Formal Problem-Solving
Collection
[ICML'26 Spotlight] Beyond Theorem Proving: Formulation, Framework and Benchmark for Formal Problem-Solving β’ 5 items β’ Updated β’ 3
Datasets:
informal_problem string | informal_answer string | header string | formal_answer string | formal_answer_type string | metainfo dict | independent_variables list | hypotheses list | conclusions list |
|---|---|---|---|---|---|---|---|---|
Find 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 : β β β |
β (a : β) (c : β), a β₯ (0 : β) β§ f = fun (x : β) => (a / (((1 : β) - (c * x : β) : β) ^ (2 : β) : β) : β)} :
Set (β β β)) | Set (β β β) | {
"benchmark": "putnam_solving",
"index": 1
} | [] | [
{
"t": "Set β β (β β β) β Prop",
"v": null,
"name": "P",
"t_type": "Type"
},
{
"t": "β s f, P s f β 0 β€ f β§ β x β s, β¨ t in Ico 0 x, f t = β(f 0 * f x)",
"v": null,
"name": "P_def",
"t_type": "Prop"
}
] | [
"((β f,\n (P (Ioi 0) f β β g β answer, EqOn f g (Ici 0)) β§\n (β e > 0, P (Ioo 0 e) f β β g β answer, EqOn f g (Ico 0 e))) β§\n β f β answer, P (Ioi 0) f β¨ (β 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 = β 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 : β β β) (n : β) (x t : β) =>
((((x - t : β) ^ (n - (1 : β) : β) : β) * (f t : β) : β) / ((β((n - (1 : β) : β)! : β) : β) * (t ^ n : β) : β) : β) | (β β β) β β β β β β β β | {
"benchmark": "putnam_solving",
"index": 3
} | [
{
"t": "β",
"v": null,
"name": "n",
"t_type": "Type"
},
{
"t": "β β β",
"v": null,
"name": "f",
"t_type": "Type"
}
] | [
{
"t": "β β (β β β) β (β β β)",
"v": null,
"name": "P",
"t_type": "Type"
},
{
"t": "P 0 = id β§ β i y, P (i + 1) y = P i (fun x β¦ x * deriv y x - i * y x)",
"v": null,
"name": "hP",
"t_type": "Prop"
},
{
"t": "0 < n",
"v": null,
"name": "hn",
"t_type": "Prop"
... | [
"((β i < n, deriv^[i] y 1 = 0) β§ (Ici 1).EqOn (P n y) f β\n β x β₯ 1, y x = β« t in (1 : β)..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) β£ (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 β",
"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 β (closure S β Set.Ioi (0 : β)))"
] |
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 : β β β | β (A : β) (k : β), (fun (u : β) => (A * (Real.sinh (k * u : β) : β) : β)) = x} : Set (β β β)) βͺ
({x : β β β | β (A : β), (fun (u : β) => (A * u : β)) = x} : Set (β β β)) :
Set (β β β)) βͺ
({x : β β β | β (A : β) (k : β), (fun (u : β) => (A * (Real.sin (k * u : β) : β) :... | Set (β β β) | {
"benchmark": "putnam_solving",
"index": 6
} | [] | [
{
"t": "β β β",
"v": null,
"name": "f",
"t_type": "Type"
}
] | [
"(f β answer β\n (ContDiff β 1 f β§ Differentiable β (deriv f) β§\n β x y : β, (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 : β) => (β
: Set (β β β)) | β β Set (β β β) | {
"benchmark": "putnam_solving",
"index": 7
} | [] | [
{
"t": "β",
"v": null,
"name": "Ξ±",
"t_type": "Type"
}
] | [
"(answer Ξ± = {f : β β β | (β x β Icc 0 1, f x > 0) β§ ContinuousOn f (Icc 0 1) β§ β« x in (0)..1, f x = 1 β§ β« x in (0)..1, x * f x = Ξ± β§ β« 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 : β) / (15 : β) : β) | β | {
"benchmark": "putnam_solving",
"index": 8
} | [] | [
{
"t": "EuclideanSpace β (Fin 2)",
"v": null,
"name": "A",
"t_type": "Type"
},
{
"t": "EuclideanSpace β (Fin 2)",
"v": null,
"name": "B",
"t_type": "Type"
},
{
"t": "EuclideanSpace β (Fin 2)",
"v": null,
"name": "C",
"t_type": "Type"
},
{
"t": "Euclide... | [
"(β 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 β permsOfFinset (Finset.Icc 1 n) | β m β Finset.Icc 2 n, β k β Finset.Ico 1 m, p m = p k + 1 β¨ 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 : β) | β | {
"benchmark": "putnam_solving",
"index": 10
} | [] | [] | [
"(Tendsto (fun n : β β¦ β« x in {x : Fin (n+1) β β | β k : Fin (n+1), x k β Set.Icc 0 1}, (Real.cos (Real.pi/(2 * (n+1)) * β k : Fin (n+1), x k))^2) atTop (π 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 | β P : Polynomial β€, P.degree = 2 β§ (β z1 z2 : Set.Ioo (0 : β) 1, z1 β z2 β§ aeval (z1 : β) P = 0 β§ aeval (z2 : β) P = 0) β§P.coeff 2 = a β§ 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 β) β (Fin 4 β β) β Prop",
"v": null,
"name": "abneq0",
"t_type": "Type"
},
{
"t": "abneq0 = (fun a b : Fin 4 β β => a 0 * b 1 - a 1 * b 0 β 0)",
"v": null,
"name": "habneq0",
"t_type": "Prop"
},
{
"t": "(Fin 4 β β) β (Fin 4 β β) β β",
"v": null,
"... | [
"((β a b : Fin 4 β β, abneq0 a b β§ numtuples a b = answer) β§ (β a b : Fin 4 β β, abneq0 a b β numtuples a b β€ 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 : β) | β | {
"benchmark": "putnam_solving",
