informal_problem string | informal_answer string | header string | formal_answer string | formal_answer_type string | metainfo dict | independent_variables list | hypotheses list | conclusions list |
|---|---|---|---|---|---|---|---|---|
Let $C = \{(x, y) \in \mathbb{R}^2 : x^2 + y^2 = 1\}$ 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 × ℝ → 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"
}
] | [
"((∃ B ⊆ setOf P, IsClosed B ∧ ∀ x y : ℝ, P (x, y) → 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 : ℕ), (((2 : ℕ), (((2 : ℕ), (3 : ℕ)) : ℕ × ℕ)) : ℕ × ℕ × ℕ)) : ℕ × ℕ × ℕ × ℕ)
({(((2 : ℕ), (((3 : ℕ), (((3 : ℕ), (2 : ℕ)) : ℕ × ℕ)) : ℕ × ℕ × ℕ)) : ℕ × ℕ × ℕ × ℕ)} : Set (ℕ × ℕ × ℕ × ℕ)) :
Set (ℕ × ℕ × ℕ × ℕ)) | Set (ℕ × ℕ × ℕ × ℕ) | {
"benchmark": "putnam_solving",
"index": 40
} | [] | [] | [
"({a : ℕ × ℕ × ℕ × ℕ | Nat.Prime a.1 ∧ Nat.Prime a.2.2.1 ∧ a.2.1 > 1 ∧ a.2.2.2 > 1 ∧ |(a.1^a.2.1 : ℤ) - 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 → ℝ) × (ℝ → ℝ)) | (ℝ → ℝ) × (ℝ → ℝ) | {
"benchmark": "putnam_solving",
"index": 41
} | [
{
"t": "ℝ",
"v": null,
"name": "r",
"t_type": "Type"
}
] | [
{
"t": "ℤ",
"v": null,
"name": "a",
"t_type": "Type"
},
{
"t": "ℤ",
"v": null,
"name": "b",
"t_type": "Type"
},
{
"t": "ℤ",
"v": null,
"name": "c",
"t_type": "Type"
},
{
"t": "ℤ",
"v": null,
"name": "d",
"t_type": "Type"
},
{
"t": "... | [
"(∃ 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 : ℕ), (1 : ℕ)) : ℕ × ℕ) | ℕ × ℕ | {
"benchmark": "putnam_solving",
"index": 42
} | [] | [] | [
"(Tendsto (fun n : ℕ => ((1 : ℝ)/n)*∑ k in Finset.Icc (1 : ℤ) 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 : ℕ),
(Insert.insert ((((0 : ℤ), (0 : ℤ)) : ℤ × ℤ) :: ([] : List (ℤ × ℤ)) : List (ℤ × ℤ))
(Insert.insert ((((2 : ℤ), (0 : ℤ)) : ℤ × ℤ) :: ([] : List (ℤ × ℤ)) : List (ℤ × ℤ))
(Insert.insert ((((0 : ℤ), (1 : ℤ)) : ℤ × ℤ) :: ([] : List (ℤ × ℤ)) : List (ℤ × ℤ))
... | ℕ × Set (List (ℤ × ℤ)) | {
"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 : ℕ) => (((Polynomial.C : ℤ →+* ℤ[X]) : ℤ → ℤ[X]) (↑(n.factorial : ℕ) : ℤ) : ℤ[X]) | ℕ → Polynomial ℤ | {
"benchmark": "putnam_solving",
"index": 44
} | [] | [] | [
"(∀ n : ℕ, ∑ k in Finset.range (n + 1), C ((-(1 : ℤ))^k * Nat.choose n k) * (X - (C (k : ℤ)))^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 → 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"... | [
"((∑ 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 → ℝ → ℝ → ℝ → 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 ≠ 0 → b ≠ 0 → c ≠ 0 → d ≠ 0 → (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 → 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 → ℝ) → (ℝ → ℝ) → (ℝ → ℝ) | {
"benchmark": "putnam_solving",
"index": 47
} | [
{
"t": "ℝ → ℝ",
"v": null,
"name": "f",
"t_type": "Type"
},
{
"t": "ℝ → ℝ",
"v": null,
"name": "g",
"t_type": "Type"
}
] | [
{
"t": "ℝ → ℝ",
"v": null,
"name": "h",
"t_type": "Type"
},
{
"t": "∀ x, f x = (h (x + 1) + h (x - 1)) / 2",
"v": null,
"name": "hf",
"t_type": "Prop"
},
{
"t": "∀ 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 = ∑' n : ℕ, 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 × 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 × ℝ) → ℝ",
"v": null,
"name": "room",
"t_type": "Type"
},
{
"t": "room = fun (a,b) ↦ mi... | [
"((∀ f : (ℝ × ℝ) → ℝ, Continuous f → (∀ 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) → (∀ 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 : ℤ) 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 × ℝ × ℝ → Prop",
"v": null,
"name": "P",
"t_type": "Type"
},
{
"t": "ℝ × ℝ × ℝ → Prop",
"v": null,
"name": "balanced",
"t_type": "Type"
},
{
"t": "ℝ × ℝ × ℝ → ℝ × ℝ × ℝ",
"v": null,
"name": "B",
"t_type": "Type"
},
{
"t": "P = fun (a, b, c) =... | [
"((∀ t : ℝ × ℝ × ℝ, P t → ∃ n : ℕ, 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 : ℕ) | ℕ | {
"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": "ℕ → ... | [
"(IsLeast {y | ∃ 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 : ℚ) | ℚ | {
"benchmark": "putnam_solving",
"index": 53
} | [] | [] | [
"((∑' i : ℕ+, ∑' j : ℕ+, (1 : ℚ) / (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 : ℕ) : ℝ[X]) : ℝ[X]) -
((4 : ℝ[X]) * ((Polynomial.X : ℝ[X]) ^ (2 : ℕ) : 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 ∧ ∀ x ∈ Icc (-1 : ℝ) 1, p.eval x ∈ Icc 0 1}",
"v": null,
"name": "hS",
"t_type": "Prop"
}
] | [
"(answer ∈ S ∧ (∀ 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 : ℕ) (3 : ℕ) : Multiset ℕ) + ({(2 : ℕ)} : Multiset ℕ) : Multiset ℕ) | Multiset ℕ | {
"benchmark": "putnam_solving",
"index": 55
} | [] | [
{
"t": "Multiset ℕ → Prop",
"v": null,
"name": "P",
"t_type": "Type"
},
{
"t": "∀ a, P a ↔ Multiset.card a > 0 ∧ (∀ i ∈ a, i > 0) ∧ a.sum = 1979",
"v": null,
"name": "hP",
"t_type": "Prop"
}
] | [
"(P answer ∧ ∀ a : Multiset ℕ, P a → answer.prod ≥ 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 ≥ (0 : ℝ) | ℝ → Prop | {
"benchmark": "putnam_solving",
"index": 56
} | [
{
"t": "ℝ",
"v": null,
"name": "k",
"t_type": "Type"
}
] | [] | [
"(answer k ↔ ∃ f : ℝ → ℝ, Continuous f ∧ ∀ 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) => ∃ m : ℤ, a = m ∧ b = m | answer = fun (x : ℝ × ℝ) =>
match x with
| ((a, b) : ℝ × ℝ) => ∃ (m : ℤ), a = (↑m : ℝ) ∧ b = (↑m : ℝ) | (ℝ × ℝ) → Prop | {
"benchmark": "putnam_solving",
"index": 57
} | [
{
"t": "ℕ → ℝ",
"v": null,
"name": "x",
"t_type": "Type"
}
] | [
{
"t": "∀ n : ℕ, x n ≠ 0 ∧ (n ≥ 3 → x n = (x (n - 2))*(x (n - 1))/(2*(x (n - 2)) - (x (n - 1))))",
"v": null,
"name": "hx",
"t_type": "Prop"
}
] | [
"((∀ m : ℕ, ∃ n : ℕ, n > m ∧ ∃ a : ℤ, 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 → ℝ) × Finset (Fin 2 → ℝ) → Prop",
"v": null,
"name": "A",
"t_type": "Type"
},
{
"t": "A = fun (R, B) => R.card = B.card ∧ R ∩ B = ∅ ∧\n ∀ u : Finset (Fin 2 → ℝ), u ⊆ R ∪ B → u.card = 3 → ¬Collinear ℝ (u : Set (Fin 2 → ℝ))",
"v": null,
"name": "hA",
"t_ty... | [
