ChristianZ97/putnambench-satp huggingface dataset:
Problem putnam_1989_b1 (probability, geometry)

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 $b$, $d$ are positive.

Solution: Show that the probability is $(4\sqrt{2}-5)/3$.

Lean 4 formalization:
import Mathlib
open Nat MeasureTheory
abbrev putnam_1989_b1_solution : ℤ × ℤ × ℤ × ℤ := sorry
theorem putnam_1989_b1
    (square Scloser perimeter: Set (EuclideanSpace ℝ (Fin 2)))
    (center : EuclideanSpace ℝ (Fin 2))
    (square_def : square = {p | ∀ i : Fin 2, p i ∈ Set.Icc 0 1})
    (perimeter_def : perimeter = {p ∈ square | p 0 = 0 ∨ p 0 = 1 ∨ p 1 = 0 ∨ p 1 = 1})
    (center_def : center =  WithLp.toLp 2 (fun i : Fin 2 => 1 / 2))
    (hScloser : Scloser = {p ∈ square | ∀ q ∈ perimeter, dist p center < dist p q}) :
    let (a, b, c, d) := putnam_1989_b1_solution;
    b > 0 ∧ d > 0 ∧ (¬∃ n : ℤ, n^2 = b) ∧
    (volume Scloser).toReal / (volume square).toReal = (a * Real.sqrt b + c) / d := by
Problem putnam_1989_b2 (abstract_algebra)

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?

Solution: Prove that $S$ must be a group.

Lean 4 formalization:
import Mathlib
open Nat
abbrev putnam_1989_b2_solution : Prop := sorry
theorem putnam_1989_b2 :
    (∀ (S : Type) [Nonempty S] [Semigroup S] [IsCancelMul S]
      (h_fin : ∀ a : S, {(a * ·)^[n] a | n : ℕ}.Finite),
      ∃ e : S, ∀ x, e * x = x ∧ x * e = x ∧ ∃ y, x * y = e ∧ y * x = e) ↔
    putnam_1989_b2_solution := by
Problem putnam_1989_b3 (analysis)

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 moment of $f$).
\begin{enumerate}
\item[a)] Express $\mu_n$ in terms of $\mu_0$.
\item[b)] Prove that the sequence $\{\mu_n \frac{3^n}{n!}\}$ always converges, and that the limit is $0$ only if $\mu_0=0$.
\end{enumerate}

Solution: Show that for each $n \geq 0$, $\mu_n = \frac{n!}{3^n} \left( \prod_{m=1}^{n}(1 - 2^{-m}) \right)^{-1} \mu_0$.

Lean 4 formalization:
import Mathlib
open Nat Filter Topology
noncomputable abbrev putnam_1989_b3_solution : ℕ → ℝ → ℝ := sorry
theorem putnam_1989_b3
    (f : ℝ → ℝ)
    (hfdiff : Differentiable ℝ f)
    (hfderiv : ∀ x > 0, deriv f x = -3 * f x + 6 * f (2 * x))
    (hdecay : ∀ x ≥ 0, |f x| ≤ Real.exp (- √x))
    (μ : ℕ → ℝ)
    (μ_def : ∀ n, μ n = ∫ x in Set.Ioi 0, x ^ n * f x) :
    (∀ n, μ n = putnam_1989_b3_solution n (μ 0)) ∧
    (∃ L, Tendsto (fun n ↦ (μ n) * 3 ^ n / n !) atTop (𝓝 L)) ∧
    (Tendsto (fun n ↦ (μ n) * 3 ^ n / n !) atTop (𝓝 0) → μ 0 = 0) := by
Problem putnam_1989_b4 (set_theory)

Can a countably infinite set have an uncountable collection of non-empty subsets such that the intersection of any two of them is finite?

Solution: Prove that such a collection exists.