"index": 13
} | [] | [
{
"t": "Set β[X]",
"v": null,
"name": "V",
"t_type": "Type"
},
{
"t": "V = {P : β[X] | P.degree = 2 β§ β x β Set.Icc 0 1, |P.eval x| β€ 1}",
"v": null,
"name": "V_def",
"t_type": "Prop"
}
] | [
"(sSup {|(derivative P).eval 0| | P β 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 β§ (β k β Set.Icc 0 P.natDegree, P.coeff k = 1 β¨ P.coeff k = -1) β§\n β z : β, P.eval z = 0 β β r : β, 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 β§ 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 : β} βͺ {Set.Ici x | x : β} βͺ {Set.Iic x | x : β} βͺ {Set.Iio x | x : β} βͺ {Set.Ioi x | x : β} βͺ {Set.univ} | open Matrix Filter Topology Set Nat | answer =
(((((({x : Set β | β (x_1 : β), ({x_1} : Set β) = x} : Set (Set β)) βͺ
({x : Set β | β (x_1 : β), (Set.Ici x_1 : Set β) = x} : Set (Set β)) :
Set (Set β)) βͺ
({x : Set β | β (x_1 : β), (Set.Iic x_1 : Set β) = x} : Set (Set β)) :
Set (S... | Set (Set β) | {
"benchmark": "putnam_solving",
"index": 16
} | [] | [] | [
"(answer = {{z : β | β x : Fin 2 β β, MvPolynomial.eval x f = z} | f : MvPolynomial (Fin 2) β})"
] |
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": "β n, P n β β (G : Type) [Group G] [Finite G],\n β H : Fin n β Subgroup G, (β i, H i < β€) β β i, (H i : Set G) < β€",
"v": null,
"name": "P_def",
"t_type": "Prop"
}
] | [
"(answer β (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": "β n : β, L n β€ (Nat.digits 10 n).length β§\n(β k : β, k < L n β (Nat.digits 10 n)[k]! = (Nat.digits 10 n)[0]!) β§\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":... | [
"((β n : β, (Nat.digits 10 (n^2))[0]! β 0 β§ L (n^2) = answer.1) β§\n(β n : β, (Nat.digits 10 (n^2))[0]! β 0 β L (n^2) β€ answer.1) β§\n(β m : β, m^2 = answer.2) β§\nL (answer.2) = answer.1 β§\n(Nat.digits 10 answer.2)[0]! β 0 β§\nβ n : β, (Nat.digits 10 (n^2))[0]! β 0 β§ 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 : β) * (Real.log (5 : β) : β) : β) - (4 : β) : β) + ((2 : β) * (Real.arctan (2 : β) : β) : β) : β) :
β) | β | {
"benchmark": "putnam_solving",
"index": 19
} | [] | [] | [
"(Tendsto (fun n => 1/(n^4) * β i in Finset.Icc (1 : β€) (2*n), ((n^2 + i^2) : β)^((1 : β)/n)) atTop (π 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 β)} : Set (Polynomial β)) | Set (Polynomial β) | {
"benchmark": "putnam_solving",
"index": 20
} | [] | [
{
"t": "Polynomial β",
"v": null,
"name": "P",
"t_type": "Type"
}
] | [
"(β P : Polynomial β, P β answer β (P.eval 0 = 0 β§ (β x : β, 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 β§ b > 0 β§ a > b",
"v": null,
"name": "hab",
"t_type": "Prop"
},
{
"t": "β€ β β€ β Prop",
"v": null,
"name": "pab",
"... | [
"(pab a b β a = answer.1 β§ 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 : β) =>
((((x ^ (3 : β) : β) - (x ^ (2 : β) : β) : β) - (1 : β) : β) / (((2 : β) * x : β) * (x - (1 : β) : β) : β) : β)} :
Set (β β β)) | Set (β β β) | {
"benchmark": "putnam_solving",
"index": 22
} | [] | [
{
"t": "Set β",
"v": null,
"name": "S",
"t_type": "Type"
},
{
"t": "S = univ \\ {0, 1}",
"v": null,
"name": "hS",
"t_type": "Prop"
},
{
"t": "(β β β) β Prop",
"v": null,
"name": "P",
"t_type": "Type"
},
{
"t": "P = fun (F : β β β) => β x β S, F x + F (... | [
"(β F β answer, P F β§ β f : β β β, P f β β F β answer, (β x β 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 : β β β |
β (A : β) (B : β), β (x : β), x β (Set.Icc (0 : β) (1 : β) : Set β) β (f x : β) = ((A * x : β) + B : β)} :
Set (β β β)) | Set (β β β) | {
"benchmark": "putnam_solving",
"index": 23
} | [] | [
{
"t": "(β β β) β Prop",
"v": null,
"name": "climit_exists",
"t_type": "Type"
},
{
"t": "(β β β) β Prop",
"v": null,
"name": "supercontinuous",
"t_type": "Type"
},
{
"t": "β x, climit_exists x β β C : β, Tendsto (fun n => (β i in Finset.range n, (x i))/(n : β)) atTop (π ... | [
"({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 : β) => (((2 : β) * s : β) / v : β) | β β β β β | {
"benchmark": "putnam_solving",
"index": 24
} | [
{
"t": "β",
"v": null,
"name": "s",
"t_type": "Type"
},
{
"t": "β",
"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": "β β (β β β) β Prop",
"v": null,
"name": "valid",
"t_type": "Type"
},
{
"t": "β t x, valid t x β\n DifferentiableOn β ... | [
"(IsGreatest {t | β x : β β β, 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 β",
"v": null,
"name": "L",
"t_type": "Type"
},
{
"t": "L.length = 8 β§ β i : Fin L.length, L[i] = 1 β¨ L[i] = -1",
"v": null,
"name": "hL",
"t_type": "Prop"
},
{
"t": "β",
"v": null,
"name": "pluses",
"t_type": "Type"
},
{
