > 0 ∧ d > 0 ∧\n (3 ^ r * 7 ^ s = Nat.lcm (Nat.l...
[ "({h : β„• Γ— β„• Γ— β„• Γ— β„• | abcdlcm h.1 h.2.1 h.2.2.1 h.2.2.2}.encard = answer r s)" ]
Evaluate $\int_0^{\pi/2}\frac{dx}{1+(\tan x)^{\sqrt{2}}}$.
$\pi / 4$
open Real
answer = ((Ο€ : ℝ) / (4 : ℝ) : ℝ)
ℝ
{ "benchmark": "putnam_solving", "index": 62 }
[]
[]
[ "(answer = ∫ x in Set.Ioo 0 (Real.pi / 2), 1 / (1 + (Real.tan x) ^ (Real.sqrt 2)))" ]
Let $C$ be the class of all real valued continuously differentiable functions $f$ on the interval $0 \leq x \leq 1$ with $f(0)=0$ and $f(1)=1$. Determine the largest real number $u$ such that $u \leq \int_0^1|f'(x)-f(x)|\,dx$ for all $f$ in $C$.
$1/e$
open Real
answer = ((1 : ℝ) / (rexp (1 : ℝ) : ℝ) : ℝ)
ℝ
{ "benchmark": "putnam_solving", "index": 63 }
[]
[ { "t": "Set (ℝ β†’ ℝ)", "v": null, "name": "C", "t_type": "Type" }, { "t": "C = {f : ℝ β†’ ℝ | ContDiffOn ℝ 1 f (Set.Icc 0 1) ∧ f 0 = 0 ∧ f 1 = 1}", "v": null, "name": "hC", "t_type": "Prop" } ]
[ "(IsGreatest {u : ℝ | βˆ€ f ∈ C, u ≀ (∫ x in Set.Ioo 0 1, |deriv f x - f x|)} answer)" ]
For which real numbers $c$ is $(e^x+e^{-x})/2 \leq e^{cx^2}$ for all real $x$?
$\{c : \mathbb{R} \mid c \geq 1/2\}$
open Real
answer = ({c : ℝ | c β‰₯ (1 / 2 : ℝ)} : Set ℝ)
Set ℝ
{ "benchmark": "putnam_solving", "index": 64 }
[ { "t": "ℝ", "v": null, "name": "c", "t_type": "Type" } ]
[]
[ "(βˆ€ x : ℝ, (exp x + exp (-x)) / 2 ≀ exp (c * x ^ 2) ↔ c ∈ answer)" ]
For which real numbers $a$ does the sequence defined by the initial condition $u_0=a$ and the recursion $u_{n+1}=2u_n-n^2$ have $u_n>0$ for all $n \geq 0$? (Express the answer in the simplest form.)
$\{a : \mathbb{R} \mid a \geq 3\}$
answer = ({a : ℝ | a β‰₯ (3 : ℝ)} : Set ℝ)
Set ℝ
{ "benchmark": "putnam_solving", "index": 65 }
[ { "t": "ℝ", "v": null, "name": "a", "t_type": "Type" } ]
[ { "t": "β„• β†’ ℝ", "v": null, "name": "u", "t_type": "Type" }, { "t": "u 0 = a ∧ (βˆ€ n : β„•, u (n + 1) = 2 * u n - n ^ 2)", "v": null, "name": "hu", "t_type": "Prop" } ]
[ "((βˆ€ n : β„•, u n > 0) ↔ a ∈ answer)" ]
A function $f$ is convex on $[0, 1]$ if and only if $$f(su + (1-s)v) \le sf(u) + (1 - s)f(v)$$ for all $s \in [0, 1]$. Let $S_t$ denote the set of all nonnegative increasing convex continuous functions $f : [0, 1] \rightarrow \mathbb{R}$ such that $$f(1) - 2f\left(\frac{2}{3}\right) + f\left(\frac{1}{3}\right) \ge t\le...
$t \le 1$
open Set
answer = fun (t : ℝ) => t ≀ (1 : ℝ)
ℝ β†’ Prop
{ "benchmark": "putnam_solving", "index": 66 }
[ { "t": "ℝ", "v": null, "name": "t", "t_type": "Type" } ]
[ { "t": "Set ℝ", "v": null, "name": "T", "t_type": "Type" }, { "t": "T = Icc 0 1", "v": null, "name": "hT", "t_type": "Prop" }, { "t": "ℝ β†’ (ℝ β†’ ℝ) β†’ Prop", "v": null, "name": "P", "t_type": "Type" }, { "t": "(ℝ β†’ ℝ) β†’ Prop", "v": null, "name": ...
[ "(answer t ↔ (βˆ€ f ∈ S t, βˆ€ g ∈ S t, f * g ∈ S t))" ]
Let $E(n)$ be the greatest integer $k$ such that $5^k$ divides $1^1 2^2 3^3 \cdots n^n$. Find $\lim_{n \rightarrow \infty} \frac{E(n)}{n^2}$.
$\frac{1}{8}$
open Topology Filter Set Polynomial Function
answer = (1 / 8 : ℝ)
ℝ
{ "benchmark": "putnam_solving", "index": 67 }
[]
[ { "t": "β„• β†’ β„• β†’ Prop", "v": null, "name": "P", "t_type": "Type" }, { "t": "βˆ€ n k, P n k ↔ 5^k ∣ ∏ m in Finset.Icc 1 n, (m^m : β„€)", "v": null, "name": "hP", "t_type": "Prop" }, { "t": "β„• β†’ β„•", "v": null, "name": "E", "t_type": "Type" }, { "t": "βˆ€ n ∈ Ic...
[ "(Tendsto (fun n : β„• => ((E n) : ℝ)/n^2) atTop (𝓝 answer))" ]
Does the limit $$lim_{t \rightarrow \infty}e^{-t}\int_{0}^{t}\int_{0}^{t}\frac{e^x - e^y}{x - y} dx dy$$exist?
False
open Topology Filter Set Polynomial Function
answer = False
Prop
{ "benchmark": "putnam_solving", "index": 68 }
[]
[ { "t": "ℝ β†’ ℝ", "v": null, "name": "f", "t_type": "Type" }, { "t": "f = fun t : ℝ => Real.exp (-t) * ∫ y in (Ico 0 t), ∫ x in (Ico 0 t), (Real.exp x - Real.exp y) / (x - y)", "v": null, "name": "hf", "t_type": "Prop" } ]
