ChristianZ97/putnambench-satp huggingface dataset:
Problem putnam_1999_a3 (algebra)

Consider the power series expansion \[\frac{1}{1-2x-x^2} = \sum_{n=0}^\infty a_n x^n.\] Prove that, for each integer $n\geq 0$, there is an integer $m$ such that \[a_n^2 + a_{n+1}^2 = a_m .\]

Solution: None.

Lean 4 formalization:
import Mathlib
open Filter Topology Metric
theorem putnam_1999_a3
    (f : ℝ → ℝ)
    (hf : f = fun x ↦ 1 / (1 - 2 * x - x ^ 2))
    (a : ℕ → ℝ)
    (hf' : ∀ᶠ x in 𝓝 0, Tendsto (fun N : ℕ ↦ ∑ n ∈ Finset.range N, (a n) * x ^ n) atTop (𝓝 (f x)))
    (n : ℕ) :
    ∃ m : ℕ, (a n) ^ 2 + (a (n + 1)) ^ 2 = a m := by
Problem putnam_1999_a4 (number_theory)

Sum the series \[\sum_{m=1}^\infty \sum_{n=1}^\infty \frac{m^2 n}{3^m(n3^m+m3^n)}.\]

Solution: Show that the solution is 9/32.

Lean 4 formalization:
import Mathlib
open Filter Topology Metric
noncomputable abbrev putnam_1999_a4_solution : ℝ := sorry
theorem putnam_1999_a4
: Tendsto (fun i => ∑ m ∈ Finset.range i, ∑' n : ℕ, (((m + 1)^2*(n+1))/(3^(m + 1) * ((n+1)*3^(m + 1) + (m + 1)*3^(n+1))) : ℝ)) atTop (𝓝 putnam_1999_a4_solution) := by
Problem putnam_1999_a5 (analysis)

Prove that there is a constant $C$ such that, if $p(x)$ is a polynomial of degree 1999, then \[|p(0)|\leq C \int_{-1}^1 |p(x)|\,dx.\]

Solution: None.

Lean 4 formalization:
import Mathlib
open Filter Topology Metric
theorem putnam_1999_a5
: ∃ C : ℝ, ∀ p : Polynomial ℝ, p.degree = 1999 → ‖p.eval 0‖ ≤ C * ∫ x in (-1)..1, ‖p.eval x‖ := by
Problem putnam_1999_a6 (algebra)

The sequence $(a_n)_{n\geq 1}$ is defined by $a_1=1, a_2=2, a_3=24,$ and, for $n\geq 4$, \[a_n = \frac{6a_{n-1}^2a_{n-3} - 8a_{n-1}a_{n-2}^2}{a_{n-2}a_{n-3}}.\] Show that, for all n, $a_n$ is an integer multiple of $n$.

Solution: None.

Lean 4 formalization:
import Mathlib
open Filter Topology Metric
theorem putnam_1999_a6
(a : ℤ → ℝ)
(ha1 : a 1 = 1)
(ha2 : a 2 = 2)
(ha3 : a 3 = 24)
(hange4 : ∀ n : ℕ, n ≥ 4 → a n = (6 * (a (n - 1))^2 * (a (n - 3)) - 8 * (a (n - 1)) * (a (n - 2))^2)/(a (n - 2) * a (n - 3)))
: ∀ n, n ≥ 1 → (∃ k : ℤ, a n = k * n) := by
Problem putnam_1999_b2 (analysis)

Let $P(x)$ be a polynomial of degree $n$ such that $P(x)=Q(x)P''(x)$, where $Q(x)$ is a quadratic polynomial and $P''(x)$ is the second derivative of $P(x)$.  Show that if $P(x)$ has at least two distinct roots then it must have $n$ distinct roots.

Solution: None.

Lean 4 formalization:
import Mathlib
open Filter Topology Metric
theorem putnam_1999_b2
  (P Q : Polynomial ℂ)
  (hQ : Q.natDegree = 2)
  (hP : ∀ x : ℂ, P.eval x = (Q.eval x) * (Polynomial.derivative^[2] P).eval x) :
  (∃ x1 x2, x1 ≠ x2 ∧ P.eval x1 = 0 ∧ P.eval x2 = 0) →
  (∃ f : Fin (P.natDegree) → ℂ,
    (∀ i j, i ≠ j → f i ≠ f j) ∧
    (∀ i, P.eval (f i) = 0)) := by
Problem putnam_1999_b3 (algebra)

Let $A=\{(x,y):0\leq x,y<1\}$.  For $(x,y)\in A$, let \[S(x,y) = \sum_{\frac{1}{2}\leq \frac{m}{n}\leq 2} x^m y^n,\] where the sum ranges over all pairs $(m,n)$ of positive integers satisfying the indicated inequalities.  Evaluate \[\lim_{(x,y)\rightarrow (1,1), (x,y)\in A} (1-xy^2)(1-x^2y)S(x,y).\]

Solution: Show that the answer is 3.