"(answer ↔\n (∀ R B, A (R, B) →\n ∃ v : Finset ((Fin 2 → ℝ) × (Fin 2 → ℝ)),\n (∀ L ∈ v, ∀ M ∈ v, L ≠ M → ∀ x ∈ Icc 0 1, ∀ 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) ≠ 0) ∧\n v.card = R.card ∧ ∀ 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 × ℝ) =>
match x with
| ((a, b) : ℝ × ℝ) =>
((Real.exp (-1 : ℝ) : ℝ) * (((b ^ b : ℝ) / (a ^ a : ℝ) : ℝ) ^ ((1 : ℝ) / (b - a : ℝ) : ℝ) : ℝ) : ℝ) | ℝ × ℝ → ℝ | {
"benchmark": "putnam_solving",
"index": 59
} | [] | [] | [
"(∀ a b : ℝ, 0 < a ∧ a < b → Tendsto (fun t : ℝ => (∫ x in Icc 0 1, (b*x + a*(1 - x))^t)^(1/t)) (𝓝[≠] 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 : ℕ ↦ (n - (1 : ℤ)) / 2 | open Set Topology Filter Polynomial | answer = fun (n : ℕ) => (((↑n : ℤ) - (1 : ℤ) : ℤ) / (2 : ℤ) : ℤ) | ℕ → ℤ | {
"benchmark": "putnam_solving",
"index": 60
} | [
{
"t": "ℕ",
"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 : ℕ => (1 + 4 * r + 6 * r ^ 2) * (1 + 4 * s + 6 * s ^ 2)) | answer = fun (r s : ℕ) =>
((((1 : ℕ) + ((4 : ℕ) * r : ℕ) : ℕ) + ((6 : ℕ) * (r ^ (2 : ℕ) : ℕ) : ℕ) : ℕ) *
(((1 : ℕ) + ((4 : ℕ) * s : ℕ) : ℕ) + ((6 : ℕ) * (s ^ (2 : ℕ) : ℕ) : ℕ) : ℕ) :
ℕ) | ℕ → ℕ → ℕ | {
"benchmark": "putnam_solving",
"index": 61
} | [
{
"t": "ℕ",
"v": null,
"name": "r",
"t_type": "Type"
},
{
"t": "ℕ",
"v": null,
"name": "s",
"t_type": "Type"
}
] | [
{
"t": "ℕ → ℕ → ℕ → ℕ → Prop",
"v": null,
"name": "abcdlcm",
"t_type": "Type"
},
{
"t": "r > 0 ∧ s > 0",
"v": null,
"name": "rspos",
"t_type": "Prop"
},
{
"t": "∀ a b c d : ℕ, abcdlcm a b c d ↔\n (a > 0 ∧ b > 0 ∧ c > 0 ∧ d > 0 ∧\n (3 ^ r * 7 ^ s = Nat.lcm (Nat.l... | [
"({h : ℕ × ℕ × ℕ × ℕ | 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 = ((π : 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 → ℝ)",
"v": null,
"name": "C",
"t_type": "Type"
},
{
"t": "C = {f : ℝ → 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 | ∀ 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 ≥ (1 / 2 : ℝ)} : Set ℝ) | Set ℝ | {
"benchmark": "putnam_solving",
"index": 64
} | [
{
"t": "ℝ",
"v": null,
"name": "c",
"t_type": "Type"
}
] | [] | [
"(∀ 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 ≥ (3 : ℝ)} : Set ℝ) | Set ℝ | {
"benchmark": "putnam_solving",
"index": 65
} | [
{
"t": "ℝ",
"v": null,
"name": "a",
"t_type": "Type"
}
] | [
{
"t": "ℕ → ℝ",
"v": null,
"name": "u",
"t_type": "Type"
},
{
"t": "u 0 = a ∧ (∀ n : ℕ, u (n + 1) = 2 * u n - n ^ 2)",
"v": null,
"name": "hu",
"t_type": "Prop"
}
] | [
"((∀ n : ℕ, 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 $$f(1) - 2f\left(\frac{2}{3}\right) + f\left(\frac{1}{3}\right) \ge t\le... | $t \le 1$ | open Set | answer = fun (t : ℝ) => t ≤ (1 : ℝ) | ℝ → Prop | {
"benchmark": "putnam_solving",
"index": 66
} | [
{
"t": "ℝ",
"v": null,
"name": "t",
"t_type": "Type"
}
] | [
{
"t": "Set ℝ",
"v": null,
"name": "T",
"t_type": "Type"
},
{
"t": "T = Icc 0 1",
"v": null,
"name": "hT",
"t_type": "Prop"
},
{
"t": "ℝ → (ℝ → ℝ) → Prop",
"v": null,
"name": "P",
"t_type": "Type"
},
{
"t": "(ℝ → ℝ) → Prop",
"v": null,
"name": ... | [
"(answer t ↔ (∀ f ∈ S t, ∀ g ∈ S t, f * g ∈ S t))"
] |
Let $E(n)$ be the greatest integer $k$ such that $5^k$ divides $1^1 2^2 3^3 \cdots n^n$. Find $\lim_{n \rightarrow \infty} \frac{E(n)}{n^2}$. | $\frac{1}{8}$ | open Topology Filter Set Polynomial Function | answer = (1 / 8 : ℝ) | ℝ | {
"benchmark": "putnam_solving",
"index": 67
} | [] | [
{
"t": "ℕ → ℕ → Prop",
"v": null,
"name": "P",
"t_type": "Type"
},
{
"t": "∀ n k, P n k ↔ 5^k ∣ ∏ m in Finset.Icc 1 n, (m^m : ℤ)",
"v": null,
"name": "hP",
"t_type": "Prop"
},
{
"t": "ℕ → ℕ",
"v": null,
"name": "E",
"t_type": "Type"
},
{
"t": "∀ n ∈ Ic... | [
"(Tendsto (fun n : ℕ => ((E n) : ℝ)/n^2) atTop (𝓝 answer))"
] |
Does the limit $$lim_{t \rightarrow \infty}e^{-t}\int_{0}^{t}\int_{0}^{t}\frac{e^x - e^y}{x - y} dx dy$$exist? | False | open Topology Filter Set Polynomial Function | answer = False | Prop | {
"benchmark": "putnam_solving",
"index": 68
} | [] | [
{
"t": "ℝ → ℝ",
"v": null,
"name": "f",
"t_type": "Type"
},
{
"t": "f = fun t : ℝ => Real.exp (-t) * ∫ y in (Ico 0 t), ∫ x in (Ico 0 t), (Real.exp x - Real.exp y) / (x - y)",
"v": null,
"name": "hf",
"t_type": "Prop"
}
] | [
"((∃ L : ℝ, Tendsto f atTop (𝓝 L)) ↔ answer)"
] |
Let $P(x)$ be a polynomial with real coefficients; let $$Q(x) = (x^2 + 1)P(x)P'(x) + x((P(x))^2 + (P'(x))^2).$$
Given that $P$ has $n$ distinct real roots all greater than $1$, prove or disprove that $Q$ must have at least $2n - 1$ distinct real roots. | True | open Topology Filter Set Polynomial Function | answer = True | Prop | {
"benchmark": "putnam_solving",
"index": 69
} | [] | [
{
"t": "Polynomial ℝ → Polynomial ℝ",
"v": null,
"name": "Q",
"t_type": "Type"
},
{
"t": "Q = fun P : Polynomial ℝ => (Polynomial.X^2 + 1) * P * (Polynomial.derivative P) + Polynomial.X * (P^2 + (Polynomial.derivative P)^2)",
"v": null,
"name": "hQ",
"t_type": "Prop"
},
{
... | [
"(answer ↔ (∀ P : Polynomial ℝ, {x : ℝ | (Q P).eval x = 0}.ncard ≥ 2*(n P) - 1))"
] |
Find the value of $$\lim_{n \rightarrow \infty} \frac{1}{n^5}\sum_{h=1}^{n}\sum_{k=1}^{n}(5h^4 - 18h^2k^2 + 5k^4).$$ | -1 | open Topology Filter Set Polynomial Function | answer = (-1 : ℝ) | ℝ | {
"benchmark": "putnam_solving",
"index": 70
} | [] | [
{
"t": "ℕ → ℝ",
"v": null,
"name": "f",
"t_type": "Type"
},
{
"t": "f = fun n : ℕ => ((1 : ℝ)/n^5) * ∑ h in Finset.Icc 1 n, ∑ k in Finset.Icc 1 n, (5*(h : ℝ)^4 - 18*h^2*k^2 + 5*k^4)",
"v": null,
"name": "hf",
"t_type": "Prop"
}
] | [
"(Tendsto f atTop (𝓝 answer))"
] |
Determine the minimum value attained by $$(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 × ℝ × ℝ → 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 × ℝ × ℝ → ℝ",
"v": null,
"name": "f",
"t_type": "Type"
},
{
"t": ... | [
"(IsLeast {y | ∃ 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) → Prop",
"v": null,
"name": "VAB",
"t_type": "Type"
},
{
"t": "Set (Matrix (Fin 5) (Fin 7) ℝ) → ℕ → Prop",
"v": null,
"name": "Vrank",
"t_type": "Type"
},
{
