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putnam_2025_a1
import Mathlib set_option pp.numericTypes true set_option pp.funBinderTypes true set_option maxHeartbeats 0 open Classical theorem putnam_2025_a1 (m n : β„• β†’ β„•) (h0 : m 0 > 0 ∧ n 0 > 0 ∧ m 0 β‰  n 0) (hm : βˆ€ k : β„•, m (k + 1) = ((2 * m k + 1) / (2 * n k + 1) : β„š).num) (hn : βˆ€ k : β„•, n (k + 1) = ((2 * m k + 1) / (2 * n k + 1) : β„š).den): {k | Β¬ (2 * m k + 1).Coprime (2 * n k + 1)}.Finite := by sorry
Let $m_0$ and $n_0$ be distinct positive integers. For every positive integer $k$, define $m_k$ and $n_k$ to be the relatively prime positive integers such that $$\frac{m_k}{n_k} = \frac{2m_{k-1} + 1}{2n_{k-1} + 1}.$$ Prove that $2m_k + 1$ and $2n_k + 1$ are relatively prime for all but finitely many positive integers $k$.
putnam_2025_a2
import Mathlib set_option pp.numericTypes true set_option pp.funBinderTypes true set_option maxHeartbeats 0 open Classical Set Real theorem putnam_2025_a2 : IsGreatest {a : ℝ | βˆ€ x ∈ Icc 0 Ο€, a * x * (Ο€ - x) ≀ sin x} (1 / Ο€) ∧ IsLeast {b : ℝ | βˆ€ x ∈ Icc 0 Ο€, b * x * (Ο€ - x) β‰₯ sin x} (4 / Ο€ ^ 2) := by sorry
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 $[0, \pi]$.
putnam_2025_a3
import Mathlib set_option pp.numericTypes true set_option pp.funBinderTypes true set_option maxHeartbeats 0 open Classical Finset abbrev State (n : β„•) : Type := (Fin n β†’ Fin 3) Γ— Finset (Fin n β†’ Fin 3) abbrev State.validNext {n : β„•} (s : State n) : Finset (State n) := {s' | s'.1 ∈ s.2 ∧ (βˆƒ (i : Fin n) (Ξ΄ : ({1, -1} : Finset β„€)), s'.1 i = s.1 i + Ξ΄.val ∧ βˆ€ j β‰  i, s'.1 j = s.1 j) ∧ s'.2 = s.2.erase s'.1} abbrev State.init (n : β„•) : State n := ⟨fun _ ↦ 0, Finset.univ.erase (fun _ ↦ 0)⟩ abbrev Strategy (n : β„•) : Type := State n β†’ State n abbrev IsRollout {n : β„•} (a b : Strategy n) (s : β„• β†’ State n) : Prop := s 0 = State.init n ∧ (βˆ€ i, Even i β†’ s (i + 1) = a (s i)) ∧ (βˆ€ i, Odd i β†’ s (i + 1) = b (s i)) theorem putnam_2025_a3 (n : β„•) (hn : 1 ≀ n): βˆƒ b : Strategy n, βˆ€ a : Strategy n, βˆ€ s : β„• β†’ State n, IsRollout a b s β†’ βˆƒ k : β„•, Even k ∧ (βˆ€ i < k, s (i + 1) ∈ (s i).validNext) ∧ s (k + 1) βˆ‰ (s k).validNext := by sorry
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.
putnam_2025_a4
import Mathlib set_option pp.numericTypes true set_option pp.funBinderTypes true set_option maxHeartbeats 0 open Classical theorem putnam_2025_a4 : IsLeast {k : β„• | k > 0 ∧ βˆƒ A : Fin 2025 β†’ Matrix (Fin k) (Fin k) ℝ, βˆ€ i j, A i * A j = A j * A i ↔ |(i : β„€) - (j : β„€)| ∈ ({0, 1, 2024} : Set β„€)} 3 := by sorry
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 $|i - j| \in \{0, 1, 2024\}$.
putnam_2025_b1
import Mathlib set_option pp.numericTypes true set_option pp.funBinderTypes true set_option maxHeartbeats 0 open EuclideanGeometry Affine Simplex theorem putnam_2025_b1 (c : EuclideanSpace ℝ (Fin 2) β†’ Bool) (hc : βˆ€ t : Triangle ℝ (EuclideanSpace ℝ (Fin 2)), c (t.points 0) = c (t.points 1) β†’ c (t.points 0) = c (t.points 2) β†’ c (t.points 0) = c (circumcenter t)): βˆƒ c0 : Bool, βˆ€ A, c A = c0 := by sorry
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.
putnam_2025_a6
import Mathlib set_option pp.numericTypes true set_option pp.funBinderTypes true set_option maxHeartbeats 0 open Classical theorem putnam_2025_a6 (b : β„• β†’ β„€) (hb0 : b 0 = 0) (hb : βˆ€ n, b (n + 1) = 2 * b n ^ 2 + b n + 1) (k : β„•) (hk : k β‰₯ 1): padicValInt 2 (b (2 ^ (k + 1)) - 2 * b (2 ^ k)) = 2 * k + 2 := by sorry
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 $2^{2k+3}$.