Lean 4 formalization:
import Mathlib
open Filter Topology Metric
abbrev putnam_1999_b3_solution : ℝ := sorry
theorem putnam_1999_b3
(A : Set (ℝ × ℝ))
(hA : A = {xy | 0 ≤ xy.1 ∧ xy.1 < 1 ∧ 0 ≤ xy.2 ∧ xy.2 < 1})
(S : ℝ → ℝ → ℝ)
(hS : S = fun x y => ∑' m : ℕ, ∑' n : ℕ, if (m > 0 ∧ n > 0 ∧ (1 : ℝ)/2 ≤ (m : ℝ)/n ∧ (m : ℝ)/n ≤ 2) then x^m * y^n else 0)
: Tendsto (fun xy : (ℝ × ℝ) => (1 - xy.1 * xy.2^2) * (1 - xy.1^2 * xy.2) * (S xy.1 xy.2)) (𝓝[A] ⟨1,1⟩) (𝓝 putnam_1999_b3_solution) := by
Problem putnam_1999_b4 (analysis)

Let $f$ be a real function with a continuous third derivative such that $f(x), f'(x), f''(x), f'''(x)$ are positive for all $x$.  Suppose that $f'''(x)\leq f(x)$ for all $x$.  Show that $f'(x)<2f(x)$ for all $x$.

Solution: None.

Lean 4 formalization:
import Mathlib
open Filter Topology Metric
theorem putnam_1999_b4
(f : ℝ → ℝ)
(hf : ContDiff ℝ 3 f)
(hpos: ∀ n ≤ 3, ∀ x : ℝ, iteratedDeriv n f x > 0)
(hle : ∀ x : ℝ, iteratedDeriv 3 f x ≤ f x)
: ∀ x : ℝ, deriv f x < 2 * (f x) := by
Problem putnam_1999_b5 (linear_algebra)

For an integer $n\geq 3$, let $\theta=2\pi/n$.  Evaluate the determinant of the $n\times n$ matrix $I+A$, where $I$ is the $n\times n$ identity matrix and $A=(a_{jk})$ has entries $a_{jk}=\cos(j\theta+k\theta)$ for all $j,k$.

Solution: Show that the answer is $(1 - n^2)/4$.

Lean 4 formalization:
import Mathlib
open Filter Topology Metric
noncomputable abbrev putnam_1999_b5_solution : ℕ → ℝ := sorry
theorem putnam_1999_b5
  (n : ℕ)
  (hn : n ≥ 3)
  (theta : ℝ)
  (htheta : theta = 2 * Real.pi / n)
  (A : Matrix (Fin n) (Fin n) ℝ)
  (hA : A = fun j k => Real.cos ((j.1 + 1) * theta + (k.1 + 1) * theta)) :
  (1 + A).det = putnam_1999_b5_solution n := by
Problem putnam_1999_b6 (number_theory)

Let $S$ be a finite set of integers, each greater than 1.  Suppose that for each integer $n$ there is some $s\in S$ such that $\gcd(s,n)=1$ or $\gcd(s,n)=s$.  Show that there exist $s,t\in S$ such that $\gcd(s,t)$ is prime.

Solution: None.

Lean 4 formalization:
import Mathlib
open Filter Topology Metric
theorem putnam_1999_b6
  (S : Finset ℤ)
  (hSgt : ∀ s, s ∈ S → s > 1)
  (hSgcd : ∀ n, ∃ s ∈ S, Int.gcd s n = 1 ∨ Int.gcd s n = s) :
  ∃ᵉ (s ∈ S) (t ∈ S), Prime (Int.gcd s t) := by
Problem putnam_2000_a1 (analysis)

Let $A$ be a positive real number. What are the possible values of $\sum_{j=0}^\infty x_j^2$, given that $x_0,x_1,\ldots$ are positive numbers for which $\sum_{j=0}^\infty x_j=A$?

Solution: Show that the possible values comprise the interval $(0,A^2)$.