"t": "pluses = {i... | [
"(answer β ((β l : β, Tendsto S atTop (π 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": "β β β",
"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 : β | 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 : β) => x | β β β | {
"benchmark": "putnam_solving",
"index": 27
} | [] | [] | [
"(ContDiff β 1 answer β§ (β x : β, 0 < deriv answer x β§ deriv answer x β€ 1) β§ answer 0 = 0 β§ (β« x in Icc 0 1, answer x)^2 = β« 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": "β S, conspiratorial S β β a β S, β b β S, β c β S, (a > 0 β§ b > 0 β§ c > 0) β§ ((a β b β§ b β c β§ a β c) β (Int.gcd a b > 1 β¨ Int.gcd b c > 1 β¨ Int.gcd a c > 1))",
"v": null,
"name": "hconspiratorial... | [
"(IsGreatest {k | β S, S β Icc 1 16 β§ conspiratorial S β§ 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 β§ p β‘ 1 [MOD 8]}, {p : β | p.Prime β§ p β‘ 5 [MOD 8]}) | open Set | answer =
((({p : β | Nat.Prime p β§ p β‘ (1 : β) [MOD (8 : β)]} : Set β),
({p : β | Nat.Prime p β§ p β‘ (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": "β p : β, p.Prime β§ p > 2 β ((β m n : β€, p = m^2 + n^2) β p β‘ 1 [MOD 4])",
"v": null,
"name": "h_assumption",
"t_type": "Prop"
}
] | [
"(β p : β,\n ((p.Prime β§ p > 2 β§ (β x y : β€, p = x^2 + 16*y^2)) β p β answer.1) β§\n ((p.Prime β§ p > 2 β§ (β x y : β€, p = 4*x^2 + 4*x*y + 5*y^2)) β p β answer.2))"
] |
Evaluate in closed form: $\frac{1}{2^{n-1}} \sum_{k < n/2} (n-2k)*{n \choose k}$. | (fun n β¦ (1 : β) / ((2 : β) ^ ((n :β) - 1)) * (n * (n - 1).choose βn / 2ββ)) | open Set Nat | answer = fun (n : β) =>
(((1 : β) / ((2 : β) ^ (n - (1 : β) : β) : β) : β) *
((βn : β) * (β((n - (1 : β) : β).choose (β(n / (2 : β) : β)ββ : β) : β) : β) : β) :
β) | β β β | {
"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) / (2 ^ (n - 1)) * β k in Finset.Icc 0 βn / 2ββ, (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 β§ (β x : β€, (10^6 : β€) β£ f.eval x)",
"v": null,
"name": "hdivnallx_def",
"t_type": "Prop"
}
] | [
"(sInf {d : β | β f : Polynomial β€, hdivnallx f β§ 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))/2 + (b^2 + (b : β))/2",
"v": null,
"name": "hnab",
"t_type"... | [
"((β n a b : β€, nab (n, a, b) β nxy (n, answer.1 (a, b), answer.2 (a, b))) β§ β n : β€, (β x y : β€, nxy (n, x, y)) β β 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 | β (b : β) (c : β), answer ((b, c) : β Γ β) = (c < (1 : β) β§ (c - b : β) > (-1 : β) β§ (c + b : β) > (-1 : β)) | (β Γ β) β Prop | {
"benchmark": "putnam_solving",
"index": 34
} | [] | [] | [
"(β (b : β) (c : β), (β z : β, (Polynomial.X^2 + (Polynomial.C (b : β)) * Polynomial.X + (Polynomial.C (c : β)) : Polynomial β).eval z = 0 β βzβ < 1) β 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 : β Γ β Γ β) =>
match x with
| ((a, ((b, c) : β Γ β)) : β Γ β Γ β) =>
((((a / b : β) ^ ((1 : β) / (b - a : β) : β) : β),
(((((1 : β) - (((a / b : β) ^ ((1 : β) / (b - a : β) : β) : β) ^ b : β) : β) ^ ((1 : β) / b : β) : β),
(0 : β)) :
β Γ... | ((β Γ β Γ β) β (β Γ β Γ β)) Γ ((β Γ β Γ β) β (β Γ β Γ β)) | {
"benchmark": "putnam_solving",
"index": 35
} | [
{
"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": "0 < a β§ a < b β§ b < c",
"v": null,
"name": "hi",
"t_type": "Prop"
},
{
"t": "(β Γ β Γ β) β Prop",
"v": null,
"name": "P",
"t_type": "Type"
},
{
"t": "(β Γ β Γ β) β β",
"v": null,
"name": "f",
"t_type": "Type"
},
{
"t": "P = fun (x, y, z) => x β₯... | [
"((P (answer.1 (a, b, c)) β§ β x y z : β, P (x, y, z) β\nf (x, y, z) β€ f (answer.1 (a, b, c))) β§\n(P (answer.2 (a, b, c)) β§ β x y z : β, 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) =>
β j β (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 β§ 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€) | β u v w : β€, h = (u*3 + v*4 + w*5, u*8 + v*(-1) + w*4)}",
"v": null,
"name": "hH",
"t_type": "Prop"
}
] | [
"((β b : β€, H = {h : (β€ Γ β€) | β u v : β€, h = (u, u*b + v*answer)}) β§ 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: β)/(Nat.factorial k) | open Polynomial Real Complex Matrix Filter Topology Multiset | answer = fun (k : β) => ((1 : β) / (β(k.factorial : β) : β) : β) | β β β | {
"benchmark": "putnam_solving",
"index": 38
} | [] | [] | [
"(β k : β, k > 0 β (β a : Multiset β, (β i β a, i > 0) β§ card a β₯ k β\n(esymm a k)/(esymm a 1)^k β€ answer k) β§\nβ M : β, M < answer k β (β a : Multiset β, (β i β a, i > 0) β§ card a β₯ k β§\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": "β Γ β 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"
}
] | [