Lean 4 formalization:
import Mathlib
open Nat Filter Topology Set
abbrev putnam_1989_b4_solution : Prop := sorry
theorem putnam_1989_b4 :
  (∃ S : Type,
    Countable S ∧ Infinite S ∧
    ∃ C : Set (Set S),
      ¬Countable C ∧
      (∀ R ∈ C, R ≠ ∅) ∧
      (∀ A ∈ C, ∀ B ∈ C, A ≠ B → (A ∩ B).Finite)
  ) ↔ putnam_1989_b4_solution := by
Problem putnam_1989_b6 (probability, analysis, algebra)

Let $(x_1,x_2,\dots,x_n)$ be a point chosen at random from the $n$-dimensional region defined by $0< x_1< x_2<\dots< x_n<1$. Let $f$ be a continuous function on $[0,1]$ with $f(1)=0$. Set $x_0=0$ and $x_{n+1}=1$. Show that the expected value of the Riemann sum $\sum_{i=0}^n (x_{i+1}-x_i)f(x_{i+1})$ is $\int_0^1 f(t)P(t)\,dt$, where $P$ is a polynomial of degree $n$, independent of $f$, with $0 \leq P(t) \leq 1$ for $0 \leq t \leq 1$.

Solution: None.

Lean 4 formalization:
import Mathlib
open Nat Filter Topology Set ProbabilityTheory
theorem putnam_1989_b6
    (n : ℕ) [NeZero n]
    (I : (Fin n → ℝ) → Fin (n + 2) → ℝ)
    (I_def : ∀ x, I x = Fin.cons 0 (Fin.snoc x 1))
    (X : Set (Fin n → ℝ))
    (X_def : ∀ x, x ∈ X ↔ 0 < x 0 ∧ x (-1) < 1 ∧ StrictMono x)
    (S : (ℝ → ℝ) → (Fin (n + 2) → ℝ) → ℝ)
    (S_def : ∀ f x, S f x = ∑ i : Fin n.succ, (x i.succ - x i.castSucc) * f (x i.succ)) :
    ∃ P : Polynomial ℝ,
      P.degree = n ∧
      (∀ t ∈ Icc 0 1, P.eval t ∈ Icc 0 1) ∧
      (∀ f : ℝ → ℝ, f 1 = 0 → ContinuousOn f (Icc 0 1) →
        ∫ x, S f (I x) ∂ℙ[|X] = ∫ t in (0)..1, f t * P.eval t) := by
Problem putnam_1990_a1 (algebra)

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.

Solution: Show that we have $T_n=n!+2^n$.

Lean 4 formalization:
import Mathlib
open Filter Topology Nat
abbrev putnam_1990_a1_solution : (ℕ → ℤ) × (ℕ → ℤ) := sorry
theorem putnam_1990_a1
    (T : ℕ → ℤ)
    (hT012 : T 0 = 2 ∧ T 1 = 3 ∧ T 2 = 6)
    (hTn : ∀ n, T (n + 3) = (n + 7) * T (n + 2) - 4 * (n + 3) * T (n + 1) + (4 * n + 4) * T n) :
    T = putnam_1990_a1_solution.1 + putnam_1990_a1_solution.2 := by
Problem putnam_1990_a2 (analysis)

Is $\sqrt{2}$ the limit of a sequence of numbers of the form $\sqrt[3]{n}-\sqrt[3]{m}$ ($n,m=0,1,2,\dots$)?

Solution: Show that the answer is yes.

Lean 4 formalization:
import Mathlib
open Filter Topology Nat
abbrev putnam_1990_a2_solution : Prop := sorry
theorem putnam_1990_a2
  (numform : ℝ → Prop)
  (hnumform : ∀ x : ℝ, numform x ↔ ∃ n m : ℕ, x = n ^ ((1 : ℝ) / 3) - m ^ ((1 : ℝ) / 3)) :
  putnam_1990_a2_solution ↔
  (∃ s : ℕ → ℝ,
    (∀ i : ℕ, numform (s i)) ∧
    Tendsto s atTop (𝓝 (Real.sqrt 2))) := by
Problem putnam_1990_a4 (set_theory, number_theory)

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?

Solution: Show that three punches are needed.