Lean 4 formalization:
import Mathlib
open Topology Filter
abbrev putnam_2000_a1_solution : ℝ → Set ℝ := sorry
theorem putnam_2000_a1
(A : ℝ)
(Apos : A > 0)
: {S : ℝ |
    ∃ x : ℕ → ℝ,
      (∀ j : ℕ, x j > 0) ∧
      (∑' j : ℕ, x j) = A ∧
      (∑' j : ℕ, (x j) ^ 2) = S}
  = putnam_2000_a1_solution A := by
Problem putnam_2000_a2 (number_theory)

Prove that there exist infinitely many integers $n$ such that $n,n+1,n+2$ are each the sum of the squares of two integers.

Solution: None.

Lean 4 formalization:
import Mathlib
open Topology Filter
theorem putnam_2000_a2 :
  ∀ n : ℕ,
    ∃ N : ℤ,
      ∃ i : Fin 6 → ℕ,
        N > n ∧
        N = (i 0)^2 + (i 1)^2 ∧
        N + 1 = (i 2)^2 + (i 3)^2 ∧
        N + 2 = (i 4)^2 + (i 5)^2 := by
Problem putnam_2000_a4 (analysis)

Show that the improper integral $\lim_{B \to \infty} \int_0^B \sin(x)\sin(x^2)\,dx$ converges.

Solution: None.

Lean 4 formalization:
import Mathlib
open Topology Filter
theorem putnam_2000_a4
: ∃ y : ℝ, Tendsto (fun B : ℝ => ∫ x in Set.Ioo 0 B, Real.sin x * Real.sin (x ^ 2)) atTop (𝓝 y) := by
Problem putnam_2000_a5 (algebra)

Three distinct points with integer coordinates lie in the plane on a circle of radius $r>0$. Show that two of these points are separated by a distance of at least $r^{1/3}$.

Solution: None.

Lean 4 formalization:
import Mathlib
open Topology Filter
theorem putnam_2000_a5
  (r : ℝ)
  (z : EuclideanSpace ℝ (Fin 2))
  (p : Fin 3 → (EuclideanSpace ℝ (Fin 2)))
  (rpos : r > 0)
  (pdiff : ∀ n m, (n ≠ m) → (p n ≠ p m))
  (pint : ∀ n i, p n i = round (p n i))
  (pcirc : ∀ n, p n ∈ Metric.sphere z r) :
  ∃ n m, (n ≠ m) ∧ (dist (p n) (p m) ≥ r ^ ((1 : ℝ) / 3)) := by
Problem putnam_2000_a6 (algebra)

Let $f(x)$ be a polynomial with integer coefficients. Define a sequence $a_0,a_1,\ldots$ of integers such that $a_0=0$ and $a_{n+1}=f(a_n)$ for all $n\geq 0$. Prove that if there exists a positive integer $m$ for which $a_m=0$ then either $a_1=0$ or $a_2=0$.

Solution: None.

Lean 4 formalization:
import Mathlib
open Topology Filter
theorem putnam_2000_a6
(f : Polynomial ℤ)
(a : ℕ → ℤ)
(ha0 : a 0 = 0)
(ha : ∀ n : ℕ, a (n + 1) = f.eval (a n))
: ((∃ m > 0, a m = 0) → (a 1 = 0 ∨ a 2 = 0)) := by
Problem putnam_2000_b1 (algebra)

Let $a_j,b_j,c_j$ be integers for $1\leq j\leq N$. Assume for each $j$, at least one of $a_j,b_j,c_j$ is odd. Show that there exist integers $r$, $s$, $t$ such that $ra_j+sb_j+tc_j$ is odd for at least $4N/7$ values of $j$, $1\leq j\leq N$.

Solution: None.

Lean 4 formalization:
import Mathlib
open Topology Filter
theorem putnam_2000_b1
(N : ℕ)
(a b c : Fin N → ℤ)
(Nge1 : N ≥ 1)
(hodd : ∀ j : Fin N, Odd (a j) ∨ Odd (b j) ∨ Odd (c j))
: (∃ r s t : ℤ, {j : Fin N | Odd (r * a j + s * b j + t * c j)}.ncard ≥ (4 * N : ℝ) / 7) := by
Problem putnam_2000_b2 (number_theory, algebra)

Prove that the expression
\[
\frac{gcd(m,n)}{n}\binom{n}{m}
\]
is an integer for all pairs of integers $n\geq m\geq 1$.

Solution: None.

Lean 4 formalization:
import Mathlib
open Topology Filter Nat
theorem putnam_2000_b2
: (∀ m n : ℕ, m ≥ 1 → n ≥ m → n ∣ Nat.gcd m n * Nat.choose n m) := by
Problem putnam_2000_b3 (analysis)

Let $f(t)=\sum_{j=1}^N a_j \sin(2\pi jt)$, where each $a_j$ is real and $a_N$ is not equal to $0$. Let $N_k$ denote the number of zeroes (including multiplicities) of $\frac{d^k f}{dt^k}$. Prove that
\[
N_0\leq N_1\leq N_2\leq \cdots \mbox{ and } \lim_{k\to\infty} N_k = 2N.
\]

Solution: None.