putnam_2025_b2
import Mathlib set_option pp.numericTypes true set_option pp.funBinderTypes true set_option maxHeartbeats 0 open BigOperators Real Nat Topology Rat MeasureTheory Set Real theorem putnam_2025_b2 (f : ℝ β†’ ℝ) (hf : βˆ€ x ∈ Icc 0 1, 0 ≀ f x) (hfm : StrictMono f) (hfc : Continuous f): let R : Set (ℝ Γ— ℝ) := { x | 0 ≀ x.1 ∧ x.1 ≀ 1 ∧ 0 ≀ x.2 ∧ x.2 ≀ f x.1 } let x₁ : ℝ := ⨍ x in R, x.1 let R' : Set (ℝ Γ— ℝ Γ— ℝ) := { x' | βˆƒ x ∈ R, βˆƒ ΞΈ : ℝ, x'.1 = x.1 ∧ x'.2.1 = x.2 * cos ΞΈ ∧ x'.2.2 = x.2 * sin ΞΈ } let xβ‚‚ : ℝ := ⨍ x' in R', x'.1 x₁ < xβ‚‚ := by sorry
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 $x_1 < x_2$.
putnam_2025_b3
import Mathlib set_option pp.numericTypes true set_option pp.funBinderTypes true set_option maxHeartbeats 0 open Classical theorem putnam_2025_b3 (S : Set β„•) (hS : S.Nonempty) (h0 : βˆ€ n ∈ S, 0 < n) (h : βˆ€ n ∈ S, βˆ€ d : β„•, 0 < d β†’ d ∣ 2025 ^ n - 15 ^ n β†’ d ∈ S) (n : β„•) (hn : 0 < n): n ∈ S := by sorry
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?
putnam_2025_b4
import Mathlib set_option pp.numericTypes true set_option pp.funBinderTypes true set_option maxHeartbeats 0 def MatrixProp (n : β„•) (A : β„• β†’ β„• β†’ β„•) : Prop := (βˆ€ i j : β„•, 1 ≀ i β†’ i ≀ n β†’ 1 ≀ j β†’ j ≀ n β†’ i + j ≀ n β†’ A i j = 0) ∧ (βˆ€ i j : β„•, 1 ≀ i β†’ i ≀ n - 1 β†’ 1 ≀ j β†’ j ≀ n β†’ A (i + 1) j = A i j ∨ A (i + 1) j = A i j + 1) ∧ (βˆ€ i j : β„•, 1 ≀ i β†’ i ≀ n β†’ 1 ≀ j β†’ j ≀ n - 1 β†’ A i (j + 1) = A i j ∨ A i (j + 1) = A i j + 1) def colSum (n : β„•) (A : β„• β†’ β„• β†’ β„•) (j : β„•) : β„• := Finset.sum (Finset.range n) (fun i => A (i + 1) j) def colNonzeros (n : β„•) (A : β„• β†’ β„• β†’ β„•) (j : β„•) : β„• := ((Finset.range n).filter (fun i => A (i + 1) j β‰  0)).card def totalSum (n : β„•) (A : β„• β†’ β„• β†’ β„•) : β„• := Finset.sum (Finset.range n) (fun j => colSum n A (j + 1)) def nonzeroCount (n : β„•) (A : β„• β†’ β„• β†’ β„•) : β„• := Finset.sum (Finset.range n) (fun j => colNonzeros n A (j + 1)) theorem putnam_2025_b4 (n : β„•) (hn : 2 ≀ n) (A : β„• β†’ β„• β†’ β„•) (h : MatrixProp n A): 3 * totalSum n A ≀ (n + 2) * nonzeroCount n A := by sorry
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 $S \leq \frac{(n+2)N}{3}$.
putnam_2025_b5
import Mathlib set_option pp.numericTypes true set_option pp.funBinderTypes true set_option maxHeartbeats 0 open Finset ZMod Classical theorem putnam_2025_b5 (p : β„•) [hp : Fact p.Prime] (hp3 : 3 < p): (#{k : ZMod p | k β‰  0 ∧ k β‰  -1 ∧ (k + 1)⁻¹.val < k⁻¹.val} : ℝ) > p / 4 - 1 := by sorry
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 $p/4 - 1$.
putnam_2025_b6
import Mathlib set_option pp.numericTypes true set_option pp.funBinderTypes true set_option maxHeartbeats 0 open BigOperators Real Nat Topology Rat theorem putnam_2025_b6 : IsGreatest {r : ℝ | βˆƒ g : β„•+ β†’ β„•+, βˆ€ n : β„•+, (g (n + 1) : ℝ) - (g n : ℝ) β‰₯ (g (g n) : ℝ) ^ r} (1/4 : ℝ) := by sorry
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 $n \in \mathbb{N}$.
putnam_2025_a5
import Mathlib set_option pp.numericTypes true set_option pp.funBinderTypes true set_option maxHeartbeats 0 open Finset def f (n : β„•) (s : Fin n β†’ β„€Λ£) : β„• := Finset.card {Οƒ : Equiv.Perm (Fin (n + 1)) | βˆ€ i : Fin n, 0 < (s i : β„€) * ((Οƒ i.succ : β„€) - (Οƒ i.castSucc : β„€))} theorem putnam_2025_a5 (n : β„•) (hn : 1 ≀ n) (s : Fin n β†’ β„€Λ£) : (βˆ€ t : Fin n β†’ β„€Λ£, f n t ≀ f n s) ↔ (βˆ€ i : Fin n, s i = (-1) ^ (i.val + 1)) ∨ (βˆ€ i : Fin n, s i = (-1) ^ i.val) := by sorry
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 $2^{2k+3}$.
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