ChristianZ97/PutnamBench-lean4 huggingface dataset:
Problem: putnam_1987_a4
Tags: algebra

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 $|B-A|=10$. Find $|C-A|$.

Solution: Prove that $|C - A| = \frac{5}{3}\sqrt{30}$.

Lean 4 formal statement:
noncomputable abbrev putnam_1987_a4_solution : ℂ := sorry
theorem putnam_1987_a4
(P : MvPolynomial (Fin 3) ℂ)
(hPreal : ∀ i : Fin 3 →₀ ℕ, (coeff i P).im = 0)
(F : ℝ → ℝ → ℝ)
(vars : ℂ → ℂ → ℂ → (Fin 3 → ℂ))
(hvars : vars = fun a b c ↦ fun i ↦ ite (i = 0) a (ite (i = 1) b c))
(h : ∀ x y z u : ℝ, eval (vars (u * x) (u * y) (u * z)) P = u ^ 2 * F (y - x) (z - x))
(hPval : eval (vars 1 0 0) P = 4 ∧ eval (vars 0 1 0) P = 5 ∧ eval (vars 0 0 1) P = 6)
(A B C : ℂ)
(hPABC : eval (vars A B C) P = 0)
(habs : ‖B - A‖ = 10)
: (‖C - A‖ = putnam_1987_a4_solution) :=
sorry
Problem: putnam_1987_a5
Tags: analysis

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 $(x,y,z) \neq (0,0,0)$;
\item[(ii)] $\text{Curl}\,\vec{F}=\vec{0}$ for all $(x,y,z) \neq (0,0,0)$;
\item[(iii)] $\vec{F}(x,y,0)=\vec{G}(x,y)$.
\end{enumerate}

Solution: Show that there is no such $\vec{F}$.

Lean 4 formal statement:
abbrev putnam_1987_a5_solution : Prop := sorry
theorem putnam_1987_a5
    (curl : ((Fin 3 → ℝ) → (Fin 3 → ℝ)) → ((Fin 3 → ℝ) → (Fin 3 → ℝ)))
    (curl_def : ∀ f x, curl f x = ![
      fderiv ℝ f x (Pi.single 1 1) 2 - fderiv ℝ f x (Pi.single 2 1) 1,
      fderiv ℝ f x (Pi.single 2 1) 0 - fderiv ℝ f x (Pi.single 0 1) 2,
      fderiv ℝ f x (Pi.single 0 1) 1 - fderiv ℝ f x (Pi.single 1 1) 0])
    (G : (Fin 2 → ℝ) → (Fin 3 → ℝ))
    (G_def : ∀ x y, G ![x, y] = ![(-y / (x ^ 2 + 4 * y ^ 2)), (x / (x ^ 2 + 4 * y ^ 2)), 0]) :
    (∃ F : (Fin 3 → ℝ) → (Fin 3 → ℝ),
      ContDiffOn ℝ 1 F {v | v ≠ ![0,0,0]} ∧
      (∀ x, x ≠ 0 → curl F x = 0) ∧
      ∀ x y, F ![x, y, 0] = G ![x, y]) ↔ putnam_1987_a5_solution :=
  sorry
Problem: putnam_1987_a6
Tags: algebra, analysis

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?

Solution: Show that for positive $x$, the series converges if and only if $x < 25$.

Lean 4 formal statement:
abbrev putnam_1987_a6_solution : Set ℝ := sorry
theorem putnam_1987_a6
(a : ℕ → ℕ)
(ha : a = fun n ↦ {i | (digits 3 n).get i = 0}.ncard)
: ({x : ℝ | x > 0 ∧ Summable (fun n ↦ x ^ (a n) / (n ^ 3))} = putnam_1987_a6_solution) :=
sorry
Problem: putnam_1987_b1
Tags: analysis

Evaluate
\[
\int_2^4 \frac{\sqrt{\ln(9-x)}\,dx}{\sqrt{\ln(9-x)}+\sqrt{\ln(x+3)}}.
\]

Solution: Prove that the integral evaluates to $1$.