Lean 4 formalization:
import Mathlib
open Topology Filter Nat Set Function
theorem putnam_2000_b3
  (N : ℕ) (hN : N > 0)
  (a : Icc 1 N → ℝ)
  (f : ℝ → ℝ)
  (mult : (ℝ → ℝ) → ℝ → ℕ)
  (M : ℕ → ℕ)
  (haN : a ⟨N, by simp; omega⟩ ≠ 0)
  (hf : ∀ t, f t = ∑ j : Icc 1 N, a j * Real.sin (2 * Real.pi * j * t))
  (hmult : ∀ g : ℝ → ℝ, ∀ t : ℝ, (∃ c : ℕ, iteratedDeriv c g t ≠ 0) → (iteratedDeriv (mult g t) g t ≠ 0 ∧ ∀ k < (mult g t), iteratedDeriv k g t = 0))
  (hM : ∀ k, M k = ∑' t : Ico (0 : ℝ) 1, mult (iteratedDeriv k f) t) :
  ((∀ i j : ℕ, i ≤ j → M i ≤ M j) ∧ Tendsto M atTop (𝓝 (2 * N))) := by
Problem putnam_2000_b4 (analysis)

Let $f(x)$ be a continuous function such that $f(2x^2-1)=2xf(x)$ for all $x$.  Show that $f(x)=0$ for $-1\leq x\leq 1$.

Solution: None.

Lean 4 formalization:
import Mathlib
open Topology Filter Nat Set Function
theorem putnam_2000_b4
(f : ℝ → ℝ)
(hfcont : Continuous f)
(hf : ∀ x, f (2 * x ^ 2 - 1) = 2 * x * f x)
: ∀ x, x ∈ Icc (-1) 1 → f x = 0 := by
Problem putnam_2000_b5 (algebra)

Let $S_0$ be a finite set of positive integers. We define finite sets $S_1,S_2,\ldots$ of positive integers as follows: the integer $a$ is in $S_{n+1}$ if and only if exactly one of $a-1$ or $a$ is in $S_n$. Show that there exist infinitely many integers $N$ for which $S_N=S_0\cup\{N+a: a\in S_0\}$.

Solution: None.

Lean 4 formalization:
import Mathlib
open Topology Filter Nat Set Function
theorem putnam_2000_b5
(S : ℕ → Set ℤ)
(hSfin : ∀ n, Set.Finite (S n))
(hSpos : ∀ n, ∀ s ∈ S n, s > 0)
(hSdef : ∀ n, ∀ a, a ∈ S (n + 1) ↔ Xor' (a - 1 ∈ S n) (a ∈ S n))
: (∀ n, ∃ N ≥ n, S N = S 0 ∪ {M : ℤ | M - N ∈ S 0}) := by
Problem putnam_2001_a1 (abstract_algebra)

Consider a set $S$ and a binary operation $*$, i.e., for each $a,b\in S$, $a*b\in S$.  Assume $(a*b)*a=b$ for all $a,b\in S$.  Prove that $a*(b*a)=b$ for all $a,b\in S$.

Solution: None.

Lean 4 formalization:
import Mathlib
open Topology Filter
theorem putnam_2001_a1
(S : Type*)
[Mul S]
(hS : ∀ a b : S, (a * b) * a = b)
: ∀ a b : S, a * (b * a) = b := by
Problem putnam_2001_a3 (algebra)

For each integer $m$, consider the polynomial
\[P_m(x)=x^4-(2m+4)x^2+(m-2)^2.\] For what values of $m$ is $P_m(x)$
the product of two non-constant polynomials with integer coefficients?

Solution: $P_m(x)$ factors into two nonconstant polynomials over
the integers if and only if $m$ is either a square or twice a square.

Lean 4 formalization:
import Mathlib
open Topology Filter Polynomial Set
abbrev putnam_2001_a3_solution : Set ℤ := sorry
theorem putnam_2001_a3
(P : ℤ → Polynomial ℤ)
(hP : P = fun m : ℤ => (Polynomial.X)^4 - (Polynomial.C (2*m + 4))*(Polynomial.X)^2 + Polynomial.C ((m - 2)^2))
: {m : ℤ | ∃ a b, P m = a * b ∧
(∃ n ∈ Ici 1, a.coeff n ≠ 0) ∧ (∃ n ∈ Ici 1, b.coeff n ≠ 0)} = putnam_2001_a3_solution := by
Problem putnam_2001_a5 (number_theory)

Prove that there are unique positive integers $a$, $n$ such that $a^{n+1}-(a+1)^n=2001$.