"t": "∀ V, VAB V = (∀ A ∈ V, ∀ B ∈ V, ∀ r s : ℝ, r • A + s • B ∈ V)",
"v... | [
"(answer ↔\n (∀ V,\n VAB V → Vrank V 0 → Vrank V 1 →\n Vrank V 2 → Vrank V 4 → Vrank V 5 →\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 ℕ → ℤ",
"v": null,
"name": "sumbits",
"t_type": "Type"
},
{
"t": "ℕ → ℤ",
"v": null,
"name": "B",
"t_type": "Type"
},
{
"t": "∀ bits : List ℕ, sumbits bits = ∑ i : Fin bits.length, (bits[i] : ℤ)",
"v": null,
"name": "hsumbits",
"t_type": "Prop"
... | [
"(answer ↔ (∃ q : ℚ, Real.exp (∑' 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": "ℕ → ℝ → ℝ",
"v": null,
"name": "B",
"t_type": "Type"
},
{
"t": "B = fun (n : ℕ) (x : ℝ) ↦ ∑ k in Finset.Icc 1 n, (k : ℝ) ^ x",
"v": null,
"name": "hB",
"t_type": "Prop"
},
{
"t": "ℕ → ℝ",
"v": null,
"name": "f",
"t_type": "Type"
},
{
"t": "f = ... | [
"(answer ↔ (∃ L : ℝ, Tendsto (fun N ↦ ∑ 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 = (((π : ℝ) / (2 : ℝ) : ℝ) * (Real.log (π : 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": "ℕ → ℕ",
"v": null,
"name": "b",
"t_type": "Type"
},
{
"t": "ℕ → 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 : ℕ => |x n|) (Ici 1)"... | [
"((Tendsto (fun n : ℕ => ∑ 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 →+* ℝ[X]) : ℝ → ℝ[X]) (π : ℝ) : ℝ[X]) *
((((2 : ℝ[X]) * (Polynomial.X : ℝ[X]) : ℝ[X]) - (1 : ℝ[X]) : ℝ[X]) ^ (2 : ℕ) : ℝ[X]) :
ℝ[X]) | Polynomial ℝ | {
"benchmark": "putnam_solving",
"index": 77
} | [] | [
{
"t": "ℝ × ℝ → ℕ",
"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 : ℤ × ℤ | 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": "ℕ → ℝ",
"v": null,
"name": "p",
"t_type": "Type"
},
{
"t": "p = fun n : ℕ => ({c : Finset.Icc 1 n × Finset.Icc 1 n | ∃ m : ℕ, m^2 = c.1 + c.2}.ncard : ℝ) / n^2",
"v": null,
"name": "hp",
"t_type": "Prop"
}
] | [
"(Tendsto (fun n : ℕ => 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 × Prop) | Prop × Prop | {
"benchmark": "putnam_solving",
"index": 79
} | [] | [
{
"t": "Finset ℤ → Prop",
"v": null,
"name": "P",
"t_type": "Type"
},
{
"t": "∀ n, P n ↔ n.Nonempty ∧ ∀ k, ∏ i in n, i ∣ ∏ i in n, (i + k)",
"v": null,
"name": "P_def",
"t_type": "Prop"
}
] | [
"(((∀ n, P n → 1 ∈ n ∨ -1 ∈ n) ↔ answer.1) ∧\n ((∀ n, P n → (∀ i ∈ n, 0 < i) → n = Finset.Icc (1 : ℤ) 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 : ℕ) | ℕ | {
"benchmark": "putnam_solving",
"index": 80
} | [] | [] | [
"({n : ℤ | 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 ↔ (∃ α : ℝ, α > 0 ∧ ∀ n : ℕ, n > 0 → Even (⌊α ^ n⌋ - 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 → 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": "ℕ+ → ℕ",
"v": null,
"name": "f",
"t_type": "Type"
},
{
"t": "f = fun (n : ℕ+) ↦\n Set.ncard {M : Multiset ℕ |\n (∀ m ∈ M, ∃ k : ℕ, m = (2 ^ k : ℤ)) ∧\n (∀ m ∈ M, M.count m ≤ 3) ∧\n (M.sum : ℤ) = n}",
"v": null,
"name": "hf",
"t_type": "Prop"
}
... | [
"(answer ↔ (∃ p : Polynomial ℝ, ∀ n : ℕ+, ⌊p.eval (n : ℝ)⌋ = 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) / (π : ℝ) : ℝ) : ℝ) | ℝ | {
"benchmark": "putnam_solving",
"index": 84
} | [] | [
{
"t": "ℝ → ℝ",
"v": null,
"name": "dist_fun",
"t_type": "Type"
},
{
"t": "dist_fun = fun (x : ℝ) ↦ min (x - ⌊x⌋) (⌈x⌉ - 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) : ℕ → 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 : ℚ) | ℚ | {
"benchmark": "putnam_solving",
"index": 85
} | [] | [] | [
"(∑' k : Set.Ici 1, (6 ^ (k : ℕ) / ((3 ^ ((k : ℕ) + 1) - 2 ^ ((k : ℕ) + 1)) * (3 ^ (k : ℕ) - 2 ^ (k : ℕ)))) = 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 : ℕ)) : MvPolynomial (Fin (3 : ℕ)) ℝ) ^ (2 : ℕ) : MvPolynomial (Fin (3 : ℕ)) ℝ) *
(((MvPolynomial.X (0 : Fin (3 : ℕ)) : MvPolynomial (Fin (3 : ℕ)) ℝ) ^ (2 : ℕ) : MvPolynomial (Fin (3 : ℕ)) ℝ) -
((MvPolynomial.X (1 : Fin (3 : ℕ)) : MvPolynomial (Fin (3 : ℕ)) ℝ) ^ ... | MvPolynomial (Fin 3) ℝ | {
"benchmark": "putnam_solving",
"index": 86
} | [] | [
{
"t": "ℕ",
"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 → 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[≠] 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 : ℕ), (((9 : ℕ), (((8 : ℕ), (((4 : ℕ), (25 : ℕ)) : ℕ × ℕ)) : ℕ × ℕ × ℕ)) : ℕ × ℕ × ℕ × ℕ)) : ℕ × ℕ × ℕ × ℕ × ℕ) | ℕ × ℕ × ℕ × ℕ × ℕ | {
"benchmark": "putnam_solving",
"index": 87
} | [] | [
{
"t": "Set (Fin 3 → ℝ)",
"v": null,
"name": "R",
"t_type": "Type"
},
{
"t": "(Fin 3 → ℝ) → ℝ",
"v": null,
"name": "w",
"t_type": "Type"
},
{
"t": "R = {p | (∀ i : Fin 3, p i ≥ 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 : ℕ) | ℕ | {
"benchmark": "putnam_solving",
"index": 88
} | [] | [
{
"t": "ℕ → ℕ",
"v": null,
"name": "f",
"t_type": "Type"
},
{
"t": "∀ n, some (f n) = (Nat.digits 10 (n !)).find? (fun d ↦ d ≠ 0)",
"v": null,
"name": "hf",
"t_type": "Prop"
},
{
"t": "ℕ → (ℕ → ℕ) → ℕ → Prop",
"v": null,
"name": "IsPeriodicFrom",
"t_type": "Ty... | [
"(∃ g : ℕ → ℕ,\n (∀ᵉ (k > 0) (a : Fin k → ℕ) (ha : Injective a), f (∑ i, 5 ^ (a i)) = g (∑ 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 × Polynomial ℝ) | Polynomial ℝ × Polynomial ℝ | {
"benchmark": "putnam_solving",
"index": 89
} | [] | [
{
"t": "ℕ → ℤ",
"v": null,
"name": "f",
"t_type": "Type"
},
{
"t": "∀ n > 0, f n = ∑ i : Set.Icc 1 n, ((i)! : ℤ)",
"v": null,
"name": "hf",
"t_type": "Prop"
}
] | [
"(∀ n ≥ 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 → ℝ → ℝ",
"v": null,
"name": "f",
"t_type": "Type"
},
{
"t": "∀ 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 | ∃ᵉ (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
} | [] | [] | [
"((∀ (F : Type*) (_ : Fintype F), Fintype.card F ≥ 2 → (∃ mul : F → F → F, ∀ x y z : F, (mul x z = mul y z → x = y) ∧ (mul x (mul y z) ≠ 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 : ℤ),
(((((Polynomial.X : Polynomial ℝ) * ((Polynomial.X : Polynomial ℝ) - (1 : Polynomial ℝ) : Polynomial ℝ) :
Polynomial ℝ) /
(2 : Polynomial ℝ) :
Polynomial ℝ),
(Polynomial.X : Polynomial ℕ)) :
Polynomial ℝ × Polynomial ℕ)) :
ℤ × P... | ℤ × Polynomial ℝ × Polynomial ℕ | {
