ChristianZ97/PutnamBench-lean4 huggingface dataset:
Problem: putnam_2025_a2
Tags: analysis

Find the largest real number $a$ and the smallest real number $b$ such that
$$ax(\pi - x) \le \sin x \le bx(\pi - x)$$
for all $x$ in the interval $[0, \pi]$.

Solution: Show that the optimal constants are $a = \tfrac{1}{\pi}$ and $b = \tfrac{4}{\pi^2}$.

Lean 4 formal statement:
noncomputable abbrev putnam_2025_a2_solution : ℝ × ℝ := sorry
theorem putnam_2025_a2 (a b : ℝ) :
  ((a, b) = putnam_2025_a2_solution) ↔
  (IsGreatest {a' : ℝ | ∀ x ∈ Set.Icc 0 π, a' * x * (π - x) ≤ sin x} a ∧
   IsLeast {b' : ℝ | ∀ x ∈ Set.Icc 0 π, sin x ≤ b' * x * (π - x)} b) :=
sorry
Problem: putnam_2025_a3
Tags: combinatorics

Alice and Bob play a game with a string of $n$ digits, each of which is restricted
to be 0, 1, or 2. Initially all the digits are 0. A legal move is to add or subtract 1
from one digit to create a new string that has not appeared before. A player with no
legal move loses, and the other player wins. Alice goes first, and the players alternate
moves. For each $n \ge 1$, determine which player has a strategy that guarantees winning.

Solution: Show that Bob has a winning strategy for all $n \ge 1$.

Lean 4 formal statement:
abbrev putnam_2025_a3_solution : ℕ → Prop := sorry
theorem putnam_2025_a3 (n : ℕ) (hn : 1 ≤ n) :
    putnam_2025_a3_solution n ↔ AliceHasWinningStrategy n := by
  sorry
Problem: putnam_2025_a4
Tags: linear_algebra

Find the minimal value of $k$ such that there exist $k$-by-$k$ real matrices
$A_1, \ldots, A_{2025}$ with the property that $A_i A_j = A_j A_i$ if and only if
$|i - j| \in \{0, 1, 2024\}$.

Solution: Show that the minimal value is $3$.

Lean 4 formal statement:
abbrev putnam_2025_a4_solution : ℕ := sorry
theorem putnam_2025_a4 :
  IsLeast {k : ℕ | ∃ A : Fin 2025 → Matrix (Fin k) (Fin k) ℝ,
    ∀ i j : Fin 2025, i ≤ j →
      (A i * A j = A j * A i ↔ j.val - i.val ∈ ({0, 1, 2024} : Set ℕ))}
  putnam_2025_a4_solution :=
sorry
Problem: putnam_2025_a5
Tags: combinatorics

Let $n$ be an integer with $n \ge 2$. For a sequence $s = (s_1, \ldots, s_{n-1})$ where each
$s_i = \pm 1$, let $f(s)$ be the number of permutations $(a_1, \ldots, a_n)$ of $\{1, 2, \ldots, n\}$
such that $s_i(a_{i+1} - a_i) > 0$ for all $i$. For each $n$, determine the sequences $s$ for which
$f(s)$ is maximal.

Solution: Show that $f(s)$ is maximized exactly when the signs alternate, that is, when $s_i = (-1)^{i+1}$ for all $i$ or $s_i = (-1)^i$ for all $i$.

Lean 4 formal statement:
abbrev putnam_2025_a5_solution : (n : ℕ) → Set (Fin n → ℤˣ) := sorry
theorem putnam_2025_a5
    (n : ℕ)
    (hn : 1 ≤ n)
    (s : Fin n → ℤˣ) :
    (∀ t : Fin n → ℤˣ, f n t ≤ f n s) ↔ s ∈ putnam_2025_a5_solution n := by
  sorry
Problem: putnam_2025_a6
Tags: number_theory

Let $b_0 = 0$ and, for $n \ge 0$, define $b_{n+1} = 2b_n^2 + b_n + 1$.
For each $k \ge 1$, show that $b_{2^{k+1}} - 2b_{2^k}$ is divisible by $2^{2k+2}$
but not by $2^{2k+3}$.

Solution: None.

Lean 4 formal statement:
theorem putnam_2025_a6 (k : ℕ) (hk : 1 ≤ k) :
    (2 ^ (2 * k + 2) : ℤ) ∣ (b (2 ^ (k + 1)) - 2 * b (2 ^ k)) ∧
    ¬((2 ^ (2 * k + 3) : ℤ) ∣ (b (2 ^ (k + 1)) - 2 * b (2 ^ k))) := by
  sorry
Problem: putnam_2025_b1
Tags: geometry, combinatorics

Suppose that each point in the plane is colored either red or green, subject to the following
condition: For every three noncollinear points $A$, $B$, $C$ of the same color, the center of the
circle passing through $A$, $B$, $C$ is also this color. Prove that all points of the plane are the
same color.

Solution: Show that all points of the plane must have the same color.

Lean 4 formal statement:
theorem putnam_2025_b1
    (color : EuclideanSpace ℝ (Fin 2) → Bool)
    (h : ∀ (s : Simplex ℝ (EuclideanSpace ℝ (Fin 2)) 2),
      (∀ i j : Fin 3, color (s.points i) = color (s.points j)) →
      color s.circumcenter = color (s.points 0)) :
    ∃ c : Bool, ∀ P : EuclideanSpace ℝ (Fin 2), color P = c := by
  sorry
Problem: putnam_2025_b2
Tags: analysis, geometry

Let $f: [0,1] \to [0, \infty)$ be strictly increasing and continuous.
Let $R$ be the region bounded by $x = 0$, $x = 1$, $y = 0$, and $y = f(x)$.
Let $x_1$ be the $x$-coordinate of the centroid of $R$.
Let $x_2$ be the $x$-coordinate of the centroid of the solid generated by rotating $R$ about the $x$-axis.
Prove that $x_1 < x_2$.