Solution: None.

Lean 4 formalization:
import Mathlib
open Topology Filter Polynomial Set
theorem putnam_2001_a5
: ∃! an : ℤ × ℕ, let (a, n) := an; a > 0 ∧ n > 0 ∧ a^(n+1) - (a+1)^n = 2001 := by
Problem putnam_2001_b1 (algebra)

Let $n$ be an even positive integer. Write the numbers $1,2,\ldots,n^2$ in the squares of an $n \times n$ grid so that the $k$-th row, from left to right, is $(k-1)n+1,(k-1)n+2,\ldots,(k-1)n+n$. Color the squares of the grid so that half of the squares in each row and in each column are red and the other half are black (a checkerboard coloring is one possibility). Prove that for each coloring, the sum of the numbers on the red squares is equal to the sum of the numbers on the black squares.

Solution: None.

Lean 4 formalization:
import Mathlib
open Topology Filter Polynomial Set
theorem putnam_2001_b1
  (n : ℕ)
  (hn : n > 0) (hn' : Even n)
  (nums : Fin n → Fin n → ℤ)
  (colors : Fin n → Fin n → Fin 2)
  (hnums : ∀ k l, nums k l = k * n + l + 1)
  (hcolorsrows : ∀ k, (∑ l, (if (colors k l = 0) then 1 else 0)) = n / 2)
  (hcolorscols : ∀ l, (∑ k, (if (colors k l = 0) then 1 else 0)) = n / 2) :
  (∑ k, ∑ l, (if (colors k l = 0) then (nums k l) else 0)) = (∑ k, ∑ l, (if (colors k l = 1) then (nums k l) else 0)) := by
Problem putnam_2001_b2 (algebra)

Find all pairs of real numbers $(x,y)$ satisfying the system of equations
\begin{align*}
\frac{1}{x}+\frac{1}{2y}&=(x^2+3y^2)(3x^2+y^2) \\
\frac{1}{x}-\frac{1}{2y}&=2(y^4-x^4).
\end{align*}

Solution: Show that $x=(3^{1/5}+1)/2$ and $y=(3^{1/5}-1)/2$ is the unique solution satisfying the given equations.

Lean 4 formalization:
import Mathlib
open Topology Filter Polynomial Set
abbrev putnam_2001_b2_solution : Set (ℝ × ℝ) := sorry
theorem putnam_2001_b2
    (x y : ℝ)
    (hx : x ≠ 0)
    (hy : y ≠ 0)
    (eq1 eq2 : Prop)
    (heq1 : eq1 ↔ (1 / x + 1 / (2 * y) = (x ^ 2 + 3 * y ^ 2) * (3 * x ^ 2 + y ^ 2)))
    (heq2 : eq2 ↔ (1 / x - 1 / (2 * y) = 2 * (y ^ 4 - x ^ 4))) :
    eq1 ∧ eq2 ↔ (x, y) ∈ putnam_2001_b2_solution := by
Problem putnam_2001_b3 (analysis)

For any positive integer $n$, let $\langle n \rangle$ denote the closest integer to $\sqrt{n}$. Evaluate $\sum_{n=1}^\infty \frac{2^{\langle n \rangle}+2^{-\langle n \rangle}}{2^n}$.

Solution: Show that the sum is $3$.

Lean 4 formalization:
import Mathlib
open Topology Filter Polynomial Set
abbrev putnam_2001_b3_solution : ℝ := sorry
theorem putnam_2001_b3
: ∑' n : Set.Ici 1, ((2 : ℝ) ^ (round (Real.sqrt n)) + (2 : ℝ) ^ (-round (Real.sqrt n))) / 2 ^ (n : ℝ) = putnam_2001_b3_solution := by
Problem putnam_2001_b4 (algebra)

Let $S$ denote the set of rational numbers different from $\{-1,0,1\}$.  Define $f:S\rightarrow S$ by $f(x)=x-1/x$. Prove or disprove that \[\bigcap_{n=1}^\infty f^{(n)}(S) = \emptyset,\] where $f^{(n)}$ denotes $f$ composed with itself $n$ times.

Solution: None.