[ "((βˆƒ L : ℝ, Tendsto f atTop (𝓝 L)) ↔ answer)" ]
Let $P(x)$ be a polynomial with real coefficients; let $$Q(x) = (x^2 + 1)P(x)P'(x) + x((P(x))^2 + (P'(x))^2).$$ Given that $P$ has $n$ distinct real roots all greater than $1$, prove or disprove that $Q$ must have at least $2n - 1$ distinct real roots.
True
open Topology Filter Set Polynomial Function
answer = True
Prop
{ "benchmark": "putnam_solving", "index": 69 }
[]
[ { "t": "Polynomial ℝ β†’ Polynomial ℝ", "v": null, "name": "Q", "t_type": "Type" }, { "t": "Q = fun P : Polynomial ℝ => (Polynomial.X^2 + 1) * P * (Polynomial.derivative P) + Polynomial.X * (P^2 + (Polynomial.derivative P)^2)", "v": null, "name": "hQ", "t_type": "Prop" }, { ...
[ "(answer ↔ (βˆ€ P : Polynomial ℝ, {x : ℝ | (Q P).eval x = 0}.ncard β‰₯ 2*(n P) - 1))" ]
Find the value of $$\lim_{n \rightarrow \infty} \frac{1}{n^5}\sum_{h=1}^{n}\sum_{k=1}^{n}(5h^4 - 18h^2k^2 + 5k^4).$$
-1
open Topology Filter Set Polynomial Function
answer = (-1 : ℝ)
ℝ
{ "benchmark": "putnam_solving", "index": 70 }
[]
[ { "t": "β„• β†’ ℝ", "v": null, "name": "f", "t_type": "Type" }, { "t": "f = fun n : β„• => ((1 : ℝ)/n^5) * βˆ‘ h in Finset.Icc 1 n, βˆ‘ k in Finset.Icc 1 n, (5*(h : ℝ)^4 - 18*h^2*k^2 + 5*k^4)", "v": null, "name": "hf", "t_type": "Prop" } ]
[ "(Tendsto f atTop (𝓝 answer))" ]
Determine the minimum value attained by $$(r - 1)^2 + (\frac{s}{r} - 1)^2 + (\frac{t}{s} - 1)^2 + (\frac{4}{t} - 1)^2$$ across all choices of real $r$, $s$, and $t$ that satisfy $1 \le r \le s \le t \le 4$.
12 - 8 * Real.sqrt 2
open Topology Filter Set Polynomial Function
answer = ((12 : ℝ) - ((8 : ℝ) * (√(2 : ℝ) : ℝ) : ℝ) : ℝ)
ℝ
{ "benchmark": "putnam_solving", "index": 71 }
[]
[ { "t": "ℝ Γ— ℝ Γ— ℝ β†’ Prop", "v": null, "name": "P", "t_type": "Type" }, { "t": "P = fun (r, s, t) => 1 ≀ r ∧ r ≀ s ∧ s ≀ t ∧ t ≀ 4", "v": null, "name": "hP", "t_type": "Prop" }, { "t": "ℝ Γ— ℝ Γ— ℝ β†’ ℝ", "v": null, "name": "f", "t_type": "Type" }, { "t": ...
[ "(IsLeast {y | βˆƒ r s t, P (r, s, t) ∧ f (r, s, t) = y} answer)" ]
Let $V$ be a set of $5$ by $7$ matrices, with real entries and with the property that $rA+sB \in V$ whenever $A,B \in V$ and $r$ and $s$ are scalars (i.e., real numbers). \emph{Prove or disprove} the following assertion: If $V$ contains matrices of ranks $0$, $1$, $2$, $4$, and $5$, then it also contains a matrix of ra...
False
open Topology Filter Set Polynomial Function
answer = False
Prop
{ "benchmark": "putnam_solving", "index": 72 }
[]
[ { "t": "Set (Matrix (Fin 5) (Fin 7) ℝ) β†’ Prop", "v": null, "name": "VAB", "t_type": "Type" }, { "t": "Set (Matrix (Fin 5) (Fin 7) ℝ) β†’ β„• β†’ Prop", "v": null, "name": "Vrank", "t_type": "Type" }, { "t": "βˆ€ V, VAB V = (βˆ€ A ∈ V, βˆ€ B ∈ V, βˆ€ r s : ℝ, r β€’ A + s β€’ B ∈ V)", "v...
[ "(answer ↔\n (βˆ€ V,\n VAB V β†’ Vrank V 0 β†’ Vrank V 1 β†’\n Vrank V 2 β†’ Vrank V 4 β†’ Vrank V 5 β†’\n Vrank V 3))" ]
Let $B(n)$ be the number of ones in the base two expression for the positive integer $n$. For example, $B(6)=B(110_2)=2$ and $B(15)=B(1111_2)=4$. Determine whether or not $\exp \left(\sum_{n=1}^\infty \frac{B(n)}{n(n+1)}\right)$ is a rational number. Here $\exp(x)$ denotes $e^x$.
True
open Topology Filter Set Polynomial Function
answer = True
Prop
{ "benchmark": "putnam_solving", "index": 73 }
[]
[ { "t": "List β„• β†’ β„€", "v": null, "name": "sumbits", "t_type": "Type" }, { "t": "β„• β†’ β„€", "v": null, "name": "B", "t_type": "Type" }, { "t": "βˆ€ bits : List β„•, sumbits bits = βˆ‘ i : Fin bits.length, (bits[i] : β„€)", "v": null, "name": "hsumbits", "t_type": "Prop" ...
[ "(answer ↔ (βˆƒ q : β„š, Real.exp (βˆ‘' n : Set.Ici 1, B n / ((n : ℝ) * ((n : ℝ) + 1))) = q))" ]
Let $B_n(x) = 1^x + 2^x + \dots + n^x$ and let $f(n) = \frac{B_n(\log_n 2)}{(n \log_2 n)^2}$. Does $f(2) + f(3) + f(4) + \dots$ converge?