Lean 4 formal statement:
abbrev putnam_1987_b1_solution : ℝ := sorry
theorem putnam_1987_b1
: (∫ x in (2)..4, sqrt (log (9 - x)) / (sqrt (log (9 - x)) + sqrt (log (x + 3))) = putnam_1987_b1_solution) :=
sorry
Problem: putnam_1987_b2
Tags: algebra

Let $r, s$ and $t$ be integers with $0 \leq r$, $0 \leq s$ and $r+s \leq t$. Prove that
\[
\frac{\binom s0}{\binom tr}
+ \frac{\binom s1}{\binom{t}{r+1}} + \cdots
+ \frac{\binom ss}{\binom{t}{r+s}}
= \frac{t+1}{(t+1-s)\binom{t-s}{r}}.
\]

Solution: None.

Lean 4 formal statement:
theorem putnam_1987_b2
(r s t : ℕ)
(hsum : r + s ≤ t)
: (∑ i : Finset.range (s + 1), (choose s i : ℚ) / (choose t (r + i)) = ((t + 1) : ℚ) / ((t + 1 - s) * choose (t - s) r)) :=
sorry
Problem: putnam_1987_b3
Tags: abstract_algebra

Let $F$ be a field in which $1+1 \neq 0$. Show that the set of solutions to the equation $x^2+y^2=1$ with $x$ and $y$ in $F$ is given by $(x,y)=(1,0)$ and $(x,y)=\left(\frac{r^2-1}{r^2+1},\frac{2r}{r^2+1}\right)$, where $r$ runs through the elements of $F$ such that $r^2 \neq -1$.

Solution: None.

Lean 4 formal statement:
theorem putnam_1987_b3
(F : Type*) [Field F]
(hF : (1 : F) + 1 ≠ 0)
: {(x, y) : F × F | x ^ 2 + y ^ 2 = 1} = {(1, 0)} ∪ {((r ^ 2 - 1) / (r ^ 2 + 1), (2 * r) / (r ^ 2 + 1)) | r ∈ {r' : F | r' ^ 2 ≠ -1}} :=
sorry
Problem: putnam_1987_b4
Tags: analysis

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.

Solution: Show that $\lim_{n \to \infty} x_n = -1$ and $\lim_{n \to \infty} y_n = 0$.

Lean 4 formal statement:
abbrev putnam_1987_b4_solution : Prop × ℝ × Prop × ℝ := sorry
theorem putnam_1987_b4
    (x y : ℕ → ℝ)
    (hxy1 : (x 1, y 1) = (0.8, 0.6))
    (hx : ∀ n ≥ 1, x (n + 1) = (x n) * cos (y n) - (y n) * sin (y n))
    (hy : ∀ n ≥ 1, y (n + 1) = (x n) * sin (y n) + (y n) * cos (y n)) :
    let (existsx, limx, existsy, limy) := putnam_1987_b4_solution
    ((∃ c : ℝ, Tendsto x atTop (𝓝 c)) → existsx) ∧
    (existsx → Tendsto x atTop (𝓝 limx)) ∧
    ((∃ c : ℝ, Tendsto y atTop (𝓝 c)) → existsy) ∧
    (existsy → Tendsto y atTop (𝓝 limy)) :=
  sorry
Problem: putnam_1987_b5
Tags: linear_algebra

Let $O_n$ be the $n$-dimensional vector $(0,0,\cdots, 0)$. Let $M$ be a $2n \times n$ matrix of complex numbers such that whenever $(z_1, z_2, \dots, z_{2n})M = O_n$, with complex $z_i$, not all zero, then at least one of the $z_i$ is not real. Prove that for arbitrary real numbers $r_1, r_2, \dots, r_{2n}$, there are complex numbers $w_1, w_2, \dots, w_n$ such that
\[
\mathrm{re}\left[ M \left( \begin{array}{c} w_1 \\ \vdots \\ w_n \end{array} \right) \right] = \left( \begin{array}{c} r_1 \\ \vdots \\ r_{2n} \end{array} \right).
\]
(Note: if $C$ is a matrix of complex numbers, $\mathrm{re}(C)$ is the matrix whose entries are the real parts of the entries of $C$.)