Lean 4 formalization:
import Mathlib
open Topology Filter Polynomial Set
abbrev putnam_2001_b4_solution : Prop := sorry
theorem putnam_2001_b4
(S : Set ℚ)
(hS : S = univ \ {-1, 0, 1})
(f : S → S)
(hf : ∀ x : S, f x = x - 1 / (x : ℚ))
: ⋂ n ∈ Ici 1, f^[n] '' univ = ∅ ↔ putnam_2001_b4_solution := by
Problem putnam_2001_b5 (analysis)

Let $a$ and $b$ be real numbers in the interval $(0,1/2)$, and let $g$ be a continuous real-valued function such that $g(g(x))=ag(x)+bx$ for all real $x$. Prove that $g(x)=cx$ for some constant $c$.

Solution: None.

Lean 4 formalization:
import Mathlib
open Topology Filter Polynomial Set
theorem putnam_2001_b5
(a b : ℝ)
(g : ℝ → ℝ)
(abint : 0 < a ∧ a < 1 / 2 ∧ 0 < b ∧ b < 1 / 2)
(gcont : Continuous g)
(hg : ∀ x : ℝ, g (g x) = a * g x + b * x)
: ∃ c : ℝ, ∀ x : ℝ, g x = c * x := by
Problem putnam_2001_b6 (analysis)

Assume that $(a_n)_{n \geq 1}$ is an increasing sequence of positive real numbers such that $\lim a_n/n=0$. Must there exist infinitely many positive integers $n$ such that $a_{n-i}+a_{n+i}<2a_n$ for $i=1,2,\ldots,n-1$?

Solution: Show that the answer is yes, there must exist infinitely many such $n$.

Lean 4 formalization:
import Mathlib
open Topology Filter Polynomial Set
abbrev putnam_2001_b6_solution : Prop := sorry
theorem putnam_2001_b6 :
  (∀ a : ℤ → ℝ,
    (∀ n ≥ 1, a n > 0 ∧ a n < a (n + 1)) →
    (Tendsto (fun n : ℤ => a (n + 1) / (n + 1)) atTop (𝓝 0)) →
    {n : ℤ | n > 0 ∧ (∀ i ∈ Set.Icc 1 (n - 1), a (n - i) + a (n + i) < 2 * a n)}.Infinite)
  ↔ putnam_2001_b6_solution := by
Problem putnam_2002_a1 (analysis, algebra)

Let $k$ be a fixed positive integer. The $n$-th derivative of $\frac{1}{x^k-1}$ has the form $\frac{P_n(x)}{(x^k-1)^{n+1}}$ where $P_n(x)$ is a polynomial. Find $P_n(1)$.

Solution: Show that $P_n(1)=(-k)^nn!$ for all $n \geq 0$.

Lean 4 formalization:
import Mathlib
open Nat
abbrev putnam_2002_a1_solution : ℕ → ℕ → ℝ := sorry
theorem putnam_2002_a1
(k : ℕ)
(P : ℕ → Polynomial ℝ)
(kpos : k > 0)
(Pderiv : ∀ n x, iteratedDeriv n (fun x' : ℝ => 1 / (x' ^ k - 1)) x = ((P n).eval x) / ((x ^ k - 1) ^ (n + 1)))
: ∀ n, (P n).eval 1 = putnam_2002_a1_solution k n := by
Problem putnam_2002_a2 (geometry)

Given any five points on a sphere, show that some four of them must lie on a closed hemisphere.

Solution: None.

Lean 4 formalization:
import Mathlib
open Nat Metric
open scoped InnerProductSpace
theorem putnam_2002_a2
(unit_sphere : Set (EuclideanSpace ℝ (Fin 3)))
(hsphere : unit_sphere = sphere 0 1)
(hemi : EuclideanSpace ℝ (Fin 3) → Set (EuclideanSpace ℝ (Fin 3)))
(hhemi : hemi = fun V ↦ {P : EuclideanSpace ℝ (Fin 3) | ⟪P, V⟫_ℝ ≥ 0})
: (∀ (S : Set (EuclideanSpace ℝ (Fin 3))), S ⊆ unit_sphere ∧ S.encard = 5 → ∃ V : EuclideanSpace ℝ (Fin 3), V ≠ 0 ∧ (S ∩ hemi V).encard ≥ 4) := by
Problem putnam_2002_a3 (algebra)

Let $n \geq 2$ be an integer and $T_n$ be the number of non-empty subsets $S$ of $\{1, 2, 3, \dots, n\}$ with the property that the average of the elements of $S$ is an integer. Prove that $T_n - n$ is always even.

Solution: None.