True
open Set Function Filter Topology Polynomial Real
answer = True
Prop
{ "benchmark": "putnam_solving", "index": 74 }
[]
[ { "t": "β„• β†’ ℝ β†’ ℝ", "v": null, "name": "B", "t_type": "Type" }, { "t": "B = fun (n : β„•) (x : ℝ) ↦ βˆ‘ k in Finset.Icc 1 n, (k : ℝ) ^ x", "v": null, "name": "hB", "t_type": "Prop" }, { "t": "β„• β†’ ℝ", "v": null, "name": "f", "t_type": "Type" }, { "t": "f = ...
[ "(answer ↔ (βˆƒ L : ℝ, Tendsto (fun N ↦ βˆ‘ j in Finset.Icc 2 N, f j) atTop (𝓝 L)))" ]
Evaluate $\int_0^{\infty} \frac{\tan^{-1}(\pi x) - \tan^{-1} x}{x} \, dx$.
$\frac{\pi}{2} \log \pi$
open Set Function Filter Topology Polynomial Real
answer = (((Ο€ : ℝ) / (2 : ℝ) : ℝ) * (Real.log (Ο€ : ℝ) : ℝ) : ℝ)
ℝ
{ "benchmark": "putnam_solving", "index": 75 }
[]
[]
[ "(Tendsto (fun t ↦ ∫ x in (0)..t, (arctan (Real.pi * x) - arctan x) / x) atTop (𝓝 answer))" ]
Let $b$ be a bijection from the positive integers to the positive integers. Also, let $x_1, x_2, x_3, \dots$ be an infinite sequence of real numbers with the following properties: \begin{enumerate} \item $|x_n|$ is a strictly decreasing function of $n$; \item $\lim_{n \rightarrow \infty} |b(n) - n| \cdot |x_n| = 0$; \i...
False
open Set Function Filter Topology Polynomial Real
answer = False
Prop
{ "benchmark": "putnam_solving", "index": 76 }
[]
[ { "t": "β„• β†’ β„•", "v": null, "name": "b", "t_type": "Type" }, { "t": "β„• β†’ ℝ", "v": null, "name": "x", "t_type": "Type" }, { "t": "BijOn b (Ici 1) (Ici 1)", "v": null, "name": "h_bij", "t_type": "Prop" }, { "t": "StrictAntiOn (fun n : β„• => |x n|) (Ici 1)"...
[ "((Tendsto (fun n : β„• => βˆ‘ k in Finset.Icc 1 n, x (b k)) atTop (𝓝 1)) ↔ answer)" ]
Let $A(x, y)$ denote the number of points $(m, n)$ with integer coordinates $m$ and $n$ where $m^2 + n^2 \le x^2 + y^2$. Also, let $g = \sum_{k = 0}^{\infty} e^{-k^2}$. Express the value $$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} A(x, y)e^{-x^2 - y^2} dx dy$$ as a polynomial in $g$.
C Real.pi * (2*X - 1)^2
open Set Function Filter Topology Polynomial Real
answer = ((((Polynomial.C : ℝ β†’+* ℝ[X]) : ℝ β†’ ℝ[X]) (Ο€ : ℝ) : ℝ[X]) * ((((2 : ℝ[X]) * (Polynomial.X : ℝ[X]) : ℝ[X]) - (1 : ℝ[X]) : ℝ[X]) ^ (2 : β„•) : ℝ[X]) : ℝ[X])
Polynomial ℝ
{ "benchmark": "putnam_solving", "index": 77 }
[]
[ { "t": "ℝ Γ— ℝ β†’ β„•", "v": null, "name": "A", "t_type": "Type" }, { "t": "ℝ", "v": null, "name": "g", "t_type": "Type" }, { "t": "ℝ", "v": null, "name": "I", "t_type": "Type" }, { "t": "A = fun (x, y) => {a : β„€ Γ— β„€ | a.1^2 + a.2^2 ≀ x^2 + y^2}.ncard", ...
[ "(I = answer.eval g)" ]
Let $p_n$ denote the probability that $c + d$ will be a perfect square if $c$ and $d$ are selected independently and uniformly at random from $\{1, 2, 3, \dots, n\}$. Express $\lim_{n \rightarrow \infty} p_n \sqrt{n}$ in the form $r(\sqrt{s} - t)$ for integers $s$ and $t$ and rational $r$.
4/3 * (Real.sqrt 2 - 1)
open Set Function Filter Topology Polynomial Real
answer = ((4 / 3 : ℝ) * ((√(2 : ℝ) : ℝ) - (1 : ℝ) : ℝ) : ℝ)
ℝ
{ "benchmark": "putnam_solving", "index": 78 }
[]
[ { "t": "β„• β†’ ℝ", "v": null, "name": "p", "t_type": "Type" }, { "t": "p = fun n : β„• => ({c : Finset.Icc 1 n Γ— Finset.Icc 1 n | βˆƒ m : β„•, m^2 = c.1 + c.2}.ncard : ℝ) / n^2", "v": null, "name": "hp", "t_type": "Prop" } ]
[ "(Tendsto (fun n : β„• => p n * Real.sqrt n) atTop (𝓝 answer))" ]
Let $n_1, n_2, \dots, n_s$ be distinct integers such that, for every integer $k$, $n_1n_2\cdots n_s$ divides $(n_1 + k)(n_2 + k) \cdots (n_s + k)$. Prove or provide a counterexample to the following claims: \begin{enumerate} \item For some $i$, $|n_i| = 1$. \item If all $n_i$ are positive, then $\{n_1, n_2, \dots, n_s\...
(True, True)
open Set Function Filter Topology Polynomial Real
answer = ((True, True) : Prop Γ— Prop)
Prop Γ— Prop
{ "benchmark": "putnam_solving", "index": 79 }