Lean 4 formalization:
import Mathlib
open Filter Topology Nat
abbrev putnam_1990_a4_solution : ℕ := sorry
theorem putnam_1990_a4
: sInf {n : ℕ | ∃ S : Set (EuclideanSpace ℝ (Fin 2)), S.encard = n ∧ ∀ Q : EuclideanSpace ℝ (Fin 2), ∃ P ∈ S, Irrational (dist P Q)} = putnam_1990_a4_solution := by
Problem putnam_1990_a5 (linear_algebra)

If $\mathbf{A}$ and $\mathbf{B}$ are square matrices of the same size such that $\mathbf{ABAB}=\mathbf{0}$, does it follow that $\mathbf{BABA}=\mathbf{0}$?

Solution: Show that the answer is no.

Lean 4 formalization:
import Mathlib
open Filter Topology Nat
abbrev putnam_1990_a5_solution : Prop := sorry
theorem putnam_1990_a5 :
  putnam_1990_a5_solution ↔
  (∀ n ≥ 1, ∀ A B : Matrix (Fin n) (Fin n) ℝ,
    A * B * A * B = 0 → B * A * B * A = 0) := by
Problem putnam_1990_a6 (algebra)

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.

Solution: Show that the number of admissible ordered pairs of subsets of $\{1,2,\dots,10\}$ equals the $22$nd Fibonacci number $F_{22}=17711$.

Lean 4 formalization:
import Mathlib
open Filter Topology Nat
abbrev putnam_1990_a6_solution : ℕ := sorry
theorem putnam_1990_a6 :
    ((Finset.univ : Finset <| Finset (Set.Icc 1 10) × Finset (Set.Icc 1 10)).filter
      fun ⟨S, T⟩ ↦ (∀ s ∈ S, T.card < s) ∧ (∀ t ∈ T, S.card < t)).card =
    putnam_1990_a6_solution := by
Problem putnam_1990_b1 (analysis)

Find all real-valued continuously differentiable functions $f$ on the real line such that for all $x$, $(f(x))^2=\int_0^x [(f(t))^2+(f'(t))^2]\,dt+1990$.

Solution: Show that there are two such functions, namely $f(x)=\sqrt{1990}e^x$, and $f(x)=-\sqrt{1990}e^x$.

Lean 4 formalization:
import Mathlib
open Filter Topology Nat
abbrev putnam_1990_b1_solution : Set (ℝ → ℝ) := sorry
theorem putnam_1990_b1
    (P : (ℝ → ℝ) → Prop)
    (P_def : ∀ f, P f ↔ ∀ x,
      (f x) ^ 2 = (∫ t in (0 : ℝ)..x, (f t) ^ 2 + (deriv f t) ^ 2) + 1990)
    (f : ℝ → ℝ) :
    (ContDiff ℝ 1 f ∧ P f) ↔ f ∈ putnam_1990_b1_solution := by
Problem putnam_1990_b2 (analysis)

Prove that for $|x|<1$, $|z|>1$, $1+\sum_{j=1}^\infty (1+x^j)P_j=0$, where $P_j$ is $\frac{(1-z)(1-zx)(1-zx^2) \cdots (1-zx^{j-1})}{(z-x)(z-x^2)(z-x^3) \cdots (z-x^j)}$.

Solution: None.

Lean 4 formalization:
import Mathlib
open Filter Topology Nat
theorem putnam_1990_b2
(x z : ℝ)
(P : ℕ → ℝ)
(xlt1 : |x| < 1)
(zgt1 : |z| > 1)
(hP : ∀ j ≥ 1, P j = (∏ i : Fin j, (1 - z * x ^ (i : ℕ))) / (∏ i : Set.Icc 1 j, (z - x ^ (i : ℕ))))
: 1 + (∑' j : Set.Ici 1, (1 + x ^ (j : ℕ)) * P j) = 0 := by
Problem putnam_1990_b3 (linear_algebra)

Let $S$ be a set of $2 \times 2$ integer matrices whose entries $a_{ij}$ (1) are all squares of integers, and, (2) satisfy $a_{ij} \leq 200$. Show that if $S$ has more than $50387$ ($=15^4-15^2-15+2$) elements, then it has two elements that commute.

Solution: None.