"((β B β setOf P, IsClosed B β§ β x y : β, P (x, y) β Xor' ((x, y) β B) ((-x, -y) β B)) β 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 β§ Nat.Prime a.2.2.1 β§ a.2.1 > 1 β§ a.2.2.2 > 1 β§ |(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 : β) => ((-1 : β) / (r + (1 : β) : β) : β), fun (r : β) => ((-(r + (1 : β) : β) : β) / r : β)) :
(β β β) Γ (β β β)) | (β β β) Γ (β β β) | {
"benchmark": "putnam_solving",
"index": 41
} | [
{
"t": "β",
"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": "... | [
"(β s : β, Polynomial.aeval s P = 0 β§ (s = answer.1 r β¨ 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).$$ Your answer should be in the form $\ln(a) - b$, where $a$ and $b$ are positive integers. | ln(4) - 1 | open Polynomial Filter Topology | answer = (((4 : β), (1 : β)) : β Γ β) | β Γ β | {
"benchmark": "putnam_solving",
"index": 42
} | [] | [] | [
"(Tendsto (fun n : β => ((1 : β)/n)*β k in Finset.Icc (1 : β€) n, (Int.floor ((2*n)/k) - 2*Int.floor (n/k))) atTop\n (π (Real.log answer.1 - answer.2)))"
] |
Let $G$ be a group generated by two elements $A$ and $B$; i.e., every element of $G$ can be expressed as a finite word $A^{n_1}B^{n_2} \cdots A^{n_{k-1}}B^{n_k}$, where the $n_i$ can assume any integer values and $A^0 = B^0 = 1$. Further assume that $A^4 = B^7 = ABA^{-1}B = 1$, but $A^2 \ne 1$ and $B \ne 1$. Find the n... | (8, {[(0, 0)], [(2, 0)], [(0, 1)], [(0, 2)], [(0, 3)], [(0, 4)], [(0, 5)], [(0, 6)]}) | open Polynomial Filter Topology | answer =
(((8 : β),
(Insert.insert ((((0 : β€), (0 : β€)) : β€ Γ β€) :: ([] : List (β€ Γ β€)) : List (β€ Γ β€))
(Insert.insert ((((2 : β€), (0 : β€)) : β€ Γ β€) :: ([] : List (β€ Γ β€)) : List (β€ Γ β€))
(Insert.insert ((((0 : β€), (1 : β€)) : β€ Γ β€) :: ([] : List (β€ Γ β€)) : List (β€ Γ β€))
... | β Γ Set (List (β€ Γ β€)) | {
"benchmark": "putnam_solving",
"index": 43
} | [] | [
{
"t": "Type*",
"v": null,
"name": "G",
"t_type": "Type"
},
{
"t": "Group G",
"v": null,
"name": null,
"t_type": "Type"
},
{
"t": "G",
"v": null,
"name": "A",
"t_type": "Type"
},
{
"t": "G",
"v": null,
"name": "B",
"t_type": "Type"
},
{... | [
"(S.ncard = answer.1 β§ S = {word w | w β answer.2})"
] |
Find $$\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]) (β(n.factorial : β) : β€) : β€[X]) | β β Polynomial β€ | {
"benchmark": "putnam_solving",
"index": 44
} | [] | [] | [
"(β n : β, β 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 : β) | β | {
"benchmark": "putnam_solving",
"index": 45
} | [] | [
{
"t": "β β β",
"v": null,
"name": "y",
"t_type": "Type"
},
{
"t": "y = fun x β¦ 2 * x ^ 4 + 7 * x ^ 3 + 3 * x - 5",
"v": null,
"name": "hy",
"t_type": "Prop"
},
{
"t": "Finset β",
"v": null,
"name": "S",
"t_type": "Type"
},
{
"t": "S.card = 4",
"v"... | [
"((β 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 : β) => d = a β§ b = (-c : β) β¨ d = b β§ a = (-c : β) β¨ d = c β§ a = (-b : β) | β β β β β β β β Prop | {
"benchmark": "putnam_solving",
"index": 46
} | [
{
"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"
}
] | [] | [
"(answer a b c d β\n a β 0 β b β 0 β c β 0 β d β 0 β (a + b + c = d β§ 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 : β β β) (x : β) =>
(((((g x : β) - (f (x - (3 : β) : β) : β) : β) + (f (x - (1 : β) : β) : β) : β) + (f (x + (1 : β) : β) : β) : β) -
(f (x + (3 : β) : β) : β) :
β) | (β β β) β (β β β) β (β β β) | {
"benchmark": "putnam_solving",
"index": 47
} | [
{
"t": "β β β",
"v": null,
"name": "f",
"t_type": "Type"
},
{
"t": "β β β",
"v": null,
"name": "g",
"t_type": "Type"
}
] | [
{
"t": "β β β",
"v": null,
"name": "h",
"t_type": "Type"
},
{
"t": "β x, f x = (h (x + 1) + h (x - 1)) / 2",
"v": null,
"name": "hf",
"t_type": "Prop"
},
{
"t": "β 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 β) / ((1 : RatFunc β) - (RatFunc.X : RatFunc β) : RatFunc β) : RatFunc β) | RatFunc β | {
"benchmark": "putnam_solving",
"index": 48
} | [] | [
{
"t": "β",
"v": null,
"name": "x",
"t_type": "Type"
},
{
"t": "x β Ioo 0 1",
"v": null,
"name": "hx",
"t_type": "Prop"
}
] | [
"(answer.eval (id β) x = β' 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 (β Γ β)",
"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": "(β Γ β) β β",
"v": null,
"name": "room",
"t_type": "Type"
},
{
"t": "room = fun (a,b) β¦ mi... | [
"((β f : (β Γ β) β β, Continuous f β (β P β X, β« x in (P.1 - room P)..(P.1 + room P), β« y in (P.2 - room P)..(P.2 + room P), f (x, y) = 0) β (β P β X, f P = 0)) β answer)"
] |
Find $\prod_{n=2}^{\infty} \frac{(n^3 - 1)}{(n^3 + 1)}$. | 2/3 | open RingHom Set Nat Filter Topology | answer = (2 / 3 : β) | β | {