[]
[ { "t": "Finset β„€ β†’ Prop", "v": null, "name": "P", "t_type": "Type" }, { "t": "βˆ€ n, P n ↔ n.Nonempty ∧ βˆ€ k, ∏ i in n, i ∣ ∏ i in n, (i + k)", "v": null, "name": "P_def", "t_type": "Prop" } ]
[ "(((βˆ€ n, P n β†’ 1 ∈ n ∨ -1 ∈ n) ↔ answer.1) ∧\n ((βˆ€ n, P n β†’ (βˆ€ i ∈ n, 0 < i) β†’ n = Finset.Icc (1 : β„€) n.card) ↔ answer.2))" ]
How many positive integers $n$ are there such that $n$ is an exact divisor of at least one of the numbers $10^{40},20^{30}$?
2301
answer = (2301 : β„•)
β„•
{ "benchmark": "putnam_solving", "index": 80 }
[]
[]
[ "({n : β„€ | n > 0 ∧ (n ∣ 10 ^ 40 ∨ n ∣ 20 ^ 30)}.encard = answer)" ]
Prove or disprove that there exists a positive real number $\alpha$ such that $[\alpha_n] - n$ is even for all integers $n > 0$. (Here $[x]$ denotes the greatest integer less than or equal to $x$.)
True
open Nat
answer = True
Prop
{ "benchmark": "putnam_solving", "index": 81 }
[]
[]
[ "(answer ↔ (βˆƒ Ξ± : ℝ, Ξ± > 0 ∧ βˆ€ n : β„•, n > 0 β†’ Even (⌊α ^ nβŒ‹ - n)))" ]
Let $T$ be the triangle with vertices $(0, 0)$, $(a, 0)$, and $(0, a)$. Find $\lim_{a \to \infty} a^4 \exp(-a^3) \int_T \exp(x^3+y^3) \, dx \, dy$.
2 / 9
open Nat Filter Topology Real
answer = (2 / 9 : ℝ)
ℝ
{ "benchmark": "putnam_solving", "index": 82 }
[]
[ { "t": "ℝ β†’ ℝ", "v": null, "name": "F", "t_type": "Type" }, { "t": "F = fun a ↦ (a ^ 4 / exp (a ^ 3)) * ∫ x in (0)..a, ∫ y in (0)..(a - x), exp (x ^ 3 + y ^ 3)", "v": null, "name": "hF", "t_type": "Prop" } ]
[ "(Tendsto F atTop (𝓝 answer))" ]
Let $f(n)$ be the number of ways of representing $n$ as a sum of powers of $2$ with no power being used more than $3$ times. For example, $f(7) = 4$ (the representations are $4 + 2 + 1$, $4 + 1 + 1 + 1$, $2 + 2 + 2 + 1$, $2 + 2 + 1 + 1 + 1$). Can we find a real polynomial $p(x)$ such that $f(n) = [p(n)]$, where $[u]$ d...
True
open Nat Filter Topology Real
answer = True
Prop
{ "benchmark": "putnam_solving", "index": 83 }
[]
[ { "t": "β„•+ β†’ β„•", "v": null, "name": "f", "t_type": "Type" }, { "t": "f = fun (n : β„•+) ↦\n Set.ncard {M : Multiset β„• |\n (βˆ€ m ∈ M, βˆƒ k : β„•, m = (2 ^ k : β„€)) ∧\n (βˆ€ m ∈ M, M.count m ≀ 3) ∧\n (M.sum : β„€) = n}", "v": null, "name": "hf", "t_type": "Prop" } ...
[ "(answer ↔ (βˆƒ p : Polynomial ℝ, βˆ€ n : β„•+, ⌊p.eval (n : ℝ)βŒ‹ = f n))" ]
Define $\left\lVert x \right\rVert$ as the distance from $x$ to the nearest integer. Find $\lim_{n \to \infty} \frac{1}{n} \int_{1}^{n} \left\lVert \frac{n}{x} \right\rVert \, dx$. You may assume that $\prod_{n=1}^{\infty} \frac{2n}{(2n-1)} \cdot \frac{2n}{(2n+1)} = \frac{\pi}{2}$.
$\log \left(\frac{4}{\pi}\right)$
open Nat Filter Topology Real
answer = (Real.log ((4 : ℝ) / (Ο€ : ℝ) : ℝ) : ℝ)
ℝ
{ "benchmark": "putnam_solving", "index": 84 }
[]
[ { "t": "ℝ β†’ ℝ", "v": null, "name": "dist_fun", "t_type": "Type" }, { "t": "dist_fun = fun (x : ℝ) ↦ min (x - ⌊xβŒ‹) (⌈xβŒ‰ - x)", "v": null, "name": "hdist_fun", "t_type": "Prop" }, { "t": "Tendsto (fun N ↦ ∏ n in Finset.Icc 1 N, (2 * n / (2 * n - 1)) * (2 * n / (2 * n + 1)) ...
[ "(Tendsto (fun n ↦ (1 / n) * ∫ x in (1)..n, dist_fun (n / x) : β„• β†’ ℝ) atTop (𝓝 answer))" ]
Express $\sum_{k=1}^\infty (6^k/(3^{k+1}-2^{k+1})(3^k-2^k))$ as a rational number.
2
answer = (2 : β„š)
β„š
{ "benchmark": "putnam_solving", "index": 85 }
[]
[]
[ "(βˆ‘' k : Set.Ici 1, (6 ^ (k : β„•) / ((3 ^ ((k : β„•) + 1) - 2 ^ ((k : β„•) + 1)) * (3 ^ (k : β„•) - 2 ^ (k : β„•)))) = answer)" ]
Let $n$ be a positive integer. Let $a,b,x$ be real numbers, with $a \neq b$, and let $M_n$ denote the $2n \times 2n$ matrix whose $(i,j)$ entry $m_{ij}$ is given by \[ m_{ij}=\begin{cases} x & \text{if }i=j, \\ a & \text{if }i \neq j\text{ and }i+j\text{ is even}, \\ b & \text{if }i \neq j\text{ and }i+j\text{ is odd}....
$(X_2)^2 \cdot ((X_0)^2 - (X_1)^2)$
open Topology Filter