"benchmark": "putnam_solving",
"index": 92
} | [] | [
{
"t": "ℕ",
"v": null,
"name": "m",
"t_type": "Type"
},
{
"t": "m > 0",
"v": null,
"name": "mpos",
"t_type": "Prop"
},
{
"t": "ℕ → ℕ",
"v": null,
"name": "d",
"t_type": "Type"
},
{
"t": "List ℕ → ℕ",
"v": null,
"name": "sumbits",
"t_type": ... | [
"(let (a, f, g) := answer;\n ∑ k : Set.Icc 0 (2 ^ m - 1), (-(1 : ℤ)) ^ (d k) * (k : ℕ) ^ 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 : ℕ), (((10 : ℕ), (((0 : ℕ), (0 : ℕ)) : ℕ × ℕ)) : ℕ × ℕ × ℕ)) : ℕ × ℕ × ℕ × ℕ) | ℕ × ℕ × ℕ × ℕ | {
"benchmark": "putnam_solving",
"index": 93
} | [] | [] | [
"(let (a, b, c, d) := answer;\n {(A1, A2, A3) : Set ℤ × Set ℤ × Set ℤ | A1 ∪ A2 ∪ A3 = Icc 1 10 ∧ A1 ∩ A2 ∩ A3 = ∅}.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 → ℝ | {
"benchmark": "putnam_solving",
"index": 94
} | [
{
"t": "ℝ",
"v": null,
"name": "d",
"t_type": "Type"
}
] | [
{
"t": "ℕ → ℕ → ℝ",
"v": null,
"name": "a",
"t_type": "Type"
},
{
"t": "∀ m : ℕ, a m 0 = d / 2 ^ m",
"v": null,
"name": "ha0",
"t_type": "Prop"
},
{
"t": "∀ m : ℕ, ∀ j : ℕ, 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 : ℕ))} : Set (Fin (100 : ℕ))) | Set (Fin 100) | {
"benchmark": "putnam_solving",
"index": 95
} | [] | [
{
"t": "ℕ → ℕ",
"v": null,
"name": "a",
"t_type": "Type"
},
{
"t": "a 1 = 3",
"v": null,
"name": "ha1",
"t_type": "Prop"
},
{
"t": "∀ i ≥ 1, a (i + 1) = 3 ^ a i",
"v": null,
"name": "ha",
"t_type": "Prop"
}
] | [
"({k : Fin 100 | ∀ N : ℕ, ∃ i ≥ 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 : ℕ) (Insert.insert (4 : ℕ) (Insert.insert (7 : ℕ) ({(8 : ℕ)} : Set ℕ) : Set ℕ) : Set ℕ) : Set ℕ) | Set ℕ | {
"benchmark": "putnam_solving",
"index": 96
} | [] | [
{
"t": "ℕ → ℝ",
"v": null,
"name": "I",
"t_type": "Type"
},
{
"t": "I = fun (m : ℕ) ↦ ∫ 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 ≠ 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 : ℕ) : 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 → ℝ",
"v": null,
"name": "Γ",
"t_type": "Type"
},
{
"t": "Polynomial ℝ",
"v": null,
"name": "f",
"t_type": "Type"
},
{
"t": "Γ = fun p ↦ ∑ k in Finset.range (p.natDegree + 1), coeff p k ^ 2",
"v": null,
"name": "hΓ",
"t_type": "Prop"
},
... | [
"(let g := answer;\n g.eval 0 = 1 ∧ ∀ n : ℕ, n ≥ 1 → Γ (f ^ n) = Γ (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 : ℤ) | open Set Filter Topology Real Polynomial Function | answer = fun (i : Fin (5 : ℕ)) => ((↑(↑i : ℕ) : ℤ) - (2 : ℤ) : ℤ) | Fin 5 → ℤ | {
"benchmark": "putnam_solving",
"index": 98
} | [] | [
{
"t": "(Fin 5 → ℤ) → (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 → ℕ",
"v": null,
"name": "numnzcoeff",
"t_type": ... | [
"((Injective answer ∧ ∀ m : Fin 5 → ℤ, Injective m → 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 : ℕ) => (if n = (101 : ℕ) then (99 : ℕ) else (0 : ℕ) : ℕ) | ℕ → ℕ | {
"benchmark": "putnam_solving",
"index": 99
} | [] | [
{
"t": "ℕ -> Polynomial ℕ",
"v": null,
"name": "f",
"t_type": "Prop"
},
{
"t": "f 0 = 1",
"v": null,
"name": "hf0x",
"t_type": "Prop"
},
{
"t": "∀ n ≥ 1, (f n).eval 0 = 0",
"v": null,
"name": "hfn0",
"t_type": "Prop"
},
{
"t": "∀ n : ℕ, 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 = ((√((π : 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)"
] |
Find, with explanation, the maximum value of $f(x)=x^3-3x$ on the set of all real numbers $x$ satisfying $x^4+36 \leq 13x^2$. | 18 | answer = (18 : ℝ) | ℝ | {
"benchmark": "putnam_solving",
"index": 101
} | [] | [
{
"t": "Set ℝ",
"v": null,
"name": "S",
"t_type": "Type"
},
{
"t": "ℝ → ℝ",
"v": null,
"name": "f",
"t_type": "Type"
},
{
"t": "S = {x : ℝ | x ^ 4 + 36 ≤ 13 * x ^ 2}",
"v": null,
"name": "hS",
"t_type": "Prop"
},
{
"t": "f = fun x ↦ x ^ 3 - 3 * x",
... | [
"(IsGreatest {f x | x ∈ S} answer)"
] | |
What is the units (i.e., rightmost) digit of
\[
\left\lfloor \frac{10^{20000}}{10^{100}+3}\right\rfloor ?
\] | 3 | answer = (3 : ℕ) | ℕ | {
"benchmark": "putnam_solving",
"index": 102
} | [] | [] | [
"(Nat.floor ((10 ^ 20000 : ℝ) / (10 ^ 100 + 3)) % 10 = answer)"
] | |
Evaluate $\sum_{n=0}^\infty \mathrm{Arccot}(n^2+n+1)$, where $\mathrm{Arccot}\,t$ for $t \geq 0$ denotes the number $\theta$ in the interval $0 < \theta \leq \pi/2$ with $\cot \theta = t$. | π / 2 | open Real | answer = ((π : ℝ) / (2 : ℝ) : ℝ) | ℝ | {
"benchmark": "putnam_solving",
"index": 103
} | [] | [
{
"t": "ℝ → ℝ",
"v": null,
"name": "cot",
"t_type": "Type"
},
{
"t": "cot = fun θ ↦ Real.cos θ / Real.sin θ",
"v": null,
"name": "fcot",
"t_type": "Prop"
},
{
"t": "ℝ → ℝ",
"v": null,
"name": "arccot",
"t_type": "Type"
},
{
"t": "∀ t : ℝ, t ≥ 0 → arcco... | [
"((∑' n : ℕ, arccot (n ^ 2 + n + 1)) = answer)"
] |
A \emph{transversal} of an $n\times n$ matrix $A$ consists of $n$ entries of $A$, no two in the same row or column. Let $f(n)$ be the number of $n \times n$ matrices $A$ satisfying the following two conditions:
\begin{enumerate}
\item[(a)] Each entry $\alpha_{i,j}$ of $A$ is in the set
$\{-1,0,1\}$.
\item[(b)] The sum ... | (1, 4, 2, 3, -4, 2, 1) | open Real Equiv | answer =
(((1 : ℚ),
(((4 : ℚ),
(((2 : ℚ), (((3 : ℚ), (((-4 : ℚ), (((2 : ℚ), (1 : ℚ)) : ℚ × ℚ)) : ℚ × ℚ × ℚ)) : ℚ × ℚ × ℚ × ℚ)) :
ℚ × ℚ × ℚ × ℚ × ℚ)) :
ℚ × ℚ × ℚ × ℚ × ℚ × ℚ)) :
ℚ × ℚ × ℚ × ℚ × ℚ × ℚ × ℚ) | ℚ × ℚ × ℚ × ℚ × ℚ × ℚ × ℚ | {
"benchmark": "putnam_solving",
"index": 104
} | [] | [
{
"t": "ℕ → ℕ",
"v": null,
"name": "f",
"t_type": "Type"
},
{
"t": "f = fun n ↦\n Set.ncard {A : Matrix (Fin n) (Fin n) ℤ |\n (∀ i j : Fin n, A i j ∈ ({-1, 0, 1} : Set ℤ)) ∧\n ∃ S : ℤ, ∀ ϕ : Perm (Fin n), ∑ i : Fin n, A i (ϕ i) = S}",
"v": null,
"name": "hf",
"t_type": "Prop"
... | [
"(let (a1, b1, a2, b2, a3, b3, a4) := answer;\n (∀ n > 0, f n = a1 * b1 ^ n + a2 * b2 ^ n + a3 * b3 ^ n + a4))"
] |
Let $a_1, a_2, \dots, a_n$ be real numbers, and let $b_1, b_2, \dots, b_n$ be distinct positive integers. Suppose that there is a polynomial $f(x)$ satisfying the identity
\[
(1-x)^n f(x) = 1 + \sum_{i=1}^n a_i x^{b_i}.