"benchmark": "putnam_solving",
"index": 50
} | [] | [] | [
"(Tendsto (fun N β¦ β n in Finset.Icc (2 : β€) N, ((n : β) ^ 3 - 1) / (n ^ 3 + 1)) atTop (π 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": "β Γ β Γ β β Prop",
"v": null,
"name": "P",
"t_type": "Type"
},
{
"t": "β Γ β Γ β β Prop",
"v": null,
"name": "balanced",
"t_type": "Type"
},
{
"t": "β Γ β Γ β β β Γ β Γ β",
"v": null,
"name": "B",
"t_type": "Type"
},
{
"t": "P = fun (a, b, c) =... | [
"((β t : β Γ β Γ β, P t β β n : β, balanced (B^[n] t)) β 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 β",
"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 | β k β 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) | β | {
"benchmark": "putnam_solving",
"index": 53
} | [] | [] | [
"((β' i : β+, β' j : β+, (1 : β) / (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 : β[X]) * ((Polynomial.X : β[X]) ^ (4 : β) : β[X]) : β[X]) -
((4 : β[X]) * ((Polynomial.X : β[X]) ^ (2 : β) : β[X]) : β[X]) :
β[X]) +
(1 : β[X]) :
β[X]) | Polynomial β | {
"benchmark": "putnam_solving",
"index": 54
} | [] | [
{
"t": "Set (Polynomial β)",
"v": null,
"name": "S",
"t_type": "Type"
},
{
"t": "S = {p : Polynomial β | p.degree = 4 β§ β x β Icc (-1 : β) 1, p.eval x β Icc 0 1}",
"v": null,
"name": "hS",
"t_type": "Prop"
}
] | [
"(answer β S β§ (β p β S, p.coeff 4 β€ 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": "β a, P a β Multiset.card a > 0 β§ (β i β a, i > 0) β§ a.sum = 1979",
"v": null,
"name": "hP",
"t_type": "Prop"
}
] | [
"(P answer β§ β 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 : β) => k β₯ (0 : β) | β β Prop | {
"benchmark": "putnam_solving",
"index": 56
} | [
{
"t": "β",
"v": null,
"name": "k",
"t_type": "Type"
}
] | [] | [
"(answer k β β f : β β β, Continuous f β§ β x : β, 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) => β m : β€, a = m β§ b = m | answer = fun (x : β Γ β) =>
match x with
| ((a, b) : β Γ β) => β (m : β€), a = (βm : β) β§ b = (βm : β) | (β Γ β) β Prop | {
"benchmark": "putnam_solving",
"index": 57
} | [
{
"t": "β β β",
"v": null,
"name": "x",
"t_type": "Type"
}
] | [
{
"t": "β n : β, x n β 0 β§ (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"
}
] | [
"((β m : β, β n : β, n > m β§ β a : β€, a = x n) β 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 β) Γ Finset (Fin 2 β β) β Prop",
"v": null,
"name": "A",
"t_type": "Type"
},
{
"t": "A = fun (R, B) => R.card = B.card β§ R β© B = β
β§\n β u : Finset (Fin 2 β β), u β R βͺ B β u.card = 3 β Β¬Collinear β (u : Set (Fin 2 β β))",
"v": null,
"name": "hA",
"t_ty... | [
"(answer β\n (β R B, A (R, B) β\n β v : Finset ((Fin 2 β β) Γ (Fin 2 β β)),\n (β L β v, β M β v, L β M β β x β Icc 0 1, β y β 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) β§\n v.card = R.card β§ β L β v, L.1 β R β§ 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 : β Γ β) =>
match x with
| ((a, b) : β Γ β) =>
((Real.exp (-1 : β) : β) * (((b ^ b : β) / (a ^ a : β) : β) ^ ((1 : β) / (b - a : β) : β) : β) : β) | β Γ β β β | {
"benchmark": "putnam_solving",
"index": 59
} | [] | [] | [
"(β a b : β, 0 < a β§ a < b β Tendsto (fun t : β => (β« x in Icc 0 1, (b*x + a*(1 - x))^t)^(1/t)) (π[β ] 0) (π (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 β¦ (n - (1 : β€)) / 2 | open Set Topology Filter Polynomial | answer = fun (n : β) => (((βn : 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 β§ s > 0",
"v": null,
"name": "rspos",
"t_type": "Prop"
},
{
"t": "β a b c d : β, abcdlcm a b c d β\n (a > 0 β§ b > 0 β§ c > 0 β§ d > 0 β§\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 : β) / (4 : β) : β) | β | {
"benchmark": "putnam_solving",
"index": 62
} | [] | [] | [
"(answer = β« 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 : β) / (rexp (1 : β) : β) : β) | β | {
"benchmark": "putnam_solving",
"index": 63
} | [] | [
{
"t": "Set (β β β)",
"v": null,
"name": "C",
"t_type": "Type"
},
{
"t": "C = {f : β β β | ContDiffOn β 1 f (Set.Icc 0 1) β§ f 0 = 0 β§ f 1 = 1}",
"v": null,
"name": "hC",
"t_type": "Prop"
}
] | [
"(IsGreatest {u : β | β f β C, u β€ (β« 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 : β | c β₯ (1 / 2 : β)} : Set β) | Set β | {
"benchmark": "putnam_solving",
"index": 64
} | [
{
"t": "β",
"v": null,
"name": "c",
"t_type": "Type"
}
] | [] | [
"(β x : β, (exp x + exp (-x)) / 2 β€ exp (c * x ^ 2) β c β 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 : β | a β₯ (3 : β)} : Set β) | Set β | {
"benchmark": "putnam_solving",
"index": 65
} | [
{
"t": "β",
"v": null,
"name": "a",
"t_type": "Type"
}