answer = (((MvPolynomial.X (2 : Fin (3 : β„•)) : MvPolynomial (Fin (3 : β„•)) ℝ) ^ (2 : β„•) : MvPolynomial (Fin (3 : β„•)) ℝ) * (((MvPolynomial.X (0 : Fin (3 : β„•)) : MvPolynomial (Fin (3 : β„•)) ℝ) ^ (2 : β„•) : MvPolynomial (Fin (3 : β„•)) ℝ) - ((MvPolynomial.X (1 : Fin (3 : β„•)) : MvPolynomial (Fin (3 : β„•)) ℝ) ^ ...
MvPolynomial (Fin 3) ℝ
{ "benchmark": "putnam_solving", "index": 86 }
[]
[ { "t": "β„•", "v": null, "name": "n", "t_type": "Type" }, { "t": "ℝ", "v": null, "name": "a", "t_type": "Type" }, { "t": "ℝ", "v": null, "name": "b", "t_type": "Type" }, { "t": "ℝ β†’ Matrix (Fin (2 * n)) (Fin (2 * n)) ℝ", "v": null, "name": "Mn", ...
[ "(Tendsto (fun x : ℝ => (Mn x).det / (x - a) ^ (2 * n - 2)) (𝓝[β‰ ] a) (𝓝 (MvPolynomial.eval polyabn answer)))" ]
Let $R$ be the region consisting of all triples $(x,y,z)$ of nonnegative real numbers satisfying $x+y+z \leq 1$. Let $w=1-x-y-z$. Express the value of the triple integral $\iiint_R x^1y^9z^8w^4\,dx\,dy\,dz$ in the form $a!b!c!d!/n!$, where $a$, $b$, $c$, $d$, and $n$ are positive integers.
(1, 9, 8, 4, 25)
open Topology Filter Nat
answer = (((1 : β„•), (((9 : β„•), (((8 : β„•), (((4 : β„•), (25 : β„•)) : β„• Γ— β„•)) : β„• Γ— β„• Γ— β„•)) : β„• Γ— β„• Γ— β„• Γ— β„•)) : β„• Γ— β„• Γ— β„• Γ— β„• Γ— β„•)
β„• Γ— β„• Γ— β„• Γ— β„• Γ— β„•
{ "benchmark": "putnam_solving", "index": 87 }
[]
[ { "t": "Set (Fin 3 β†’ ℝ)", "v": null, "name": "R", "t_type": "Type" }, { "t": "(Fin 3 β†’ ℝ) β†’ ℝ", "v": null, "name": "w", "t_type": "Type" }, { "t": "R = {p | (βˆ€ i : Fin 3, p i β‰₯ 0) ∧ p 0 + p 1 + p 2 ≀ 1}", "v": null, "name": "hR", "t_type": "Prop" }, { ...
[ "(answer = (a, b, c, d, n))" ]
Let $n$ be a positive integer, and let $f(n)$ denote the last nonzero digit in the decimal expansion of $n!$. For instance, $f(5)=2$. \begin{enumerate} \item[(a)] Show that if $a_1,a_2,\dots,a_k$ are \emph{distinct} nonnegative integers, then $f(5^{a_1}+5^{a_2}+\dots+5^{a_k})$ depends only on the sum $a_1+a_2+\dots+a_k...
4
open Topology Filter Function Nat
answer = (4 : β„•)
β„•
{ "benchmark": "putnam_solving", "index": 88 }
[]
[ { "t": "β„• β†’ β„•", "v": null, "name": "f", "t_type": "Type" }, { "t": "βˆ€ n, some (f n) = (Nat.digits 10 (n !)).find? (fun d ↦ d β‰  0)", "v": null, "name": "hf", "t_type": "Prop" }, { "t": "β„• β†’ (β„• β†’ β„•) β†’ β„• β†’ Prop", "v": null, "name": "IsPeriodicFrom", "t_type": "Ty...
[ "(βˆƒ g : β„• β†’ β„•,\n (βˆ€α΅‰ (k > 0) (a : Fin k β†’ β„•) (ha : Injective a), f (βˆ‘ i, 5 ^ (a i)) = g (βˆ‘ i, a i)) ∧\n P 1 g answer)" ]
Let $n$ be a positive integer, and define $f(n)=1!+2!+\dots+n!$. Find polynomials $P(x)$ and $Q(x)$ such that $f(n+2)=P(n)f(n+1)+Q(n)f(n)$ for all $n \geq 1$.
$(x + 3, -x - 2)$
open Topology Filter Nat
answer = ((((Polynomial.X : Polynomial ℝ) + (3 : Polynomial ℝ) : Polynomial ℝ), ((-(Polynomial.X : Polynomial ℝ) : Polynomial ℝ) - (2 : Polynomial ℝ) : Polynomial ℝ)) : Polynomial ℝ Γ— Polynomial ℝ)
Polynomial ℝ Γ— Polynomial ℝ
{ "benchmark": "putnam_solving", "index": 89 }
[]
[ { "t": "β„• β†’ β„€", "v": null, "name": "f", "t_type": "Type" }, { "t": "βˆ€ n > 0, f n = βˆ‘ i : Set.Icc 1 n, ((i)! : β„€)", "v": null, "name": "hf", "t_type": "Prop" } ]
[ "(βˆ€ n β‰₯ 1, f (n + 2) = (answer.1).eval (n : ℝ) * f (n + 1) + (answer.2).eval (n : ℝ) * f n)" ]
Find the minimum value of $(u-v)^2+(\sqrt{2-u^2}-\frac{9}{v})^2$ for $0<u<\sqrt{2}$ and $v>0$.
8
open Topology Filter Nat
answer = (8 : ℝ)
ℝ
{ "benchmark": "putnam_solving", "index": 90 }
[]
[ { "t": "ℝ β†’ ℝ β†’ ℝ", "v": null, "name": "f", "t_type": "Type" }, { "t": "βˆ€ u v : ℝ, f u v = (u - v) ^ 2 + (Real.sqrt (2 - u ^ 2) - 9 / v) ^ 2", "v": null, "name": "hf", "t_type": "Prop" } ]
[ "(IsLeast {y | βˆƒα΅‰ (u : Set.Ioo 0 √2) (v > 0), f u v = y} answer)" ]