\]
Find a simple expression (not involving any sums) for $f(1)$ in terms of $b_1, b_2, \dots, b_n$ a... | fun b n ↦ (∏ i : Finset.Icc 1 n, b i) / Nat.factorial n | open Real Equiv | answer = fun (b : ℕ → ℕ) (n : ℕ) =>
((↑(∏ i : { x : ℕ // x ∈ (Finset.Icc (1 : ℕ) n : Finset ℕ) }, (b (↑i : ℕ) : ℕ)) : ℝ) / (↑(n.factorial : ℕ) : ℝ) : ℝ) | (ℕ → ℕ) → ℕ → ℝ | {
"benchmark": "putnam_solving",
"index": 105
} | [
{
"t": "ℕ",
"v": null,
"name": "n",
"t_type": "Type"
},
{
"t": "ℕ → ℕ",
"v": null,
"name": "b",
"t_type": "Type"
}
] | [
{
"t": "n > 0",
"v": null,
"name": "npos",
"t_type": "Prop"
},
{
"t": "ℕ → ℝ",
"v": null,
"name": "a",
"t_type": "Type"
},
{
"t": "∀ i ∈ Finset.Icc 1 n, b i > 0",
"v": null,
"name": "bpos",
"t_type": "Prop"
},
{
"t": "∀ i ∈ Finset.Icc 1 n, ∀ j ∈ Finset... | [
"(f.eval 1 = answer b n)"
] |
Inscribe a rectangle of base $b$ and height $h$ and an isosceles triangle of base $b$ (against a corresponding side of the rectangle and pointed in the other direction) in a circle of radius one. For what value of $h$ do the rectangle and triangle have the same area? | $2/5$ | open Real Equiv | answer = (2 / 5 : ℝ) | ℝ | {
"benchmark": "putnam_solving",
"index": 106
} | [] | [
{
"t": "ℝ",
"v": null,
"name": "b",
"t_type": "Type"
},
{
"t": "ℝ",
"v": null,
"name": "h",
"t_type": "Type"
},
{
"t": "b > 0 ∧ h > 0 ∧ b ^ 2 + h ^ 2 = 2 ^ 2",
"v": null,
"name": "hbh",
"t_type": "Prop"
},
{
"t": "b * h = 0.5 * b * (1 - h / 2)",
"v... | [
"(h = answer)"
] |
Prove that there are only a finite number of possibilities for the ordered triple $T=(x-y,y-z,z-x)$, where $x,y,z$ are complex numbers satisfying the simultaneous equations
\[
x(x-1)+2yz = y(y-1)+2zx = z(z-1)+2xy,
\]
and list all such triples $T$. | {(0, 0, 0), (0, -1, 1), (1, 0, -1), (-1, 1, 0)} | open Real Equiv | answer =
(Insert.insert (((0 : ℂ), (((0 : ℂ), (0 : ℂ)) : ℂ × ℂ)) : ℂ × ℂ × ℂ)
(Insert.insert (((0 : ℂ), (((-1 : ℂ), (1 : ℂ)) : ℂ × ℂ)) : ℂ × ℂ × ℂ)
(Insert.insert (((1 : ℂ), (((0 : ℂ), (-1 : ℂ)) : ℂ × ℂ)) : ℂ × ℂ × ℂ)
({(((-1 : ℂ), (((1 : ℂ), (0 : ℂ)) : ℂ × ℂ)) : ℂ × ℂ × ℂ)} : Finset (ℂ ... | Finset (ℂ × ℂ × ℂ) | {
"benchmark": "putnam_solving",
"index": 107
} | [] | [] | [
"({T : ℂ × ℂ × ℂ | ∃ x y z : ℂ, T = (x - y, y - z, z - x) ∧ x * (x - 1) + 2 * y * z = y * (y - 1) + 2 * z * x ∧ y * (y - 1) + 2 * z * x = z * (z - 1) + 2 * x * y} = answer)"
] |
For a positive real number $r$, let $G(r)$ be the minimum value of $|r - \sqrt{m^2+2n^2}|$ for all integers $m$ and $n$. Prove or disprove the assertion that $\lim_{r\to \infty}G(r)$ exists and equals $0$. | True | open Real Equiv Polynomial Filter Topology | answer = True | Prop | {
"benchmark": "putnam_solving",
"index": 108
} | [] | [
{
"t": "ℝ → ℝ",
"v": null,
"name": "G",
"t_type": "Type"
},
{
"t": "∀ r : ℝ, ∃ m n : ℤ, G r = |r - Real.sqrt (m ^ 2 + 2 * n ^ 2)|",
"v": null,
"name": "hGeq",
"t_type": "Prop"
},
{
"t": "∀ r : ℝ, ∀ m n : ℤ, G r ≤ |r - Real.sqrt (m ^ 2 + 2 * n ^ 2)|",
"v": null,
"n... | [
"(answer ↔ Tendsto G atTop (𝓝 0))"
] |
Let $f(x,y,z) = x^2+y^2+z^2+xyz$. Let $p(x,y,z), q(x,y,z)$, $r(x,y,z)$ be polynomials with real coefficients satisfying
\[
f(p(x,y,z), q(x,y,z), r(x,y,z)) = f(x,y,z).
\]
Prove or disprove the assertion that the sequence $p,q,r$ consists of some permutation of $\pm x, \pm y, \pm z$, where the number of minus signs is $0... | False | open Real Equiv Polynomial Filter Topology MvPolynomial | answer = False | Prop | {
"benchmark": "putnam_solving",
"index": 109
} | [] | [
{
"t": "MvPolynomial (Fin 3) ℝ",
"v": null,
"name": "f",
"t_type": "Type"
},
{
"t": "Set (Set (MvPolynomial (Fin 3) ℝ))",
"v": null,
"name": "perms",
"t_type": "Type"
},
{
"t": "f = (X 0) ^ 2 + (X 1) ^ 2 + (X 2) ^ 2 + (X 0) * (X 1) * (X 2)",
"v": null,
"name": "hf... | [
"(answer ↔ (∀ pqr : Fin 3 → MvPolynomial (Fin 3) ℝ,\n (∀ xyz : Fin 3 → ℝ, MvPolynomial.eval (fun i ↦ MvPolynomial.eval xyz (pqr i)) f = MvPolynomial.eval xyz f) →\n ({pqr 0, pqr 1, pqr 2} ∈ perms)))"
] |
The sequence of digits $123456789101112131415161718192021 \dots$ is obtained by writing the positive integers in order. If the $10^n$-th digit in this sequence occurs in the part of the sequence in which the $m$-digit numbers are placed, define $f(n)$ to be $m$. For example, $f(2)=2$ because the $100$th digit enters th... | 1984 | answer = (1984 : ℕ) | ℕ | {
"benchmark": "putnam_solving",
"index": 110
} | [] | [
{
"t": "ℕ → ℕ",
"v": null,
"name": "seqind",
"t_type": "Type"
},
{
"t": "ℕ → ℕ",
"v": null,
"name": "seqsize",
"t_type": "Type"
},
{
"t": "ℕ → ℕ",
"v": null,
"name": "f",
"t_type": "Type"
},
{
"t": "seqind 1 = 1 ∧ ∀ i ≥ 2, seqind i = seqind (i - 1) + (... | [
"(f 1987 = answer)"
] | |
Let $P$ be a polynomial, with real coefficients, in three variables and $F$ be a function of two variables such that
\[
P(ux, uy, uz) = u^2 F(y-x,z-x) \quad \mbox{for all real $x,y,z,u$},
\]
and such that $P(1,0,0)=4$, $P(0,1,0)=5$, and $P(0,0,1)=6$. Also let $A,B,C$ be complex numbers with $P(A,B,C)=0$ and $|B-A|=10$.... | $\frac{5}{3}\sqrt{30}$ | open MvPolynomial Real | answer = ((5 / 3 : ℂ) * (↑(√(30 : ℝ) : ℝ) : ℂ) : ℂ) | ℂ | {
"benchmark": "putnam_solving",
"index": 111
} | [] | [
{
"t": "MvPolynomial (Fin 3) ℂ",
"v": null,
"name": "P",
"t_type": "Type"
},
{
"t": "∀ i : Fin 3 →₀ ℕ, (coeff i P).im = 0",
"v": null,
"name": "hPreal",
"t_type": "Prop"
},
{
"t": "ℝ → ℝ → ℝ",
"v": null,
"name": "F",
"t_type": "Type"
},
{
"t": "ℂ → ℂ →... | [
"(‖C - A‖ = answer)"
] |
Let $\vec{G}(x,y)=\left(\frac{-y}{x^2+4y^2},\frac{x}{x^2+4y^2},0\right)$. Prove or disprove that there is a vector-valued function $\vec{F}(x,y,z)=(M(x,y,z),N(x,y,z),P(x,y,z))$ with the following properties:
\begin{enumerate}
\item[(i)] $M$, $N$, $P$ have continuous partial derivatives for all $(x,y,z) \neq (0,0,0)$;
\... | False | answer = False | Prop | {
"benchmark": "putnam_solving",
"index": 112
} | [] | [
{
"t": "ℝ → ℝ → (Fin 2 → ℝ)",
"v": null,
"name": "vec2",
"t_type": "Type"
},
{
"t": "ℝ → ℝ → ℝ → (Fin 3 → ℝ)",
"v": null,
"name": "vec3",
"t_type": "Type"
},
{
"t": "(Fin 2 → ℝ) → (Fin 3 → ℝ)",
"v": null,
"name": "G",
"t_type": "Type"
},
{
"t": "G = (f... | [
"((∃ F : Fin 3 → ((Fin 3 → ℝ) → ℝ), Fprop1 F ∧ Fprop2 F ∧ Fprop3 F) ↔ answer)"
] | |
For each positive integer $n$, let $a(n)$ be the number of zeroes in the base $3$ representation of $n$. For which positive real numbers $x$ does the series
\[
\sum_{n=1}^\infty \frac{x^{a(n)}}{n^3}
\]
converge? | the set of positive real numbers $x$ such that $0 < x < 25$ | open MvPolynomial Real Nat | answer = ({x : ℝ | x > (0 : ℝ) ∧ x < (25 : ℝ)} : Set ℝ) | Set ℝ | {
"benchmark": "putnam_solving",
"index": 113
} | [] | [
{
"t": "ℕ → ℕ",
"v": null,
"name": "a",
"t_type": "Type"
},
{
"t": "a = fun n ↦ {i | (digits 3 n).get i = 0}.ncard",
"v": null,
"name": "ha",
"t_type": "Prop"
}
] | [
"({x : ℝ | x > 0 ∧ Summable (fun n ↦ x ^ (a n) / (n ^ 3))} = answer)"
] |
Evaluate
\[
\int_2^4 \frac{\sqrt{\ln(9-x)}\,dx}{\sqrt{\ln(9-x)}+\sqrt{\ln(x+3)}}.