Lean 4 formalization:
import Mathlib
open Nat
theorem putnam_2002_a3
(n Tn : ℤ)
(hn : n ≥ 2)
(hTn : Tn = Set.ncard {S : Set ℤ | S ⊆ Set.Icc 1 n ∧ Nonempty S ∧ ∃ k : ℤ, k = ((1 : ℝ) / S.ncard) * (∑' s : S, s.1)})
: Even (Tn - n) := by
Problem putnam_2002_a5 (number_theory, algebra)

Define a sequence by $a_0=1$, together with the rules
$a_{2n+1} = a_n$ and $a_{2n+2} = a_n + a_{n+1}$ for each
integer $n \geq 0$. Prove that every positive rational number
appears in the set
\[
\left\{ \frac{a_{n-1}}{a_n}: n \geq 1 \right\} =
\left\{ \frac{1}{1}, \frac{1}{2}, \frac{2}{1}, \frac{1}{3},
\frac{3}{2}, \dots \right\}.
\]

Solution: None.

Lean 4 formalization:
import Mathlib
open Nat Set
theorem putnam_2002_a5
(a : ℕ → ℚ)
(ha : a 0 = 1 ∧ ∀ n : ℕ, a (2*n + 1) = a n ∧ a (2*n + 2) = a n + a (n + 1))
: ∀ q : ℚ, q > 0 → q ∈ {a (n - 1) / a n | n ∈ Ici 1} := by
Problem putnam_2002_a6 (analysis, number_theory)

Fix an integer $b \geq 2$. Let $f(1) = 1$, $f(2) = 2$, and for each
$n \geq 3$, define $f(n) = n f(d)$, where $d$ is the number of
base-$b$ digits of $n$. For which values of $b$ does
\[
\sum_{n=1}^\infty \frac{1}{f(n)}
\]
converge?

Solution: The sum converges for $b=2$ and diverges for $b \geq 3$.

Lean 4 formalization:
import Mathlib
open Nat Set Topology Filter
abbrev putnam_2002_a6_solution : Set ℕ := sorry
theorem putnam_2002_a6
(f : ℕ → ℕ → ℝ)
(hf : ∀ b : ℕ, f b 1 = 1 ∧ f b 2 = 2 ∧ ∀ n ∈ Ici 3, f b n = n * f b (Nat.digits b n).length)
: {b ∈ Ici 2 | ∃ L : ℝ, Tendsto (fun m : ℕ => ∑ n ∈ Finset.Icc 1 m, 1/(f b n)) atTop (𝓝 L)} = putnam_2002_a6_solution := by
Problem putnam_2002_b3 (algebra)

Show that, for all integers $n > 1$,
\[
\frac{1}{2ne} < \frac{1}{e} - \left( 1 - \frac{1}{n} \right)^n
< \frac{1}{ne}.
\]

Solution: None.

Lean 4 formalization:
import Mathlib
open Nat Set Topology Filter
theorem putnam_2002_b3
(e : ℝ)
(he : e = Real.exp 1)
(f : ℤ → ℝ)
(hf : f = fun n : ℤ => 1/e - (1 - 1/n)^n)
: ∀ n : ℤ, n > 1 → 1/(2*n*e) < f n ∧ f n < 1/(n*e) := by
Problem putnam_2002_b5 (number_theory)

A palindrome in base $b$ is a positive integer whose base-$b$
digits read the same backwards and forwards; for example,
$2002$ is a 4-digit palindrome in base 10. Note that 200 is not
a palindrome in base 10, but it is the 3-digit palindrome
242 in base 9, and 404 in base 7. Prove that there is an integer
which is a 3-digit palindrome in base $b$ for at least 2002
different values of $b$.

Solution: None.

Lean 4 formalization:
import Mathlib
open Nat Set Topology Filter
theorem putnam_2002_b5
: ∃ n : ℕ, {b : ℕ | b ≥ 1 ∧ (Nat.digits b n).length = 3 ∧ List.Palindrome (Nat.digits b n)}.ncard ≥ 2002 := by
Problem putnam_2002_b6 (linear_algebra, number_theory, algebra)

Let $p$ be a prime number. Prove that the determinant of the matrix
\[
\begin{pmatrix}
x & y & z \\
x^p & y^p & z^p \\
x^{p^2} & y^{p^2} & z^{p^2}
\end{pmatrix}
\]
is congruent modulo $p$ to a product of polynomials of the form
$ax+by+cz$, where $a,b,c$ are integers. (We say two integer
polynomials are congruent modulo $p$ if corresponding coefficients
are congruent modulo $p$.)

Solution: None.