Prove or disprove the following statement: If $F$ is a finite set with two or more elements, then there exists a binary operation $*$ on F such that for all $x,y,z$ in $F$, \begin{enumerate} \item[(i)] $x*z=y*z$ implies $x=y$ (right cancellation holds), and \item[(ii)] $x*(y*z) \neq (x*y)*z$ (\emph{no} case of associat...
True
open Topology Filter Nat
answer = True
Prop
{ "benchmark": "putnam_solving", "index": 91 }
[]
[]
[ "((βˆ€ (F : Type*) (_ : Fintype F), Fintype.card F β‰₯ 2 β†’ (βˆƒ mul : F β†’ F β†’ F, βˆ€ x y z : F, (mul x z = mul y z β†’ x = y) ∧ (mul x (mul y z) β‰  mul (mul x y) z))) ↔ answer)" ]
For each nonnegative integer $k$, let $d(k)$ denote the number of $1$'s in the binary expansion of $k$ (for example, $d(0)=0$ and $d(5)=2$). Let $m$ be a positive integer. Express $\sum_{k=0}^{2^m-1} (-1)^{d(k)}k^m$ in the form $(-1)^ma^{f(m)}(g(m))!$, where $a$ is an integer and $f$ and $g$ are polynomials.
(2, (Polynomial.X * (Polynomial.X - 1)) / 2, Polynomial.X)
open Topology Filter Nat
answer = (((2 : β„€), (((((Polynomial.X : Polynomial ℝ) * ((Polynomial.X : Polynomial ℝ) - (1 : Polynomial ℝ) : Polynomial ℝ) : Polynomial ℝ) / (2 : Polynomial ℝ) : Polynomial ℝ), (Polynomial.X : Polynomial β„•)) : Polynomial ℝ Γ— Polynomial β„•)) : β„€ Γ— P...
β„€ Γ— Polynomial ℝ Γ— Polynomial β„•
{ "benchmark": "putnam_solving", "index": 92 }
[]
[ { "t": "β„•", "v": null, "name": "m", "t_type": "Type" }, { "t": "m > 0", "v": null, "name": "mpos", "t_type": "Prop" }, { "t": "β„• β†’ β„•", "v": null, "name": "d", "t_type": "Type" }, { "t": "List β„• β†’ β„•", "v": null, "name": "sumbits", "t_type": ...
[ "(let (a, f, g) := answer;\n βˆ‘ k : Set.Icc 0 (2 ^ m - 1), (-(1 : β„€)) ^ (d k) * (k : β„•) ^ m = (-1) ^ m * (a : ℝ) ^ (f.eval (m : ℝ)) * (g.eval m)!)" ]
Determine, with proof, the number of ordered triples $(A_1, A_2, A_3)$ of sets which have the property that \begin{enumerate} \item[(i)] $A_1 \cup A_2 \cup A_3 = \{1,2,3,4,5,6,7,8,9,10\}$, and \item[(ii)] $A_1 \cap A_2 \cap A_3 = \emptyset$. \end{enumerate} Express your answer in the form $2^a 3^b 5^c 7^d$, where $a,b,...
(10, 10, 0, 0)
open Set
answer = (((10 : β„•), (((10 : β„•), (((0 : β„•), (0 : β„•)) : β„• Γ— β„•)) : β„• Γ— β„• Γ— β„•)) : β„• Γ— β„• Γ— β„• Γ— β„•)
β„• Γ— β„• Γ— β„• Γ— β„•
{ "benchmark": "putnam_solving", "index": 93 }
[]
[]
[ "(let (a, b, c, d) := answer;\n {(A1, A2, A3) : Set β„€ Γ— Set β„€ Γ— Set β„€ | A1 βˆͺ A2 βˆͺ A3 = Icc 1 10 ∧ A1 ∩ A2 ∩ A3 = βˆ…}.ncard = 2 ^ a * 3 ^ b * 5 ^ c * 7 ^ d)" ]
Let $d$ be a real number. For each integer $m \geq 0$, define a sequence $\{a_m(j)\}$, $j=0,1,2,\dots$ by the condition \begin{align*} a_m(0) &= d/2^m, \\ a_m(j+1) &= (a_m(j))^2 + 2a_m(j), \qquad j \geq 0. \end{align*} Evaluate $\lim_{n \to \infty} a_n(n)$.
$e^d - 1$
open Set Filter Topology Real
answer = fun (d : ℝ) => ((rexp d : ℝ) - (1 : ℝ) : ℝ)
ℝ β†’ ℝ
{ "benchmark": "putnam_solving", "index": 94 }
[ { "t": "ℝ", "v": null, "name": "d", "t_type": "Type" } ]
[ { "t": "β„• β†’ β„• β†’ ℝ", "v": null, "name": "a", "t_type": "Type" }, { "t": "βˆ€ m : β„•, a m 0 = d / 2 ^ m", "v": null, "name": "ha0", "t_type": "Prop" }, { "t": "βˆ€ m : β„•, βˆ€ j : β„•, a m (j + 1) = (a m j) ^ 2 + 2 * a m j", "v": null, "name": "ha", "t_type": "Prop" } ]
[ "(Tendsto (fun n ↦ a n n) atTop (𝓝 (answer d)))" ]
Define a sequence $\{a_i\}$ by $a_1=3$ and $a_{i+1}=3^{a_i}$ for $i \geq 1$. Which integers between $00$ and $99$ inclusive occur as the last two digits in the decimal expansion of infinitely many $a_i$?
{87}
open Set Filter Topology Real
answer = ({(87 : Fin (100 : β„•))} : Set (Fin (100 : β„•)))
Set (Fin 100)
{ "benchmark": "putnam_solving", "index": 95 }
[]
[ { "t": "β„• β†’ β„•", "v": null, "name": "a", "t_type": "Type" }, { "t": "a 1 = 3", "v": null, "name": "ha1", "t_type": "Prop" }, { "t": "βˆ€ i β‰₯ 1, a (i + 1) = 3 ^ a i", "v": null, "name": "ha", "t_type": "Prop" } ]
[ "({k : Fin 100 | βˆ€ N : β„•, βˆƒ i β‰₯ N, a i % 100 = k} = answer)" ]