Solution: None.

Lean 4 formal statement:
theorem putnam_2025_b2
    (f : ℝ → ℝ)
    (hf_cont : ContinuousOn f (Icc 0 1))
    (hf_mono : StrictMonoOn f (Icc 0 1))
    (hf_nonneg : ∀ x ∈ Icc (0 : ℝ) 1, 0 ≤ f x) :
    (∫ x in (0:ℝ)..1, x * f x) / (∫ x in (0:ℝ)..1, f x) <
    (∫ x in (0:ℝ)..1, x * (f x) ^ 2) / (∫ x in (0:ℝ)..1, (f x) ^ 2) := by
  sorry
Problem: putnam_2025_b3
Tags: number_theory

Suppose $S$ is a nonempty set of positive integers with the property that if $n$ is in $S$,
then every positive divisor of $2025^n - 15^n$ is in $S$. Must $S$ contain all positive integers?

Solution: Yes, $S$ must contain all positive integers.

Lean 4 formal statement:
abbrev putnam_2025_b3_solution : Prop := sorry
theorem putnam_2025_b3 :
    putnam_2025_b3_solution ↔
    ∀ S : Set ℕ,
      S.Nonempty →
      (∀ n ∈ S, 0 < n) →
      (∀ n ∈ S, ∀ d : ℕ, 0 < d → d ∣ (2025 ^ n - 15 ^ n) → d ∈ S) →
      S = {n : ℕ | 0 < n} := by
  sorry
Problem: putnam_2025_b4
Tags: linear_algebra, combinatorics

For $n \geq 2$, let $A = [a_{i,j}]_{i,j=1}^n$ be an $n$-by-$n$ matrix of nonnegative integers such that:
(a) $a_{i,j} = 0$ when $i + j \leq n$;
(b) $a_{i+1,j} \in \{a_{i,j}, a_{i,j} + 1\}$ when $1 \leq i \leq n-1$ and $1 \leq j \leq n$; and
(c) $a_{i,j+1} \in \{a_{i,j}, a_{i,j} + 1\}$ when $1 \leq i \leq n$ and $1 \leq j \leq n-1$.

Let $S$ be the sum of the entries of $A$, and let $N$ be the number of nonzero entries of $A$.
Prove that $S \leq \frac{(n+2)N}{3}$.

Solution: None.

Lean 4 formal statement:
theorem putnam_2025_b4
    (m : ℕ)
    (A : Matrix (Fin (m + 2)) (Fin (m + 2)) ℕ)
    (ha : ∀ (i j : Fin (m + 2)), (i : ℕ) + (j : ℕ) ≤ m → A i j = 0)
    (hb : ∀ (i : Fin (m + 1)) (j : Fin (m + 2)),
      A (Fin.succ i) j = A (Fin.castSucc i) j ∨
      A (Fin.succ i) j = A (Fin.castSucc i) j + 1)
    (hc : ∀ (i : Fin (m + 2)) (j : Fin (m + 1)),
      A i (Fin.succ j) = A i (Fin.castSucc j) ∨
      A i (Fin.succ j) = A i (Fin.castSucc j) + 1)
    (S : ℕ)
    (hS : S = ∑ i : Fin (m + 2), ∑ j : Fin (m + 2), A i j)
    (N : ℕ)
    (hN : N = #{p : Fin (m + 2) × Fin (m + 2) | A p.1 p.2 ≠ 0}) :
    3 * S ≤ (m + 4) * N := by
  sorry
Problem: putnam_2025_b5
Tags: number_theory

Let $p$ be a prime number greater than 3. For each $k \in \{1, \ldots, p-1\}$,
let $I(k) \in \{1, 2, \ldots, p-1\}$ be such that $k \cdot I(k) \equiv 1 \pmod{p}$.
Prove that the number of integers $k \in \{1, \ldots, p-2\}$ such that
$I(k+1) < I(k)$ is greater than $p/4 - 1$.

Solution: None.

Lean 4 formal statement:
theorem putnam_2025_b5
    (p : ℕ)
    (hp_prime : p.Prime)
    (hp_gt : 3 < p) :
    (p : ℚ) / 4 - 1 < descentCount p := by
  sorry
Problem: putnam_2025_b6
Tags: analysis, number_theory

Let $\mathbb{N} = \{1, 2, 3, \ldots\}$. Find the largest real constant $r$ such that
there exists a function $g: \mathbb{N} \to \mathbb{N}$ such that
$$g(n+1) - g(n) \geq (g(g(n)))^r$$
for all $n \in \mathbb{N}$.

Solution: Show that the largest possible value of $r$ is $1/4$.

Lean 4 formal statement:
noncomputable abbrev putnam_2025_b6_solution : ℝ := sorry
theorem putnam_2025_b6 :
    IsGreatest
      {r : ℝ | ∃ g : ℕ → ℕ, (∀ n : ℕ, 0 < n → 0 < g n) ∧
        ∀ n : ℕ, 0 < n → ((g (g n) : ℝ) ^ r) ≤ (g (n + 1) : ℝ) - (g n : ℝ)}
      putnam_2025_b6_solution := by
  sorry