] | [
{
"t": "β β β",
"v": null,
"name": "u",
"t_type": "Type"
},
{
"t": "u 0 = a β§ (β n : β, u (n + 1) = 2 * u n - n ^ 2)",
"v": null,
"name": "hu",
"t_type": "Prop"
}
] | [
"((β n : β, u n > 0) β a β 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 $$f(1) - 2f\left(\frac{2}{3}\right) + f\left(\frac{1}{3}\right) \ge t\le... | $t \le 1$ | open Set | answer = fun (t : β) => t β€ (1 : β) | β β Prop | {
"benchmark": "putnam_solving",
"index": 66
} | [
{
"t": "β",
"v": null,
"name": "t",
"t_type": "Type"
}
] | [
{
"t": "Set β",
"v": null,
"name": "T",
"t_type": "Type"
},
{
"t": "T = Icc 0 1",
"v": null,
"name": "hT",
"t_type": "Prop"
},
{
"t": "β β (β β β) β Prop",
"v": null,
"name": "P",
"t_type": "Type"
},
{
"t": "(β β β) β Prop",
"v": null,
"name": ... | [
"(answer t β (β f β S t, β g β S t, f * g β S t))"
] |
Let $E(n)$ be the greatest integer $k$ such that $5^k$ divides $1^1 2^2 3^3 \cdots n^n$. Find $\lim_{n \rightarrow \infty} \frac{E(n)}{n^2}$. | $\frac{1}{8}$ | open Topology Filter Set Polynomial Function | answer = (1 / 8 : β) | β | {
"benchmark": "putnam_solving",
"index": 67
} | [] | [
{
"t": "β β β β Prop",
"v": null,
"name": "P",
"t_type": "Type"
},
{
"t": "β n k, P n k β 5^k β£ β m in Finset.Icc 1 n, (m^m : β€)",
"v": null,
"name": "hP",
"t_type": "Prop"
},
{
"t": "β β β",
"v": null,
"name": "E",
"t_type": "Type"
},
{
"t": "β n β Ic... | [
"(Tendsto (fun n : β => ((E n) : β)/n^2) atTop (π answer))"
] |
Does the limit $$lim_{t \rightarrow \infty}e^{-t}\int_{0}^{t}\int_{0}^{t}\frac{e^x - e^y}{x - y} dx dy$$exist? | False | open Topology Filter Set Polynomial Function | answer = False | Prop | {
"benchmark": "putnam_solving",
"index": 68
} | [] | [
{
"t": "β β β",
"v": null,
"name": "f",
"t_type": "Type"
},
{
"t": "f = fun t : β => Real.exp (-t) * β« y in (Ico 0 t), β« x in (Ico 0 t), (Real.exp x - Real.exp y) / (x - y)",
"v": null,
"name": "hf",
"t_type": "Prop"
}
] | [
"((β L : β, Tendsto f atTop (π L)) β answer)"
] |
Let $P(x)$ be a polynomial with real coefficients; let $$Q(x) = (x^2 + 1)P(x)P'(x) + x((P(x))^2 + (P'(x))^2).$$
Given that $P$ has $n$ distinct real roots all greater than $1$, prove or disprove that $Q$ must have at least $2n - 1$ distinct real roots. | True | open Topology Filter Set Polynomial Function | answer = True | Prop | {
"benchmark": "putnam_solving",
"index": 69
} | [] | [
{
"t": "Polynomial β β Polynomial β",
"v": null,
"name": "Q",
"t_type": "Type"
},
{
"t": "Q = fun P : Polynomial β => (Polynomial.X^2 + 1) * P * (Polynomial.derivative P) + Polynomial.X * (P^2 + (Polynomial.derivative P)^2)",
"v": null,
"name": "hQ",
"t_type": "Prop"
},
{
... | [
"(answer β (β P : Polynomial β, {x : β | (Q P).eval x = 0}.ncard β₯ 2*(n P) - 1))"
] |
Find the value of $$\lim_{n \rightarrow \infty} \frac{1}{n^5}\sum_{h=1}^{n}\sum_{k=1}^{n}(5h^4 - 18h^2k^2 + 5k^4).$$ | -1 | open Topology Filter Set Polynomial Function | answer = (-1 : β) | β | {
"benchmark": "putnam_solving",
"index": 70
} | [] | [
{
"t": "β β β",
"v": null,
"name": "f",
"t_type": "Type"
},
{
"t": "f = fun n : β => ((1 : β)/n^5) * β h in Finset.Icc 1 n, β k in Finset.Icc 1 n, (5*(h : β)^4 - 18*h^2*k^2 + 5*k^4)",
"v": null,
"name": "hf",
"t_type": "Prop"
}
] | [
"(Tendsto f atTop (π answer))"
] |
Determine the minimum value attained by $$(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 : β) - ((8 : β) * (β(2 : β) : β) : β) : β) | β | {
"benchmark": "putnam_solving",
"index": 71
} | [] | [
{
"t": "β Γ β Γ β β Prop",
"v": null,
"name": "P",
"t_type": "Type"
},
{
"t": "P = fun (r, s, t) => 1 β€ r β§ r β€ s β§ s β€ t β§ t β€ 4",
"v": null,
"name": "hP",
"t_type": "Prop"
},
{
"t": "β Γ β Γ β β β",
"v": null,
"name": "f",
"t_type": "Type"
},
{
"t": ... | [
"(IsLeast {y | β r s t, P (r, s, t) β§ 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) β) β Prop",
"v": null,
"name": "VAB",
"t_type": "Type"
},
{
"t": "Set (Matrix (Fin 5) (Fin 7) β) β β β Prop",
"v": null,
"name": "Vrank",
"t_type": "Type"
},
{
"t": "β V, VAB V = (β A β V, β B β V, β r s : β, r β’ A + s β’ B β V)",
"v... | [
"(answer β\n (β 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": "β bits : List β, sumbits bits = β i : Fin bits.length, (bits[i] : β€)",
"v": null,
"name": "hsumbits",
"t_type": "Prop"
... | [
"(answer β (β q : β, Real.exp (β' n : Set.Ici 1, B n / ((n : β) * ((n : β) + 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 β β β",
"v": null,
"name": "B",
"t_type": "Type"
},
{
"t": "B = fun (n : β) (x : β) β¦ β k in Finset.Icc 1 n, (k : β) ^ x",
"v": null,
"name": "hB",
"t_type": "Prop"