Lean 4 formalization:
import Mathlib
open Filter Topology Nat
theorem putnam_1990_b3
(S : Set (Matrix (Fin 2) (Fin 2) ℕ))
(hS : ∀ A ∈ S, ∀ i j : Fin 2, (∃ x : ℤ, A i j = x ^ 2) ∧ A i j ≤ 200)
: (S.encard > 50387) → (∃ A ∈ S, ∃ B ∈ S, A ≠ B ∧ A * B = B * A) := by
Problem putnam_1990_b4 (abstract_algebra)

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{itemize}

Solution: Show that such a sequence does exist.

Lean 4 formalization:
import Mathlib
open Filter Topology Nat
abbrev putnam_1990_b4_solution : Prop := sorry
theorem putnam_1990_b4
: (∀ (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)
  ∧ (∀ i : Fin (2 * n), (g ((i + 1) % (2 * n)) = g i * a) ∨ (g ((i + 1) % (2 * n)) = g i * b))) ↔ putnam_1990_b4_solution) := by
Problem putnam_1990_b5 (algebra, analysis)

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?

Solution: Show that the answer is yes, such an infinite sequence exists.

Lean 4 formalization:
import Mathlib
open Filter Polynomial Topology Nat
abbrev putnam_1990_b5_solution : Prop := sorry
theorem putnam_1990_b5 :
    (∃ a : ℕ → ℝ, (∀ i, a i ≠ 0) ∧
      (∀ n ≥ 1, (∑ i ∈ Finset.Iic n, a i • X ^ i : Polynomial ℝ).roots.toFinset.card = n)) ↔
    putnam_1990_b5_solution := by
Problem putnam_1991_a2 (linear_algebra)

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?

Solution: Show that the answer is no.

Lean 4 formalization:
import Mathlib
open Filter Topology
abbrev putnam_1991_a2_solution : Prop := sorry
theorem putnam_1991_a2
  (n : ℕ) (hn : 1 ≤ n) :
  putnam_1991_a2_solution ↔ (∃ A B : Matrix (Fin n) (Fin n) ℝ,
    A ≠ B ∧ A ^ 3 = B ^ 3 ∧
    A ^ 2 * B = B ^ 2 * A ∧
    Nonempty (Invertible (A ^ 2 + B ^ 2))) := by
Problem putnam_1991_a3 (algebra, analysis)

Find all real polynomials $p(x)$ of degree $n \geq 2$ for which there exist real numbers $r_1< r_2<\cdots< r_n$ such that
\begin{enumerate}
\item $p(r_i)=0, \qquad i=1,2,\dots,n$, and
\item $p'(\frac{r_i+r_{i+1}}{2})=0 \qquad i=1,2,\dots,n-1$,
\end{enumerate}
where $p'(x)$ denotes the derivative of $p(x)$.

Solution: Show that the real polynomials with the required property are exactly those that are of degree $2$ with $2$ distinct real zeros.

Lean 4 formalization:
import Mathlib
open Filter Topology
abbrev putnam_1991_a3_solution : Set (Polynomial ℝ) := sorry
theorem putnam_1991_a3
  (p : Polynomial ℝ)
  (n : ℕ)
  (hn : n = p.degree)
  (hge : n ≥ 2) :
  p ∈ putnam_1991_a3_solution ↔
    (∃ r : ℕ → ℝ, (∀ i : Fin (n - 1), r i < r (i + 1)) ∧
    (∀ i : Fin n, p.eval (r i) = 0) ∧
    (∀ i : Fin (n - 1), (Polynomial.derivative p).eval ((r i + r (i + 1)) / 2) = 0)) := by
Problem putnam_1991_a4 (geometry, analysis)

Does there exist an infinite sequence of closed discs $D_1,D_2,D_3,\dots$ in the plane, with centers $c_1,c_2,c_3,\dots$, respectively, such that
\begin{enumerate}
\item the $c_i$ have no limit point in the finite plane,
\item the sum of the areas of the $D_i$ is finite, and
\item every line in the plane intersects at least one of the $D_i$?
\end{enumerate}

Solution: Show that the answer is yes, such a sequence of closed discs exists.