\] | 1 | open MvPolynomial Real Nat | answer = (1 : ℝ) | ℝ | {
"benchmark": "putnam_solving",
"index": 114
} | [] | [] | [
"(answer = ∫ x in (2)..4, sqrt (log (9 - x)) / (sqrt (log (9 - x)) + sqrt (log (x + 3))))"
] |
Let $(x_1,y_1) = (0.8, 0.6)$ and let $x_{n+1} = x_n \cos y_n - y_n \sin y_n$ and $y_{n+1}= x_n \sin y_n + y_n \cos y_n$ for $n=1,2,3,\dots$. For each of $\lim_{n\to \infty} x_n$ and $\lim_{n \to \infty} y_n$, prove that the limit exists and find it or prove that the limit does not exist. | (True, -1, True, 0) | open MvPolynomial Real Nat Filter Topology | answer = ((True, (((-1 : ℝ), ((True, (0 : ℝ)) : Prop × ℝ)) : ℝ × Prop × ℝ)) : Prop × ℝ × Prop × ℝ) | Prop × ℝ × Prop × ℝ | {
"benchmark": "putnam_solving",
"index": 115
} | [] | [
{
"t": "ℕ → ℝ",
"v": null,
"name": "x",
"t_type": "Type"
},
{
"t": "ℕ → ℝ",
"v": null,
"name": "y",
"t_type": "Type"
},
{
"t": "(x 1, y 1) = (0.8, 0.6)",
"v": null,
"name": "hxy1",
"t_type": "Prop"
},
{
"t": "∀ n ≥ 1, x (n + 1) = (x n) * cos (y n) - (y... | [
"(let (existsx, limx, existsy, limy) := answer\n((∃ c : ℝ, Tendsto x atTop (𝓝 c)) → existsx) ∧\n(existsx → Tendsto x atTop (𝓝 limx)) ∧\n((∃ c : ℝ, Tendsto y atTop (𝓝 c)) → existsy) ∧\n(existsy → Tendsto y atTop (𝓝 limy)))"
] |
Let $R$ be the region consisting of the points $(x,y)$ of the cartesian plane satisfying both $|x|-|y| \leq 1$ and $|y| \leq 1$. Find the area of $R$. | 6 | open MeasureTheory | answer = (6 : ℝ) | ℝ | {
"benchmark": "putnam_solving",
"index": 116
} | [] | [
{
"t": "Set (Fin 2 → ℝ)",
"v": null,
"name": "R",
"t_type": "Type"
},
{
"t": "R = {p | |p 0| - |p 1| ≤ 1 ∧ |p 1| ≤ 1}",
"v": null,
"name": "hR",
"t_type": "Prop"
}
] | [
"((volume R).toReal = answer)"
] |
A not uncommon calculus mistake is to believe that the product rule for derivatives says that $(fg)' = f'g'$. If $f(x)=e^{x^2}$, determine, with proof, whether there exists an open interval $(a,b)$ and a nonzero function $g$ defined on $(a,b)$ such that this wrong product rule is true for $x$ in $(a,b)$. | True | open Set | answer = True | Prop | {
"benchmark": "putnam_solving",
"index": 117
} | [] | [
{
"t": "ℝ → ℝ",
"v": null,
"name": "f",
"t_type": "Type"
},
{
"t": "f = fun x ↦ Real.exp (x ^ 2)",
"v": null,
"name": "hf",
"t_type": "Prop"
}
] | [
"(answer ↔\n (∃ a b : ℝ,\n a < b ∧\n ∃ g : ℝ → ℝ,\n (∃ x ∈ Ioo a b, g x ≠ 0) ∧\n DifferentiableOn ℝ g (Ioo a b) ∧\n ∀ x ∈ Ioo a b, deriv (fun y ↦ f y * g y) x = (deriv f x) * (deriv g x)))"
] |
Determine, with proof, the set of real numbers $x$ for which
\[
\sum_{n=1}^\infty \left( \frac{1}{n} \csc \frac{1}{n} - 1 \right)^x
\]
converges. | {x | x > 1 / 2} | open Set Filter Topology | answer = ({x : ℝ | x > (1 / 2 : ℝ)} : Set ℝ) | Set ℝ | {
"benchmark": "putnam_solving",
"index": 118
} | [] | [] | [
"(answer = {x : ℝ | ∃ L : ℝ, Tendsto (fun t ↦ ∑ n in Finset.Icc (1 : ℕ) t, (((1 / n) / Real.sin (1 / n) - 1) ^ x)) atTop (𝓝 L)})"
] |
\begin{enumerate}
\item[(a)] If every point of the plane is painted one of three colors, do there necessarily exist two points of the same color exactly one inch apart?
\item[(b)] What if ``three'' is replaced by ``nine''?