Solution: None.

Lean 4 formal statement:
theorem putnam_1987_b5
(n : ℕ)
(npos : n > 0)
(M : Matrix (Fin (2 * n)) (Fin n) ℂ)
(hM : ∀ z : Matrix (Fin 1) (Fin (2 * n)) ℂ, z * M = 0 → (¬∀ i : Fin (2 * n), z 0 i = 0) → ∃ i : Fin (2 * n), (z 0 i).im ≠ 0)
: (∀ r : Matrix (Fin (2 * n)) (Fin 1) ℝ, ∃ w : Matrix (Fin n) (Fin 1) ℂ, ∀ i : (Fin (2 * n)), ((M * w) i 0).re = r i 0) :=
sorry
Problem: putnam_1987_b6
Tags: abstract_algebra

Let $F$ be the field of $p^2$ elements, where $p$ is an odd prime. Suppose $S$ is a set of $(p^2-1)/2$ distinct nonzero elements of $F$ with the property that for each $a\neq 0$ in $F$, exactly one of $a$ and $-a$ is in $S$. Let $N$ be the number of elements in the intersection $S \cap \{2a: a \in S\}$. Prove that $N$ is even.

Solution: None.

Lean 4 formal statement:
theorem putnam_1987_b6
    (p : ℕ)
    (F : Type*) [Field F] [Fintype F]
    (S : Set F)
    (hp : Odd p ∧ Nat.Prime p)
    (Fcard : Fintype.card F = p ^ 2)
    (Snz : ∀ x ∈ S, x ≠ 0)
    (Scard : S.ncard = ((p : ℤ) ^ 2 - 1) / 2)
    (hS : ∀ a : F, a ≠ 0 → Xor' (a ∈ S) (-a ∈ S)) :
    (Even ((S ∩ {x | ∃ a ∈ S, x = 2 * a}).ncard)) :=
  sorry
Problem: putnam_1988_a1
Tags: geometry

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 $R$.

Solution: Show that the area of $R$ is $6$.

Lean 4 formal statement:
abbrev putnam_1988_a1_solution : ℝ := sorry
theorem putnam_1988_a1
    (R : Set (Fin 2 → ℝ))
    (hR : R = {p | |p 0| - |p 1| ≤ 1 ∧ |p 1| ≤ 1}) :
    (volume R).toReal = putnam_1988_a1_solution :=
  sorry
Problem: putnam_1988_a2
Tags: analysis

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 $(a,b)$.

Solution: Show that such $(a,b)$ and $g$ exist.

Lean 4 formal statement:
abbrev putnam_1988_a2_solution : Prop := sorry
theorem putnam_1988_a2
    (f : ℝ → ℝ)
    (hf : f = fun x ↦ Real.exp (x ^ 2)) :
    putnam_1988_a2_solution ↔
    (∃ a b : ℝ,
      a < b ∧
      ∃ g : ℝ → ℝ,
        (∃ x ∈ Ioo a b, g x ≠ 0) ∧
        DifferentiableOn ℝ g (Ioo a b) ∧
        ∀ x ∈ Ioo a b, deriv (fun y ↦ f y * g y) x = (deriv f x) * (deriv g x)) :=
  sorry
Problem: putnam_1988_a3
Tags: analysis

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.

Solution: Show that the series converges if and only if $x > \frac{1}{2}$.