Lean 4 formalization:
import Mathlib
open Filter Metric Topology
abbrev putnam_1991_a4_solution : Prop := sorry
theorem putnam_1991_a4 :
    (∃ (c : ℕ → EuclideanSpace ℝ (Fin 2)) (r : ℕ → ℝ),
      (¬ ∃ p, MapClusterPt p atTop c) ∧
      (Summable <| fun i ↦ r i ^ 2) ∧
      (∀ L : AffineSubspace ℝ (EuclideanSpace ℝ (Fin 2)),
        Module.finrank ℝ L.direction = 1 → ∃ i, (↑L ∩ closedBall (c i) (r i)).Nonempty)) ↔
    putnam_1991_a4_solution := by
Problem putnam_1991_a5 (analysis)

Find the maximum value of $\int_0^y \sqrt{x^4+(y-y^2)^2}\,dx$ for $0 \leq y \leq 1$.

Solution: Show that the maximum value of the integral is $1/3$.

Lean 4 formalization:
import Mathlib
open Filter Topology
noncomputable abbrev putnam_1991_a5_solution : ℝ := sorry
theorem putnam_1991_a5
  (f : Set.Icc (0 : ℝ) 1 → ℝ)
  (hf : ∀ y : Set.Icc 0 1, f y = ∫ x in Set.Ioo 0 y, Real.sqrt (x ^ 4 + (y - y ^ 2) ^ 2)) :
  IsGreatest (f '' (Set.Icc 0 1)) putnam_1991_a5_solution := by
Problem putnam_1991_a6 (algebra)

Let $A(n)$ denote the number of sums of positive integers $a_1+a_2+\cdots+a_r$ which add up to $n$ with $a_1>a_2+a_3,a_2>a_3+a_4,\dots,a_{r-2}>a_{r-1}+a_r,a_{r-1}>a_r$. Let $B(n)$ denote the number of $b_1+b_2+\cdots+b_s$ which add up to $n$, with
\begin{enumerate}
\item $b_1 \geq b_2 \geq \dots \geq b_s$,
\item each $b_i$ is in the sequence $1,2,4,\dots,g_j,\dots$ defined by $g_1=1$, $g_2=2$, and $g_j=g_{j-1}+g_{j-2}+1$, and
\item if $b_1=g_k$ then every element in $\{1,2,4,\dots,g_k\}$ appears at least once as a $b_i$.
\end{enumerate}
Prove that $A(n)=B(n)$ for each $n \geq 1$. (For example, $A(7)=5$ because the relevant sums are $7,6+1,5+2,4+3,4+2+1$, and $B(7)=5$ because the relevant sums are $4+2+1,2+2+2+1,2+2+1+1+1,2+1+1+1+1+1,1+1+1+1+1+1+1$.)

Solution: None.

Lean 4 formalization:
import Mathlib
open Filter Topology
theorem putnam_1991_a6
  (nabsum : ℕ → ℕ × (ℕ → ℕ) → Prop)
  (agt bge bg1 bg2 : ℕ × (ℕ → ℕ) → Prop)
  (A g B: ℕ → ℕ)
  (hnabsum : ∀ᵉ (n ≥ 1) (ab), nabsum n ab ↔
    (ab.1 ≥ 1 ∧ (∀ i < ab.1, ab.2 i > 0) ∧
    (∀ i ≥ ab.1, ab.2 i = 0) ∧ (∑ i : Fin ab.1, ab.2 i) = n))
  (hA : ∀ n ≥ 1, A n = {a | nabsum n a ∧
    (∀ i : Fin (a.1 - 2), a.2 i > a.2 (i + 1) + a.2 (i + 2)) ∧ 1 < a.1 → a.2 (a.1 - 2) > a.2 (a.1 - 1)}.encard)
  (hg : g 0 = 1 ∧ g 1 = 2 ∧ (∀ j ≥ 2, g j = g (j - 1) + g (j - 2) + 1))
  (hB : ∀ n ≥ 1, B n = {b | nabsum n b ∧
    (∀ i : Fin (b.1 - 1), b.2 i ≥ b.2 (i + 1)) ∧
    (∀ i : Fin b.1, ∃ j : ℕ, b.2 i = g j) ∧
    (∃ k : ℕ, b.2 0 = g k ∧ (∀ j ≤ k, ∃ i : Fin b.1, b.2 i = g j))}.encard) :
  ∀ n ≥ 1, (A n) = (B n) := by
Problem putnam_1991_b1 (algebra)

For each integer $n \geq 0$, let $S(n)=n-m^2$, where $m$ is the greatest integer with $m^2 \leq n$. Define a sequence $(a_k)_{k=0}^\infty$ by $a_0=A$ and $a_{k+1}=a_k+S(a_k)$ for $k \geq 0$. For what positive integers $A$ is this sequence eventually constant?