Let $I_m = \int_0^{2\pi} \cos(x)\cos(2x)\cdots \cos(mx)\,dx$. For which integers $m$, $1 \leq m \leq 10$ is $I_m \neq 0$?
{3, 4, 7, 8}
open Set Filter Topology Real
answer = (Insert.insert (3 : β„•) (Insert.insert (4 : β„•) (Insert.insert (7 : β„•) ({(8 : β„•)} : Set β„•) : Set β„•) : Set β„•) : Set β„•)
Set β„•
{ "benchmark": "putnam_solving", "index": 96 }
[]
[ { "t": "β„• β†’ ℝ", "v": null, "name": "I", "t_type": "Type" }, { "t": "I = fun (m : β„•) ↦ ∫ x in (0)..(2 * Real.pi), ∏ k in Finset.Icc 1 m, cos (k * x)", "v": null, "name": "hI", "t_type": "Prop" } ]
[ "({m ∈ Finset.Icc 1 10 | I m β‰  0} = answer)" ]
If $p(x)= a_0 + a_1 x + \cdots + a_m x^m$ is a polynomial with real coefficients $a_i$, then set \[ \Gamma(p(x)) = a_0^2 + a_1^2 + \cdots + a_m^2. \] Let $F(x) = 3x^2+7x+2$. Find, with proof, a polynomial $g(x)$ with real coefficients such that \begin{enumerate} \item[(i)] $g(0)=1$, and \item[(ii)] $\Gamma(f(x)^n) = \G...
6x^2 + 5x + 1
open Set Filter Topology Real Polynomial
answer = ((((6 : ℝ[X]) * ((Polynomial.X : ℝ[X]) ^ (2 : β„•) : ℝ[X]) : ℝ[X]) + ((5 : ℝ[X]) * (Polynomial.X : ℝ[X]) : ℝ[X]) : ℝ[X]) + (1 : ℝ[X]) : ℝ[X])
Polynomial ℝ
{ "benchmark": "putnam_solving", "index": 97 }
[]
[ { "t": "Polynomial ℝ β†’ ℝ", "v": null, "name": "Ξ“", "t_type": "Type" }, { "t": "Polynomial ℝ", "v": null, "name": "f", "t_type": "Type" }, { "t": "Ξ“ = fun p ↦ βˆ‘ k in Finset.range (p.natDegree + 1), coeff p k ^ 2", "v": null, "name": "hΞ“", "t_type": "Prop" }, ...
[ "(let g := answer;\n g.eval 0 = 1 ∧ βˆ€ n : β„•, n β‰₯ 1 β†’ Ξ“ (f ^ n) = Ξ“ (g ^ n))" ]
Let $k$ be the smallest positive integer for which there exist distinct integers $m_1, m_2, m_3, m_4, m_5$ such that the polynomial \[ p(x) = (x-m_1)(x-m_2)(x-m_3)(x-m_4)(x-m_5) \] has exactly $k$ nonzero coefficients. Find, with proof, a set of integers $m_1, m_2, m_3, m_4, m_5$ for which this minimum $k$ is achieved.
fun i : Fin 5 ↦ ↑i - (2 : β„€)
open Set Filter Topology Real Polynomial Function
answer = fun (i : Fin (5 : β„•)) => ((↑(↑i : β„•) : β„€) - (2 : β„€) : β„€)
Fin 5 β†’ β„€
{ "benchmark": "putnam_solving", "index": 98 }
[]
[ { "t": "(Fin 5 β†’ β„€) β†’ (Polynomial ℝ)", "v": null, "name": "p", "t_type": "Type" }, { "t": "p = fun m ↦ ∏ i : Fin 5, ((X : Polynomial ℝ) - m i)", "v": null, "name": "hp", "t_type": "Prop" }, { "t": "Polynomial ℝ β†’ β„•", "v": null, "name": "numnzcoeff", "t_type": ...
[ "((Injective answer ∧ βˆ€ m : Fin 5 β†’ β„€, Injective m β†’ numnzcoeff (p answer) ≀ numnzcoeff (p m)))" ]
Define polynomials $f_n(x)$ for $n \geq 0$ by $f_0(x)=1$, $f_n(0)=0$ for $n \geq 1$, and \[ \frac{d}{dx} f_{n+1}(x) = (n+1)f_n(x+1) \] for $n \geq 0$. Find, with proof, the explicit factorization of $f_{100}(1)$ into powers of distinct primes.
99 if $n = 101$, otherwise 0
open Set Filter Topology Real Polynomial Function
answer = fun (n : β„•) => (if n = (101 : β„•) then (99 : β„•) else (0 : β„•) : β„•)
β„• β†’ β„•
{ "benchmark": "putnam_solving", "index": 99 }
[]
[ { "t": "β„• -> Polynomial β„•", "v": null, "name": "f", "t_type": "Prop" }, { "t": "f 0 = 1", "v": null, "name": "hf0x", "t_type": "Prop" }, { "t": "βˆ€ n β‰₯ 1, (f n).eval 0 = 0", "v": null, "name": "hfn0", "t_type": "Prop" }, { "t": "βˆ€ n : β„•, derivative (f (...
[ "(Nat.factorization ((f 100).eval 1) = answer)" ]
Evaluate $\int_0^\infty t^{-1/2}e^{-1985(t+t^{-1})}\,dt$. You may assume that $\int_{-\infty}^\infty e^{-x^2}\,dx = \sqrt{\pi}$.
$\sqrt{\pi / 1985} \cdot e^{-3970}$
open Set Filter Topology Real Polynomial Function
answer = ((√((Ο€ : ℝ) / (1985 : ℝ) : ℝ) : ℝ) * (rexp (-3970 : ℝ) : ℝ) : ℝ)
ℝ
{ "benchmark": "putnam_solving", "index": 100 }
[]
[ { "t": "∫ x in Set.univ, Real.exp (- x ^ 2) = Real.sqrt Real.pi", "v": null, "name": "fact", "t_type": "Prop" } ]
[ "(∫ t in Set.Ioi 0, t ^ (- (1 : ℝ) / 2) * Real.exp (-1985 * (t + t ^ (-(1 : ℝ)))) = answer)" ]
End of preview. Expand in Data Studio