},
{
"t": "β β β",
"v": null,
"name": "f",
"t_type": "Type"
},
{
"t": "f = ... | [
"(answer β (β L : β, Tendsto (fun N β¦ β j in Finset.Icc 2 N, f j) atTop (π 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 : β) / (2 : β) : β) * (Real.log (Ο : β) : β) : β) | β | {
"benchmark": "putnam_solving",
"index": 75
} | [] | [] | [
"(Tendsto (fun t β¦ β« x in (0)..t, (arctan (Real.pi * x) - arctan x) / x) atTop (π 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 β",
"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 => β k in Finset.Icc 1 n, x (b k)) atTop (π 1)) β 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 : β β+* β[X]) : β β β[X]) (Ο : β) : β[X]) *
((((2 : β[X]) * (Polynomial.X : β[X]) : β[X]) - (1 : β[X]) : β[X]) ^ (2 : β) : β[X]) :
β[X]) | Polynomial β | {
"benchmark": "putnam_solving",
"index": 77
} | [] | [
{
"t": "β Γ β β β",
"v": null,
"name": "A",
"t_type": "Type"
},
{
"t": "β",
"v": null,
"name": "g",
"t_type": "Type"
},
{
"t": "β",
"v": null,
"name": "I",
"t_type": "Type"
},
{
"t": "A = fun (x, y) => {a : β€ Γ β€ | a.1^2 + a.2^2 β€ 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 : β) * ((β(2 : β) : β) - (1 : β) : β) : β) | β | {
"benchmark": "putnam_solving",
"index": 78
} | [] | [
{
"t": "β β β",
"v": null,
"name": "p",
"t_type": "Type"
},
{
"t": "p = fun n : β => ({c : Finset.Icc 1 n Γ Finset.Icc 1 n | β m : β, m^2 = c.1 + c.2}.ncard : β) / n^2",
"v": null,
"name": "hp",
"t_type": "Prop"
}
] | [
"(Tendsto (fun n : β => p n * Real.sqrt n) atTop (π 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": "β n, P n β n.Nonempty β§ β k, β i in n, i β£ β i in n, (i + k)",
"v": null,
"name": "P_def",
"t_type": "Prop"
}
] | [
"(((β n, P n β 1 β n β¨ -1 β n) β answer.1) β§\n ((β n, P n β (β i β n, 0 < i) β n = Finset.Icc (1 : β€) n.card) β 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 β§ (n β£ 10 ^ 40 β¨ n β£ 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 β (β Ξ± : β, Ξ± > 0 β§ β n : β, n > 0 β Even (βΞ± ^ nβ - 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 : β) | β | {
"benchmark": "putnam_solving",
"index": 82
} | [] | [
{
"t": "β β β",
"v": null,
"name": "F",
"t_type": "Type"
},
{
"t": "F = fun a β¦ (a ^ 4 / exp (a ^ 3)) * β« x in (0)..a, β« y in (0)..(a - x), exp (x ^ 3 + y ^ 3)",
"v": null,
"name": "hF",
"t_type": "Prop"
}
] | [
"(Tendsto F atTop (π 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+) β¦\n Set.ncard {M : Multiset β |\n (β m β M, β k : β, m = (2 ^ k : β€)) β§\n (β m β M, M.count m β€ 3) β§\n (M.sum : β€) = n}",
"v": null,
"name": "hf",
"t_type": "Prop"
}
... | [
"(answer β (β p : Polynomial β, β n : β+, βp.eval (n : β)β = 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 : β) / (Ο : β) : β) : β) | β | {
"benchmark": "putnam_solving",
"index": 84
} | [] | [
{
"t": "β β β",
"v": null,
"name": "dist_fun",
"t_type": "Type"
},
{
"t": "dist_fun = fun (x : β) β¦ min (x - βxβ) (βxβ - x)",
"v": null,
"name": "hdist_fun",
"t_type": "Prop"
},
{
"t": "Tendsto (fun N β¦ β n in Finset.Icc 1 N, (2 * n / (2 * n - 1)) * (2 * n / (2 * n + 1)) ... | [
"(Tendsto (fun n β¦ (1 / n) * β« x in (1)..n, dist_fun (n / x) : β β β) atTop (π 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) | β | {
"benchmark": "putnam_solving",
"index": 85
} | [] | [] | [
"(β' 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)) β) ^ (2 : β) : MvPolynomial (Fin (3 : β)) β) *
(((MvPolynomial.X (0 : Fin (3 : β)) : MvPolynomial (Fin (3 : β)) β) ^ (2 : β) : MvPolynomial (Fin (3 : β)) β) -
((MvPolynomial.X (1 : Fin (3 : β)) : MvPolynomial (Fin (3 : β)) β) ^ ... | MvPolynomial (Fin 3) β | {
"benchmark": "putnam_solving",
"index": 86
} | [] | [
{
"t": "β",
"v": null,
"name": "n",
"t_type": "Type"
},
{
"t": "β",
"v": null,
"name": "a",
"t_type": "Type"
},
{
"t": "β",
"v": null,
"name": "b",
"t_type": "Type"
},
{
"t": "β β Matrix (Fin (2 * n)) (Fin (2 * n)) β",
"v": null,
"name": "Mn",
... | [
"(Tendsto (fun x : β => (Mn x).det / (x - a) ^ (2 * n - 2)) (π[β ] a) (π (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 β)",
"v": null,
"name": "R",
"t_type": "Type"
},
{
"t": "(Fin 3 β β) β β",
"v": null,
"name": "w",
"t_type": "Type"
},
{
"t": "R = {p | (β i : Fin 3, p i β₯ 0) β§ p 0 + p 1 + p 2 β€ 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": "β n, some (f n) = (Nat.digits 10 (n !)).find? (fun d β¦ 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... | [
"(β g : β β β,\n (βα΅ (k > 0) (a : Fin k β β) (ha : Injective a), f (β i, 5 ^ (a i)) = g (β i, a i)) β§\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 β) + (3 : Polynomial β) : Polynomial β),