Lean 4 formal statement:
abbrev putnam_1988_a3_solution : Set ℝ := sorry
theorem putnam_1988_a3
: {x : ℝ | ∃ L : ℝ, Tendsto (fun t ↦ ∑ n ∈ Finset.Icc (1 : ℕ) t, (((1 / n) / Real.sin (1 / n) - 1) ^ x)) atTop (𝓝 L)} = putnam_1988_a3_solution :=
sorry
Problem: putnam_1988_a4
Tags: geometry, combinatorics

\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}

Solution: Prove that the points must exist with three colors, but not necessarily with nine.

Lean 4 formal statement:
abbrev putnam_1988_a4_solution : Prop × Prop := sorry
theorem putnam_1988_a4
    (p : ℕ → Prop)
    (hp : ∀ n, p n ↔
      ∀ color : (EuclideanSpace ℝ (Fin 2)) → Fin n,
        ∃ p q : EuclideanSpace ℝ (Fin 2),
          color p = color q ∧ dist p q = 1) :
    (let (a, b) := putnam_1988_a4_solution; (p 3 ↔ a) ∧ (p 9 ↔ b)) :=
  sorry
Problem: putnam_1988_a5
Tags: analysis

Prove that there exists a \emph{unique} function $f$ from the set $\mathrm{R}^+$ of positive real numbers to $\mathrm{R}^+$ such that
\[
f(f(x)) = 6x-f(x)
\]
and
\[
f(x)>0
\]
for all $x>0$.

Solution: None.

Lean 4 formal statement:
theorem putnam_1988_a5
: (∃ f : ℝ → ℝ, (∀ x > 0, f (f x) = 6 * x - f x ∧ f x > 0) ∧ (∀ g : ℝ → ℝ, (∀ x > 0, g (g x) = 6 * x - g x ∧ g x > 0) → (∀ x > 0, f x = g x))) :=
sorry
Problem: putnam_1988_a6
Tags: linear_algebra

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.

Solution: Show that the answer is yes, $A$ must be a scalar multiple of the identity.

Lean 4 formal statement:
abbrev putnam_1988_a6_solution : Prop := sorry
theorem putnam_1988_a6
: (∀ (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).encard = n) → LinearIndependent F sevecs)) → (∃ c : F, A = c • LinearMap.id)) ↔ putnam_1988_a6_solution :=
sorry
Problem: putnam_1988_b1
Tags: number_theory, algebra

A \emph{composite} (positive integer) is a product $ab$ with $a$ and $b$ not necessarily distinct integers in $\{2,3,4,\dots\}$. Show that every composite is expressible as $xy+xz+yz+1$, with $x,y,z$ positive integers.

Solution: None.

Lean 4 formal statement:
theorem putnam_1988_b1
: ∀ a ≥ 2, ∀ b ≥ 2, ∃ x y z : ℤ, x > 0 ∧ y > 0 ∧ z > 0 ∧ a * b = x * y + x * z + y * z + 1 :=
sorry
Problem: putnam_1988_b2
Tags: algebra

Prove or disprove: If $x$ and $y$ are real numbers with $y \geq 0$ and $y(y+1) \leq (x+1)^2$, then $y(y-1) \leq x^2$.

Solution: Show that this is true.

Lean 4 formal statement:
abbrev putnam_1988_b2_solution : Prop := sorry
theorem putnam_1988_b2
: (∀ x y : ℝ, (y ≥ 0 ∧ y * (y + 1) ≤ (x + 1) ^ 2) → (y * (y - 1) ≤ x ^ 2)) ↔ putnam_1988_b2_solution :=
sorry
Problem: putnam_1988_b3
Tags: algebra

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 $n \in N$.

Solution: Show that the smallest such $g$ is $(1+\sqrt{3})/2$.