Solution: Show that this sequence is eventually constant if and only if $A$ is a perfect square.

Lean 4 formalization:
import Mathlib
open Filter Topology
abbrev putnam_1991_b1_solution : Set ℤ := sorry
theorem putnam_1991_b1
  (m S : ℤ → ℤ)
  (A : ℤ)
  (a : ℕ → ℤ)
  (hm : ∀ n, 0 ≤ n → (m n) ^ 2 ≤ n ∧ (∀ m' : ℤ, m' ^ 2 ≤ n → m' ≤ m n))
  (hS : ∀ n, 0 ≤ n → S n = n - (m n) ^ 2)
  (ha : a 0 = A ∧ (∀ k, a (k + 1) = a k + S (a k)))
  (hA : A > 0) :
  (∃ (K : ℕ) (c : ℕ), ∀ k ≥ K, a k = c) ↔ A ∈ putnam_1991_b1_solution := by
Problem putnam_1991_b2 (analysis)

Suppose $f$ and $g$ are non-constant, differentiable, real-valued functions defined on $(-\infty,\infty)$. Furthermore, suppose that for each pair of real numbers $x$ and $y$,
\begin{align*}
f(x+y)&=f(x)f(y)-g(x)g(y), \\
g(x+y)&=f(x)g(y)+g(x)f(y).
\end{align*}
If $f'(0)=0$, prove that $(f(x))^2+(g(x))^2=1$ for all $x$.

Solution: None.

Lean 4 formalization:
import Mathlib
open Filter Topology
theorem putnam_1991_b2
  (f g : ℝ → ℝ)
  (fgnconst : ¬∃ c, f = Function.const ℝ c ∨ g = Function.const ℝ c)
  (fgdiff : Differentiable ℝ f ∧ Differentiable ℝ g)
  (fadd : ∀ x y, f (x + y) = f x * f y - g x * g y)
  (gadd : ∀ x y, g (x + y) = f x * g y + g x * f y) :
  (deriv f 0 = 0) → (∀ x, (f x) ^ 2 + (g x) ^ 2 = 1) := by
Problem putnam_1991_b4 (number_theory, algebra)

Suppose $p$ is an odd prime. Prove that $\sum_{j=0}^p \binom{p}{j}\binom{p+j}{j} \equiv 2^p+1 \pmod{p^2}$.

Solution: None.

Lean 4 formalization:
import Mathlib
open Filter Topology
theorem putnam_1991_b4
(p : ℕ)
(podd : Odd p)
(pprime : Prime p)
: (∑ j : Fin (p + 1), (p.choose j) * ((p + j).choose j)) ≡ (2 ^ p + 1) [MOD (p ^ 2)] := by
Problem putnam_1991_b5 (number_theory)

Let $p$ be an odd prime and let $\mathbb{Z}_p$ denote (the field of) integers modulo $p$. How many elements are in the set $\{x^2:x \in \mathbb{Z}_p\} \cap \{y^2+1:y \in \mathbb{Z}_p\}$?

Solution: Show that the number of elements in the intersection is $\lceil p/4 \rceil$.