Lean 4 formalization:
import Mathlib
open Nat Set Topology Filter Matrix MvPolynomial
theorem putnam_2002_b6
(p : ℕ)
(hp : Nat.Prime p)
(M : Matrix (Fin 3) (Fin 3) (MvPolynomial (Fin 3) ℤ))
(hM : M = fun (r c : Fin 3) => ((X c)^(p^(r : ℕ)) : MvPolynomial (Fin 3) ℤ))
(cong : ℕ → MvPolynomial (Fin 3) ℤ × MvPolynomial (Fin 3) ℤ → Prop)
(hcong : cong = fun p : ℕ => fun (f, g) => ∀ n : Fin 3 →₀ ℕ, Int.ModEq p (f.coeff n) (g.coeff n))
: ∃ S : Finset (MvPolynomial (Fin 3) ℤ), cong p ((det M), (∏ s ∈ S, s)) ∧ ∀ s ∈ S, (∃ a b c : ℤ, s = (C a)*(X 0) + (C b)*(X 1) + (C c)*(X 2)) := by
Problem putnam_2003_a1 (algebra)

Let $n$ be a fixed positive integer. How many ways are there to write $n$ as a sum of positive integers, \[ n = a_1 + a_2 + \dots + a_k, \] with $k$ an arbitrary positive integer and $a_1 \leq a_2 \leq \dots \leq a_k \leq a_1 + 1$? For example, with $n = 4$, there are four ways: $4, 2 + 2, 1 + 1 + 2, 1 + 1 + 1 + 1$

Solution: Show that there are $n$ such sums.

Lean 4 formalization:
import Mathlib
open MvPolynomial
abbrev putnam_2003_a1_solution : ℕ → ℕ := sorry
theorem putnam_2003_a1
(n : ℕ) (hn : n > 0)
: Set.encard {a : ℕ → ℤ |
  ∃ k > 0, (∑ i : Fin k, a i = n) ∧
    (∀ i : Fin k, a i > 0) ∧
    (∀ i : Fin (k - 1), a i ≤ a (i + 1)) ∧
    a (k - 1) ≤ a 0 + 1 ∧ (∀ i ≥ k, a i = 0)}
  = putnam_2003_a1_solution n := by
Problem putnam_2003_a2 (algebra)

Let $a_1,a_2,\dots,a_n$ and $b_1,b_2,\dots,b_n$ be nonnegative real numbers. Show that $(a_1a_2 \cdots a_n)^{1/n}+(b_1b_2 \cdots b_n)^{1/n} \leq [(a_1+b_1)(a_2+b_2) \cdots (a_n+b_n)]^{1/n}$.

Solution: None.

Lean 4 formalization:
import Mathlib
open MvPolynomial
theorem putnam_2003_a2
    (n : ℕ)
    (hn : 0 < n)
    (a b : Fin n → ℝ)
    (abnneg : ∀ i, a i ≥ 0 ∧ b i ≥ 0) :
    (∏ i, a i) ^ ((1 : ℝ) / n) +
    (∏ i, b i) ^ ((1 : ℝ) / n) ≤
    (∏ i, (a i + b i)) ^ ((1 : ℝ) / n) := by
Problem putnam_2003_a3 (analysis)

Find the minimum value of $|\sin x+\cos x+\tan x+\cot x+\sec x+\csc x|$ for real numbers $x$.

Solution: Show that the minimum is $2\sqrt{2}-1$.

Lean 4 formalization:
import Mathlib
open Set
noncomputable abbrev putnam_2003_a3_solution : ℝ := sorry
theorem putnam_2003_a3
    (f : ℝ → ℝ)
    (hf : ∀ x : ℝ, f x = |Real.sin x + Real.cos x + Real.tan x + 1 / Real.tan x + 1 / Real.cos x + 1 / Real.sin x|) :
    IsLeast (Set.range f) putnam_2003_a3_solution := by
Problem putnam_2003_a4 (algebra)

Suppose that $a,b,c,A,B,C$ are real numbers, $a \ne 0$ and $A \ne 0$, such that $|ax^2+bx+c| \leq |Ax^2+Bx+C|$ for all real numbers $x$. Show that $|b^2-4ac| \leq |B^2-4AC|$.

Solution: None.

Lean 4 formalization:
import Mathlib
open MvPolynomial
theorem putnam_2003_a4
(a b c A B C : ℝ)
(aAne0 : a ≠ 0 ∧ A ≠ 0)
(hle : ∀ x : ℝ, |a * x ^ 2 + b * x + c| ≤ |A * x ^ 2 + B * x + C|)
: |b ^ 2 - 4 * a * c| ≤ |B ^ 2 - 4 * A * C| := by