Lean 4 formal statement:
noncomputable abbrev putnam_1988_b3_solution : ℝ := sorry
theorem putnam_1988_b3
    (r : ℤ → ℝ)
    (hr : ∀ n ≥ 1,
      (∃ c d : ℤ,
        (c ≥ 0 ∧ d ≥ 0) ∧
        c + d = n ∧ r n = |c - d * Real.sqrt 3|) ∧
        (∀ c d : ℤ, (c ≥ 0 ∧ d ≥ 0 ∧ c + d = n) → |c - d * Real.sqrt 3| ≥ r n))
    : IsLeast {g : ℝ | g > 0 ∧ (∀ n : ℤ, n ≥ 1 → r n ≤ g)} putnam_1988_b3_solution :=
  sorry
Problem: putnam_1988_b4
Tags: analysis

Prove that if $\sum_{n=1}^\infty a_n$ is a convergent series of positive real numbers, then so is $\sum_{n=1}^\infty (a_n)^{n/(n+1)}$.

Solution: None.

Lean 4 formal statement:
theorem putnam_1988_b4
    (a : ℕ → ℝ)
    (IsPosConv : (ℕ → ℝ) → Prop)
    (IsPosConv_def : ∀ a' : ℕ → ℝ, IsPosConv a' ↔
      (∀ n ≥ 1, a' n > 0) ∧
      (∃ s : ℝ, Tendsto (fun N : ℕ => ∑ n : Set.Icc 1 N, a' n) atTop (𝓝 s))) :
    (IsPosConv a) → IsPosConv (fun n : ℕ => (a n) ^ ((n : ℝ) / (n + 1))) :=
  sorry
Problem: putnam_1988_b5
Tags: linear_algebra

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 is the largest $k$ such that there is a $k \times k$ submatrix with nonzero determinant.) One may note that
\begin{align*}
M_1&=\begin{pmatrix} 0 & -1 & 1 \\ 1 & 0 & -1 \\ -1 & 1 & 0 \end{pmatrix} \\
M_2&=\begin{pmatrix} 0 & -1 & -1 & 1 & 1 \\ 1 & 0 & -1 & -1 & 1 \\ 1 & 1 & 0 & -1 & -1 \\ -1 & 1 & 1 & 0 & -1 \\ -1 & -1 & 1 & 1 & 0 \end{pmatrix}.
\end{align*}

Solution: Show that the rank of $M_n$ equals $2n$.

Lean 4 formal statement:
abbrev putnam_1988_b5_solution : ℕ → ℕ := sorry
theorem putnam_1988_b5
    (n : ℕ) (hn : n > 0)
    (Mn : Matrix (Fin (2 * n + 1)) (Fin (2 * n + 1)) ℝ)
    (Mnskewsymm : ∀ i j, Mn i j = -(Mn j i))
    (hMn1 : ∀ i j, (1 ≤ (i.1 : ℤ) - j.1 ∧ (i.1 : ℤ) - j.1 ≤ n) → Mn i j = 1)
    (hMnn1 : ∀ i j, (i.1 : ℤ) - j.1 > n → Mn i j = -1) :
    Mn.rank = putnam_1988_b5_solution n :=
  sorry
Problem: putnam_1988_b6
Tags: number_theory, algebra

Prove that there exist an infinite number of ordered pairs $(a,b)$ of integers such that for every positive integer $t$, the number $at+b$ is a triangular number if and only if $t$ is a triangular number. (The triangular numbers are the $t_n=n(n+1)/2$ with $n$ in $\{0,1,2,\dots\}$.)

Solution: None.

Lean 4 formal statement:
theorem putnam_1988_b6
(trinums : Set ℤ)
(htrinums : trinums = {t : ℤ | ∃ n : ℤ, t ≥ 0 ∧ t = (n * (n + 1)) / 2})
: {(a, b) : ℤ × ℤ | ∀ t > 0, (a * t + b) ∈ trinums ↔ t ∈ trinums}.encard = ⊤ :=
sorry
Problem: putnam_1989_a1
Tags: algebra, number_theory

How many primes among the positive integers, written as usual in base $10$, are alternating $1$'s and $0$'s, beginning and ending with $1$?

Solution: Show that there is only one such prime.