Lean 4 formalization:
import Mathlib
open Filter Topology
abbrev putnam_1991_b5_solution : ℕ → ℕ := sorry
theorem putnam_1991_b5
(p : ℕ)
(podd : Odd p)
(pprime : Prime p)
: ({z : ZMod p | ∃ x : ZMod p, z = x ^ 2} ∩ {z : ZMod p | ∃ y : ZMod p, z = y ^ 2 + 1}).encard = putnam_1991_b5_solution p := by
Problem putnam_1991_b6 (analysis)

Let $a$ and $b$ be positive numbers. Find the largest number $c$, in terms of $a$ and $b$, such that $a^xb^{1-x} \leq a\frac{\sinh ux}{\sinh u}+b\frac{\sinh u(1-x)}{\sinh u}$ for all $u$ with $0<|u| \leq c$ and for all $x$, $0< x<1$. (Note: $\sinh u=(e^u-e^{-u})/2$.)

Solution: Show that the largest $c$ for which the inequality holds for $0<|u| \leq c$ is $c=|\ln(a/b)|$.

Lean 4 formalization:
import Mathlib
open Filter Topology
noncomputable abbrev putnam_1991_b6_solution : ℝ → ℝ → ℝ := sorry
theorem putnam_1991_b6
  (a b : ℝ)
  (abpos : a > 0 ∧ b > 0) :
  IsGreatest {c | ∀ u,
    (0 < |u| ∧ |u| ≤ c) →
    (∀ x ∈ Set.Ioo 0 1, a ^ x * b ^ (1 - x) ≤ a * (Real.sinh (u * x) / Real.sinh u) + b * (Real.sinh (u * (1 - x)) / Real.sinh u))}
  (putnam_1991_b6_solution a b) := by
Problem putnam_1992_a1 (algebra)

Prove that $f(n) = 1-n$ is the only integer-valued function defined on the integers that satisfies the following conditions.
\begin{itemize}
\item[(i)] $f(f(n)) = n$, for all integers $n$;
\item[(ii)] $f(f(n+2)+2) = n$ for all integers $n$;
\item[(iii)] $f(0) = 1$.
\end{itemize}

Solution: None.

Lean 4 formalization:
import Mathlib
open Topology Filter
theorem putnam_1992_a1
  (f : ℤ → ℤ) :
  (f = fun n ↦ 1 - n) ↔
  (∀ n : ℤ, f (f n) = n) ∧ (∀ n : ℤ, f (f (n + 2) + 2) = n) ∧ (f 0 = 1) := by
Problem putnam_1992_a2 (analysis, algebra)

Define $C(\alpha)$ to be the coefficient of $x^{1992}$ in the power series about $x=0$ of $(1 + x)^\alpha$. Evaluate
\[
\int_0^1 \left( C(-y-1) \sum_{k=1}^{1992} \frac{1}{y+k} \right)\,dy.
\]

Solution: Prove that the integral evaluates to $1992$.

Lean 4 formalization:
import Mathlib
open Topology Filter
abbrev putnam_1992_a2_solution : ℝ := sorry
theorem putnam_1992_a2
(C : ℝ → ℝ)
(hC : C = fun α ↦ taylorCoeffWithin (fun x ↦ (1 + x) ^ α) 1992 Set.univ 0)
: (∫ y in (0)..1, C (-y - 1) * ∑ k ∈ Finset.Icc (1 : ℕ) 1992, 1 / (y + k) = putnam_1992_a2_solution) := by
Problem putnam_1992_a3 (algebra, number_theory)

For a given positive integer $m$, find all triples $(n, x, y)$ of positive integers, with $n$ relatively prime to $m$, which satisfy
\[
(x^2 + y^2)^m = (xy)^n.
\]

Solution: Prove that if $m$ is odd, there are no solutions, and if $m$ is even, the only solution is
$(n, x, y) = (m + 1, 2 ^ {m/2}, 2 ^{m/2})$.

Lean 4 formalization:
import Mathlib
open Topology Filter Nat
abbrev putnam_1992_a3_solution : ℕ → Set (ℕ × ℕ × ℕ) := sorry
theorem putnam_1992_a3
(m : ℕ)
(mpos : m > 0)
(S : Set (ℕ × ℕ × ℕ))
(hS : ∀ n x y : ℕ, (n, x, y) ∈ S ↔ n > 0 ∧ x > 0 ∧ y > 0 ∧ Coprime n m ∧ (x ^ 2 + y ^ 2) ^ m = (x * y) ^ n)
: (S = putnam_1992_a3_solution m) := by
Problem putnam_1992_a4 (analysis)

Let $f$ be an infinitely differentiable real-valued function defined on the real numbers. If
\[
f\left( \frac{1}{n} \right) = \frac{n^2}{n^2 + 1}, \qquad n = 1, 2, 3, \dots,
\]
compute the values of the derivatives $f^{(k)}(0), k = 1, 2, 3, \dots$.