((-(Polynomial.X : Polynomial β) : Polynomial β) - (2 : Polynomial β) : Polynomial β)) :
Polynomial β Γ Polynomial β) | Polynomial β Γ Polynomial β | {
"benchmark": "putnam_solving",
"index": 89
} | [] | [
{
"t": "β β β€",
"v": null,
"name": "f",
"t_type": "Type"
},
{
"t": "β n > 0, f n = β i : Set.Icc 1 n, ((i)! : β€)",
"v": null,
"name": "hf",
"t_type": "Prop"
}
] | [
"(β n β₯ 1, f (n + 2) = (answer.1).eval (n : β) * f (n + 1) + (answer.2).eval (n : β) * 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 : β) | β | {
"benchmark": "putnam_solving",
"index": 90
} | [] | [
{
"t": "β β β β β",
"v": null,
"name": "f",
"t_type": "Type"
},
{
"t": "β u v : β, f u v = (u - v) ^ 2 + (Real.sqrt (2 - u ^ 2) - 9 / v) ^ 2",
"v": null,
"name": "hf",
"t_type": "Prop"
}
] | [
"(IsLeast {y | βα΅ (u : Set.Ioo 0 β2) (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
} | [] | [] | [
"((β (F : Type*) (_ : Fintype F), Fintype.card F β₯ 2 β (β mul : F β F β F, β x y z : F, (mul x z = mul y z β x = y) β§ (mul x (mul y z) β mul (mul x y) z))) β 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 β) * ((Polynomial.X : Polynomial β) - (1 : Polynomial β) : Polynomial β) :
Polynomial β) /
(2 : Polynomial β) :
Polynomial β),
(Polynomial.X : Polynomial β)) :
Polynomial β Γ Polynomial β)) :
β€ Γ P... | β€ Γ Polynomial β 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 β k : Set.Icc 0 (2 ^ m - 1), (-(1 : β€)) ^ (d k) * (k : β) ^ m = (-1) ^ m * (a : β) ^ (f.eval (m : β)) * (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 βͺ A2 βͺ A3 = Icc 1 10 β§ A1 β© A2 β© A3 = β
}.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 : β) => ((rexp d : β) - (1 : β) : β) | β β β | {
"benchmark": "putnam_solving",
"index": 94
} | [
{
"t": "β",
"v": null,
"name": "d",
"t_type": "Type"
}
] | [
{
"t": "β β β β β",
"v": null,
"name": "a",
"t_type": "Type"
},
{
"t": "β m : β, a m 0 = d / 2 ^ m",
"v": null,
"name": "ha0",
"t_type": "Prop"
},
{
"t": "β m : β, β j : β, a m (j + 1) = (a m j) ^ 2 + 2 * a m j",
"v": null,
"name": "ha",
"t_type": "Prop"
}
] | [
"(Tendsto (fun n β¦ a n n) atTop (π (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": "β i β₯ 1, a (i + 1) = 3 ^ a i",
"v": null,
"name": "ha",
"t_type": "Prop"
}
] | [
"({k : Fin 100 | β N : β, β 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 β",
"v": null,
"name": "I",
"t_type": "Type"
},
{
"t": "I = fun (m : β) β¦ β« x in (0)..(2 * Real.pi), β k in Finset.Icc 1 m, cos (k * x)",
"v": null,
"name": "hI",
"t_type": "Prop"
}
] | [
"({m β 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 : β[X]) * ((Polynomial.X : β[X]) ^ (2 : β) : β[X]) : β[X]) + ((5 : β[X]) * (Polynomial.X : β[X]) : β[X]) :
β[X]) +
(1 : β[X]) :
β[X]) | Polynomial β | {
"benchmark": "putnam_solving",
"index": 97
} | [] | [
{
"t": "Polynomial β β β",
"v": null,
"name": "Ξ",
"t_type": "Type"
},
{
"t": "Polynomial β",
"v": null,
"name": "f",
"t_type": "Type"
},
{
"t": "Ξ = fun p β¦ β 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 β§ β 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 β¦ βi - (2 : β€) | open Set Filter Topology Real Polynomial Function | answer = fun (i : Fin (5 : β)) => ((β(βi : β) : β€) - (2 : β€) : β€) | Fin 5 β β€ | {
"benchmark": "putnam_solving",
"index": 98
} | [] | [
{
"t": "(Fin 5 β β€) β (Polynomial β)",
"v": null,
"name": "p",
"t_type": "Type"
},
{
"t": "p = fun m β¦ β i : Fin 5, ((X : Polynomial β) - m i)",
"v": null,
"name": "hp",
"t_type": "Prop"
},
{
"t": "Polynomial β β β",
"v": null,
"name": "numnzcoeff",
"t_type": ... | [
"((Injective answer β§ β m : Fin 5 β β€, Injective m β numnzcoeff (p answer) β€ 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": "β n β₯ 1, (f n).eval 0 = 0",
"v": null,
"name": "hfn0",
"t_type": "Prop"
},
{
"t": "β 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 = ((β((Ο : β) / (1985 : β) : β) : β) * (rexp (-3970 : β) : β) : β) | β | {
"benchmark": "putnam_solving",
"index": 100
} | [] | [
{
"t": "β« x in Set.univ, Real.exp (- x ^ 2) = Real.sqrt Real.pi",
"v": null,
"name": "fact",
"t_type": "Prop"
}
] | [
"(β« t in Set.Ioi 0, t ^ (- (1 : β) / 2) * Real.exp (-1985 * (t + t ^ (-(1 : β)))) = 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:
- Paper: https://openreview.net/forum?id=hgMZraPlSv
- Project: https://github.com/Purewhite2019/formal_problem_solving_main
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 Leans.t./iffblock 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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