Lean 4 formal statement:
abbrev putnam_1989_a1_solution : ℕ∞ := sorry
theorem putnam_1989_a1
    (pdigalt : List ℕ → Prop)
    (hpdigalt : ∀ l, pdigalt l ↔ Odd l.length ∧ (∀ i, l.get i = if Even (i : ℕ) then 1 else 0)) :
    {p : ℕ | p.Prime ∧ pdigalt (Nat.digits 10 p)}.encard = putnam_1989_a1_solution :=
  sorry
Problem: putnam_1989_a2
Tags: analysis

Evaluate $\int_0^a \int_0^b e^{\max\{b^2x^2,a^2y^2\}}\,dy\,dx$ where $a$ and $b$ are positive.

Solution: Show that the value of the integral is $(e^{a^2b^2}-1)/(ab)$.

Lean 4 formal statement:
noncomputable abbrev putnam_1989_a2_solution : ℝ → ℝ → ℝ := sorry
theorem putnam_1989_a2
(a b : ℝ)
(abpos : a > 0 ∧ b > 0)
: ∫ x in Set.Ioo 0 a, ∫ y in Set.Ioo 0 b, Real.exp (max (b ^ 2 * x ^ 2) (a ^ 2 * y ^ 2)) = putnam_1989_a2_solution a b :=
sorry
Problem: putnam_1989_a3
Tags: algebra

Prove that if
\[
11z^{10}+10iz^9+10iz-11=0,
\]
then $|z|=1.$ (Here $z$ is a complex number and $i^2=-1$.)

Solution: None.

Lean 4 formal statement:
theorem putnam_1989_a3
(z : ℂ)
(hz : 11 * z ^ 10 + 10 * I * z ^ 9 + 10 * I * z - 11 = 0)
: (‖z‖ = 1) :=
sorry
Problem: putnam_1989_a6
Tags: algebra, abstract_algebra

Let $\alpha=1+a_1x+a_2x^2+\cdots$ be a formal power series with coefficients in the field of two elements. Let
\[
a_n =
\begin{cases}
1 & \parbox{2in}{if every block of zeros in the binary expansion of $n$ has an even number of zeros in the block} \\[.3in]
0 & \text{otherwise.}
\end{cases}
\]
(For example, $a_{36}=1$ because $36=100100_2$ and $a_{20}=0$ because $20=10100_2.$) Prove that $\alpha^3+x\alpha+1=0.$

Solution: None.

Lean 4 formal statement:
theorem putnam_1989_a6
(F : Type*) [Field F] [Fintype F]
(hF : Fintype.card F = 2)
(α : PowerSeries F)
(hα : ∀ n : ℕ, let bin := [1] ++ (digits 2 n) ++ [1]; PowerSeries.coeff n α = ite (∀ i j : Fin bin.length, i < j → bin.get i = 1 → bin.get j = 1 → (∀ k, i < k → k < j → bin.get k = 0) → Even ((j : ℕ) - (i : ℕ) - 1)) 1 0)
: (α ^ 3 + PowerSeries.X * α + 1 = 0) :=
sorry
Problem: putnam_1989_b1
Tags: 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 formal statement:
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 :=
  sorry
Problem: putnam_1989_b2
Tags: 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 formal statement:
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 :=
  sorry
Problem: putnam_1989_b3
Tags: 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 formal statement:
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) :=
  sorry
Problem: putnam_1989_b4
Tags: 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 formal statement:
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 :=
sorry
Problem: putnam_1989_b6
Tags: 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 formal statement:
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) :=
  sorry
Problem: putnam_1990_a1
Tags: 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 formal statement:
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 :=
  sorry
Problem: putnam_1990_a2
Tags: 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 formal statement:
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))) :=
sorry
Problem: putnam_1990_a4
Tags: 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 formal statement:
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 :=
sorry
Problem: putnam_1990_a5
Tags: 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 formal statement:
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) :=
sorry