Dataset Card for Formal Problem-Solving Benchmarks

This dataset is part of the official implementation of Beyond Theorem Proving: Formulation, Framework and Benchmark for Formal Problem-Solving, accepted as an ICML 2026 Spotlight paper.

Links:

Overview

The benchmark supports three evaluation settings:

  • Formal Problem-Solving (FPS): Given a formal problem, generate a formal solution. The formal solution should solve all goals and provide a direct answer.
  • Deductive Formal Problem-Solving (D-FPS): Given a formal problem, generate a forward solution and, optionally, a backward proof. The forward solution should derive a direct answer and prove its completeness; the backward proof should prove the answer's soundness.
  • Formal Theorem Proving (FTP): Given a formal problem and its ground-truth answer, generate a formal proof of the ground-truth answer's correctness.

Benchmarks

JSONL File Benchmark Size
formal_math500.jsonl FormalMath500 387
minif2f_solving.jsonl MiniF2F-Solving 375
putnam_solving.jsonl PutnamBench-Solving 324

FormalMath500 is a formalized subset of MATH500 with 387 samples covering Algebra, Intermediate Algebra, Number Theory, Prealgebra, and Precalculus.

MiniF2F-Solving refactors MiniF2F propositions into problem-solving instances with detached direct answers. It contains 375 samples from AIME, MATH-Algebra, AMC, IMO, and MATH-Number Theory.

PutnamBench-Solving refactors a subset of PutnamBench into 324 undergraduate-level problem-solving instances across algebra, analysis, number theory, geometry, linear algebra, combinatorics, abstract algebra, probability, and set theory.

Dataset Structure

Each JSONL record contains the following core fields:

  • informal_problem: The natural-language problem statement, including LaTeX when applicable.
  • informal_answer: The natural-language ground-truth answer, including LaTeX when applicable.
  • header: Lean openings/options used before the formal problem.
  • independent_variables: Problem-level independent variables.
  • hypotheses: Assumptions rendered in the Lean s.t. / iff block before the conclusion.
  • conclusions: Lean propositions that the queried answer must satisfy.
  • formal_answer: The formal ground-truth answer predicate in Lean 4.
  • formal_answer_type: The Lean type of the direct answer.
  • metainfo: Benchmark name and sample index.

The released JSONL files intentionally omit empty draft fields and Lean-only export override fields. The structured JSONL files can be exported to the Lean benchmark project with common/export_fps_benchmark.py in the code repository.

License

This dataset is released under the Apache 2.0 license.

Citation

If you find this dataset useful, please cite:

@inproceedings{
liu2026beyond,
title={Beyond Theorem Proving: Formulation, Framework and Benchmark for Formal Problem-Solving},
author={Qi Liu and Xinhao Zheng and Renqiu Xia and Xingzhi Qi and Qinxiang Cao and Junchi Yan},
booktitle={Forty-third International Conference on Machine Learning},
year={2026},
url={https://openreview.net/forum?id=hgMZraPlSv}
}
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