Solution: Prove that
\[
f^{(k)}(0) =
\begin{cases}
(-1)^{k/2}k! & \text{if $k$ is even;} \\
0 & \text{if $k$ is odd.} \\
\end{cases}
\]

Lean 4 formalization:
import Mathlib
open Topology Filter Nat Function
abbrev putnam_1992_a4_solution : ℕ → ℝ := sorry
theorem putnam_1992_a4
(f : ℝ → ℝ)
(hfdiff : ContDiff ℝ ⊤ f)
(hf : ∀ n : ℕ, n > 0 → f (1 / n) = n ^ 2 / (n ^ 2 + 1))
: (∀ k : ℕ, k > 0 → iteratedDeriv k f 0 = putnam_1992_a4_solution k) := by
Problem putnam_1992_a5 (algebra)

For each positive integer $n$, let $a_n = 0$ (or $1$) if the number of $1$'s in the binary representation of $n$ is even (or odd), respectively. Show that there do not exist positive integers $k$ and $m$ such that
\[
a_{k+j} = a_{k+m+j} = a_{k+2m+j},
\]
for $0 \leq j \leq m-1$.

Solution: None.

Lean 4 formalization:
import Mathlib
open Topology Filter Nat Function
theorem putnam_1992_a5
  (a : ℕ → ℕ)
  (ha : a = fun n ↦ ite (Even {i | (digits 2 n).get i = 1}.ncard) 0 1) :
  ¬∃ k > 0, ∃ m > 0, ∀ j ≤ m - 1,
    a (k + j) = a (k + m + j) ∧ a (k + m + j) = a (k + 2 * m + j) := by
Problem putnam_1992_b1 (algebra)

Let $S$ be a set of $n$ distinct real numbers. Let $A_S$ be the set of numbers that occur as averages of two distinct elements of $S$. For a given $n \geq 2$, what is the smallest possible number of elements in $A_S$?

Solution: Show that the answer is $2n - 3$.

Lean 4 formalization:
import Mathlib
open Topology Filter Nat Function
abbrev putnam_1992_b1_solution : ℕ → ℤ := sorry
theorem putnam_1992_b1
  (n : ℕ) (hn : n ≥ 2)
  (A : Finset ℝ → Set ℝ)
  (hA : A = fun S ↦ {x | ∃ a ∈ S, ∃ b ∈ S, a ≠ b ∧ (a + b) / 2 = x}) :
  IsLeast {k : ℤ | ∃ S : Finset ℝ, S.card = n ∧ k = (A S).ncard} (putnam_1992_b1_solution n) := by
Problem putnam_1992_b2 (algebra)

For nonnegative integers $n$ and $k$, define $Q(n, k)$ to be the coefficient of $x^k$ in the expansion of $(1 + x + x^2 + x^3)^n$. Prove that
\[
Q(n, k) = \sum_{j=0}^k \binom{n}{j} \binom{n}{k-2j},
\]
where $\binom{a}{b}$ is the standard binomial coefficient. (Reminder: For integers $a$ and $b$ with $a \geq 0$, $\binom{a}{b} = \frac{a!}{b!(a-b)!}$ for $0 \leq b \leq a$, with $\binom{a}{b} = 0$ otherwise.)

Solution: None.

Lean 4 formalization:
import Mathlib
open Topology Filter Nat Function Polynomial
theorem putnam_1992_b2
    (Q : ℕ → ℕ → ℕ)
    (hQ : Q = fun n k ↦ coeff ((1 + X + X ^ 2 + X ^ 3) ^ n) k)
    (n k : ℕ) :
    Q n k = ∑ j ∈ Finset.Iic k, choose n j * (if 2 * j ≤ k then choose n (k - 2 * j) else 0) := by