\end{enumerate} | (True, False) | open Set Filter Topology | answer = ((True, False) : Prop × Prop) | Prop × Prop | {
"benchmark": "putnam_solving",
"index": 119
} | [] | [
{
"t": "ℕ → Prop",
"v": null,
"name": "p",
"t_type": "Type"
},
{
"t": "∀ n, p n ↔\n ∀ color : (EuclideanSpace ℝ (Fin 2)) → Fin n,\n ∃ p q : EuclideanSpace ℝ (Fin 2),\n color p = color q ∧ dist p q = 1",
"v": null,
"name": "hp",
"t_type": "Prop"
}
] | [
"(let (a, b) := answer; (p 3 ↔ a) ∧ (p 9 ↔ b))"
] |
If a linear transformation $A$ on an $n$-dimensional vector space has $n+1$ eigenvectors such that any $n$ of them are linearly independent, does it follow that $A$ is a scalar multiple of the identity? Prove your answer. | True | open Set Filter Topology | answer = True | Prop | {
"benchmark": "putnam_solving",
"index": 120
} | [] | [] | [
"((∀ (F V : Type*) (_ : Field F) (_ : AddCommGroup V) (_ : Module F V) (_ : FiniteDimensional F V) (n : ℕ) (A : Module.End F V) (evecs : Set V), (n = Module.finrank F V ∧ evecs ⊆ {v : V | ∃ f : F, A.HasEigenvector f v} ∧ evecs.encard = n + 1 ∧ (∀ sevecs : Fin n → V, (Set.range sevecs ⊆ evecs ∧ (Set.range sevecs).en... |
Prove or disprove: If $x$ and $y$ are real numbers with $y \geq 0$ and $y(y+1) \leq (x+1)^2$, then $y(y-1) \leq x^2$. | True | open Set Filter Topology | answer = True | Prop | {
"benchmark": "putnam_solving",
"index": 121
} | [] | [] | [
"((∀ x y : ℝ, (y ≥ 0 ∧ y * (y + 1) ≤ (x + 1) ^ 2) → (y * (y - 1) ≤ x ^ 2)) ↔ answer)"
] |
For every $n$ in the set $N=\{1,2,\dots\}$ of positive integers, let $r_n$ be the minimum value of $|c-d \sqrt{3}|$ for all nonnegative integers $c$ and $d$ with $c+d=n$. Find, with proof, the smallest positive real number $g$ with $r_n \leq g$ for all $n \in N$. | (1 + √3) / 2 | open Set Filter Topology | answer = (((1 : ℝ) + (√(3 : ℝ) : ℝ) : ℝ) / (2 : ℝ) : ℝ) | ℝ | {
"benchmark": "putnam_solving",
"index": 122
} | [] | [
{
"t": "ℤ → ℝ",
"v": null,
"name": "r",
"t_type": "Type"
},
{
"t": "∀ n ≥ 1,\n (∃ c d : ℤ,\n (c ≥ 0 ∧ d ≥ 0) ∧\n c + d = n ∧ r n = |c - d * Real.sqrt 3|) ∧\n (∀ c d : ℤ, (c ≥ 0 ∧ d ≥ 0 ∧ c + d = n) → |c - d * Real.sqrt 3| ≥ r n)",
"v": null,
"name": "hr",
... | [
"(IsLeast {g : ℝ | g > 0 ∧ (∀ n : ℤ, n ≥ 1 → r n ≤ g)} answer)"
] |
For positive integers $n$, let $M_n$ be the $2n+1$ by $2n+1$ skew-symmetric matrix for which each entry in the first $n$ subdiagonals below the main diagonal is $1$ and each of the remaining entries below the main diagonal is $-1$. Find, with proof, the rank of $M_n$. (According to one definition, the rank of a matrix ... | 2n | open Set Filter Topology | answer = fun (n : ℕ) => ((2 : ℕ) * n : ℕ) | ℕ → ℕ | {
"benchmark": "putnam_solving",
"index": 123
} | [
{
"t": "ℕ",
"v": null,
"name": "n",
"t_type": "Type"
}
] | [
{
"t": "n > 0",
"v": null,
"name": "hn",
"t_type": "Prop"
},
{
"t": "Matrix (Fin (2 * n + 1)) (Fin (2 * n + 1)) ℝ",
"v": null,
"name": "Mn",
"t_type": "Type"
},
{
"t": "∀ i j, Mn i j = -(Mn j i)",
"v": null,
"name": "Mnskewsymm",
"t_type": "Prop"
},
{
... | [
"(Mn.rank = answer n)"
] |
How many primes among the positive integers, written as usual in base $10$, are alternating $1$'s and $0$'s, beginning and ending with $1$? | 1 | answer = (1 : ℕ∞) | ℕ∞ | {
"benchmark": "putnam_solving",
"index": 124
} | [] | [
{
"t": "List ℕ → Prop",
"v": null,
"name": "pdigalt",
"t_type": "Type"
},
{
"t": "∀ l, pdigalt l ↔ Odd l.length ∧ (∀ i, l.get i = if Even (i : ℕ) then 1 else 0)",
"v": null,
"name": "hpdigalt",
"t_type": "Prop"
}
] | [
"({p : ℕ | p.Prime ∧ pdigalt (Nat.digits 10 p)}.encard = answer)"
] | |
Evaluate $\int_0^a \int_0^b e^{\max\{b^2x^2,a^2y^2\}}\,dy\,dx$ where $a$ and $b$ are positive. | $\frac{e^{a^2 b^2} - 1}{a b}$ | open Real | answer = fun (a b : ℝ) => (((rexp ((a ^ (2 : ℕ) : ℝ) * (b ^ (2 : ℕ) : ℝ) : ℝ) : ℝ) - (1 : ℝ) : ℝ) / (a * b : ℝ) : ℝ) | ℝ → ℝ → ℝ | {
"benchmark": "putnam_solving",
"index": 125
} | [
{
"t": "ℝ",
"v": null,
"name": "a",
"t_type": "Type"
},
{
"t": "ℝ",
"v": null,
"name": "b",
"t_type": "Type"
}
] | [
{
"t": "a > 0 ∧ b > 0",
"v": null,
"name": "abpos",
"t_type": "Prop"
}
] | [
"(∫ x in Set.Ioo 0 a, ∫ y in Set.Ioo 0 b, Real.exp (max (b ^ 2 * x ^ 2) (a ^ 2 * y ^ 2)) = answer a b)"
] |
A dart, thrown at random, hits a square target. Assuming that any two parts of the target of equal area are equally likely to be hit, find the probability that the point hit is nearer to the center than to any edge. Express your answer in the form $(a\sqrt{b}+c)/d$, where $a$, $b$, $c$, $d$ are integers and $b$, $d$ ar... | (4, 2, -5, 3) | open Nat MeasureTheory | answer = (((4 : ℤ), (((2 : ℤ), (((-5 : ℤ), (3 : ℤ)) : ℤ × ℤ)) : ℤ × ℤ × ℤ)) : ℤ × ℤ × ℤ × ℤ) | ℤ × ℤ × ℤ × ℤ | {
"benchmark": "putnam_solving",
"index": 126
} | [] | [
{
"t": "Set (EuclideanSpace ℝ (Fin 2))",
"v": null,
"name": "square",
"t_type": "Type"
},
{
"t": "Set (EuclideanSpace ℝ (Fin 2))",
"v": null,
"name": "Scloser",
"t_type": "Type"
},
{
"t": "Set (EuclideanSpace ℝ (Fin 2))",
"v": null,
"name": "perimeter",
"t_typ... | [
"(let (a, b, c, d) := answer;\n b > 0 ∧ d > 0 ∧ (¬∃ n : ℤ, n^2 = b) ∧\n (volume Scloser).toReal / (volume square).toReal = (a * Real.sqrt b + c) / d)"
] |
Let $S$ be a non-empty set with an associative operation that is left and right cancellative ($xy=xz$ implies $y=z$, and $yx=zx$ implies $y=z$). Assume that for every $a$ in $S$ the set $\{a^n:\,n=1, 2, 3, \ldots\}$ is finite. Must $S$ be a group? | True | open Nat | answer = True | Prop | {
"benchmark": "putnam_solving",
"index": 127
} | [] | [
{
"t": "(S : Type) → ℕ → S → S",
"v": null,
"name": "pow",
"t_type": "Type"
},
{
"t": "∀ (S : Type) (_ : Semigroup S), ∀ s : S, pow S 1 s = s",
"v": null,
"name": "hpow1",
"t_type": "Prop"
},
{
"t": "∀ (S : Type) (_ : Semigroup S), ∀ s : S, ∀ n > 0, pow S (n + 1) s = s * ... | [
"(((∀ (S : Type) (_ : Nonempty S) (_ : Semigroup S) (_ : IsCancelMul S), (∀ a : S, Finite {x | ∃ n : ℕ, n > 0 ∧ pow S n a = x}) → ∃ (_ : Group S), True) ↔ answer))"
] |
Let $f$ be a function on $[0,\infty)$, differentiable and satisfying
\[
f'(x)=-3f(x)+6f(2x)
\]
for $x>0$. Assume that $|f(x)|\le e^{-\sqrt{x}}$ for $x\ge 0$ (so that $f(x)$ tends rapidly to $0$ as $x$ increases). For $n$ a non-negative integer, define
\[
\mu_n=\int_0^\infty x^n f(x)\,dx
\]
(sometimes called the $n$th m... | fun n c ↦ c * n ! / (3 ^ n * ∏ m in Finset.Icc (1 : ℤ) n, (1 - 2 ^ (-m))) | open Nat Filter Topology | answer = fun (n : ℕ) (c : ℝ) =>
((c * (↑(n ! : ℕ) : ℝ) : ℝ) /
(((3 : ℝ) ^ n : ℝ) * ∏ m ∈ (Finset.Icc (1 : ℤ) (↑n : ℤ) : Finset ℤ), ((1 : ℝ) - ((2 : ℝ) ^ (-m : ℤ) : ℝ) : ℝ) :
ℝ) :
ℝ) | ℕ → ℝ → ℝ | {
"benchmark": "putnam_solving",
"index": 128
} | [] | [
{
"t": "ℝ → ℝ",
"v": null,
"name": "f",
"t_type": "Type"
},
{
"t": "Differentiable ℝ f",
"v": null,
"name": "hfdiff",
"t_type": "Prop"
},
{
"t": "∀ x > 0, deriv f x = -3 * f x + 6 * f (2 * x)",
"v": null,
"name": "hfderiv",
"t_type": "Prop"
},
{
"t": "... | [
"((∀ n, μ n = answer n (μ 0)) ∧\n (∃ L, Tendsto (fun n ↦ (μ n) * 3 ^ n / n !) atTop (𝓝 L)) ∧\n (Tendsto (fun n ↦ (μ n) * 3 ^ n / n !) atTop (𝓝 0) → μ 0 = 0))"
] |
Can a countably infinite set have an uncountable collection of non-empty subsets such that the intersection of any two of them is finite? | True | open Nat Filter Topology Set | answer = True | Prop | {
"benchmark": "putnam_solving",
"index": 129
} | [] | [] | [
"(answer ↔\n (∃ S : Type,\n Countable S ∧ Infinite S ∧\n ∃ C : Set (Set S),\n ¬Countable C ∧\n (∀ R ∈ C, R ≠ ∅) ∧\n (∀ A ∈ C, ∀ B ∈ C, A ≠ B → (A ∩ B).Finite)\n ))"
] |
Let $T_0=2,T_1=3,T_2=6$, and for $n \geq 3$, $T_n=(n+4)T_{n-1}-4nT_{n-2}+(4n-8)T_{n-3}$. The first few terms are $2,3,6,14,40,152,784,5168,40576$. Find, with proof, a formula for $T_n$ of the form $T_n=A_n+B_n$, where $\{A_n\}$ and $\{B_n\}$ are well-known sequences. | fun n : ℕ => (n)! + 2 ^ n | open Filter Topology Nat | answer = fun (n : ℕ) => ((↑(n ! : ℕ) : ℤ) + ((2 : ℤ) ^ n : ℤ) : ℤ) | ℕ → ℤ | {
"benchmark": "putnam_solving",
"index": 130
} | [] | [
{
"t": "ℕ → ℤ",
"v": null,
"name": "T",
"t_type": "Type"
},
{
"t": "T 0 = 2 ∧ T 1 = 3 ∧ T 2 = 6",
"v": null,
"name": "hT012",
"t_type": "Prop"
},
{
"t": "∀ n, T (n + 3) = (n + 7) * T (n + 2) - 4 * (n + 3) * T (n + 1) + (4 * n + 4) * T n",
"v": null,
"name": "hTn",... | [
"(T = answer)"
] |
Is $\sqrt{2}$ the limit of a sequence of numbers of the form $\sqrt[3]{n}-\sqrt[3]{m}$ ($n,m=0,1,2,\dots$)? | True | open Filter Topology Nat | answer = True | Prop | {
"benchmark": "putnam_solving",
"index": 131
} | [] | [
{
"t": "ℝ → Prop",
"v": null,
"name": "numform",
"t_type": "Type"
},
{
"t": "∀ x : ℝ, numform x ↔ ∃ n m : ℕ, x = n ^ ((1 : ℝ) / 3) - m ^ ((1 : ℝ) / 3)",
"v": null,
"name": "hnumform",
"t_type": "Prop"
}
] | [
"(answer ↔ (∃ s : ℕ → ℝ, (∀ i : ℕ, numform (s i)) ∧ Tendsto s atTop (𝓝 (Real.sqrt 2))))"
] |
Consider a paper punch that can be centered at any point of the plane and that, when operated, removes from the plane precisely those points whose distance from the center is irrational. How many punches are needed to remove every point? | 3 | open Filter Topology Nat | answer = (3 : ℕ) | ℕ | {
"benchmark": "putnam_solving",
"index": 132
} | [] | [] | [
"(sInf {n : ℕ | ∃ S : Set (EuclideanSpace ℝ (Fin 2)), S.encard = n ∧ ∀ Q : EuclideanSpace ℝ (Fin 2), ∃ P ∈ S, Irrational (dist P Q)} = answer)"
] |
If $\mathbf{A}$ and $\mathbf{B}$ are square matrices of the same size such that $\mathbf{ABAB}=\mathbf{0}$, does it follow that $\mathbf{BABA}=\mathbf{0}$? | False | open Filter Topology Nat | answer = False | Prop | {
"benchmark": "putnam_solving",
"index": 133
} | [] | [] | [
"(answer ↔\n (∀ n ≥ 1, ∀ A B : Matrix (Fin n) (Fin n) ℝ,\n A * B * A * B = 0 → B * A * B * A = 0))"
] |
If $X$ is a finite set, let $|X|$ denote the number of elements in $X$. Call an ordered pair $(S,T)$ of subsets of $\{1,2,\dots,n\}$ \emph{admissible} if $s>|T|$ for each $s \in S$, and $t>|S|$ for each $t \in T$. How many admissible ordered pairs of subsets of $\{1,2,\dots,10\}$ are there? Prove your answer. | 17711 | open Filter Topology Nat | answer = (17711 : ℕ) | ℕ | {
"benchmark": "putnam_solving",
"index": 134
} | [] | [] | [
"(((Finset.univ : Finset <| Finset (Set.Icc 1 10) × Finset (Set.Icc 1 10)).filter\n fun ⟨S, T⟩ ↦ (∀ s ∈ S, T.card < s) ∧ (∀ t ∈ T, S.card < t)).card =\n answer)"
] |
Find all real-valued continuously differentiable functions $f$ on the real line such that for all $x$, $(f(x))^2=\int_0^x [(f(t))^2+(f'(t))^2]\,dt+1990$. | $\{\sqrt{1990} e^x, -\sqrt{1990} e^x\}$ | open Filter Topology Nat | answer =
(Insert.insert (fun (x : ℝ) => ((√(1990 : ℝ) : ℝ) * (Real.exp x : ℝ) : ℝ))
({fun (x : ℝ) => ((-(√(1990 : ℝ) : ℝ) : ℝ) * (Real.exp x : ℝ) : ℝ)} : Set (ℝ → ℝ)) :
Set (ℝ → ℝ)) | Set (ℝ → ℝ) | {
"benchmark": "putnam_solving",
"index": 135
} | [] | [
{
"t": "(ℝ → ℝ) → Prop",
"v": null,
"name": "P",
"t_type": "Type"
},
{
"t": "∀ f, P f ↔ ∀ x,\n (f x) ^ 2 = (∫ t in (0 : ℝ)..x, (f t) ^ 2 + (deriv f t) ^ 2) + 1990",
"v": null,
"name": "P_def",
"t_type": "Prop"
},
{
"t": "ℝ → ℝ",
"v": null,
"name": "f",
"t... | [
"(f ∈ answer ↔ (ContDiff ℝ 1 f ∧ P f))"
] |
Let $G$ be a finite group of order $n$ generated by $a$ and $b$. Prove or disprove: there is a sequence $g_1,g_2,g_3,\dots,g_{2n}$ such that
\begin{itemize}
\item[(1)] every element of $G$ occurs exactly twice, and
\item[(2)] $g_{i+1}$ equals $g_ia$ or $g_ib$ for $i=1,2,\dots,2n$. (Interpret $g_{2n+1}$ as $g_1$.)
\end{... | True | open Filter Topology Nat | answer = True | Prop | {
"benchmark": "putnam_solving",
"index": 136
} | [] | [] | [
"(∀ (G : Type*) (_ : Fintype G) (_ : Group G) (n : ℕ) (a b : G), (n = Fintype.card G ∧ G = Subgroup.closure {a, b} ∧ G ≠ Subgroup.closure {a} ∧ G ≠ Subgroup.closure {b}) → (∃ g : ℕ → G, (∀ x : G, {i : Fin (2 * n) | g i = x}.encard = 2)\n ∧ (∀ i : Fin (2 * n), (g ((i + 1) % (2 * n)) = g i * a) ∨ (g ((i + 1) % (2 * ... |
Is there an infinite sequence $a_0,a_1,a_2,\dots$ of nonzero real numbers such that for $n=1,2,3,\dots$ the polynomial $p_n(x)=a_0+a_1x+a_2x^2+\cdots+a_nx^n$ has exactly $n$ distinct real roots? | True | open Filter Polynomial Topology Nat | answer = True | Prop | {
"benchmark": "putnam_solving",
"index": 137
} | [] | [] | [
"((∃ a : ℕ → ℝ, (∀ i, a i ≠ 0) ∧\n (∀ n ≥ 1, (∑ i in Finset.Iic n, a i • X ^ i : Polynomial ℝ).roots.toFinset.card = n)) ↔\n answer)"
] |
Let $\mathbf{A}$ and $\mathbf{B}$ be different $n \times n$ matrices with real entries. If $\mathbf{A}^3=\mathbf{B}^3$ and $\mathbf{A}^2\mathbf{B}=\mathbf{B}^2\mathbf{A}$, can $\mathbf{A}^2+\mathbf{B}^2$ be invertible? | False | open Filter Topology | answer = False | Prop | {
"benchmark": "putnam_solving",
"index": 138
} | [] | [
{
"t": "ℕ",
"v": null,
"name": "n",
"t_type": "Type"
},
{
"t": "1 ≤ n",
"v": null,
"name": "hn",
"t_type": "Prop"
}
] | [
"(answer ↔ (∃ A B : Matrix (Fin n) (Fin n) ℝ,\n A ≠ B ∧ A ^ 3 = B ^ 3 ∧\n A ^ 2 * B = B ^ 2 * A ∧\n Nonempty (Invertible (A ^ 2 + B ^ 2))))"
] |
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