ChristianZ97/PutnamBench-lean4 huggingface dataset:
Problem: putnam_1990_a6
Tags: 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 formal statement:
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 :=
  sorry
Problem: putnam_1990_b1
Tags: 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 formal statement:
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 :=
  sorry
Problem: putnam_1990_b2
Tags: 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 formal statement:
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 :=
sorry
Problem: putnam_1990_b3
Tags: 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 formal statement:
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) :=
sorry
Problem: putnam_1990_b4
Tags: 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 formal statement:
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) :=
sorry
Problem: putnam_1990_b5
Tags: 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 formal statement:
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 :=
  sorry
Problem: putnam_1991_a2
Tags: 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 formal statement:
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))) :=
sorry
Problem: putnam_1991_a3
Tags: 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 formal statement:
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)) :=
sorry
Problem: putnam_1991_a4
Tags: 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 formal statement:
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 :=
  sorry
Problem: putnam_1991_a5
Tags: 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 formal statement:
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 :=
sorry
Problem: putnam_1991_a6
Tags: 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 formal statement:
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) :=
sorry
Problem: putnam_1991_b1
Tags: 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 formal statement:
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 :=
sorry
Problem: putnam_1991_b2
Tags: 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 formal statement:
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) :=
sorry
Problem: putnam_1991_b4
Tags: 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 formal statement:
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)] :=
sorry
Problem: putnam_1991_b5
Tags: 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 formal statement:
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 :=
sorry
Problem: putnam_1991_b6
Tags: 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 formal statement:
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) :=
sorry
Problem: putnam_1992_a1
Tags: 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 formal statement:
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) :=
sorry
Problem: putnam_1992_a2
Tags: 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 formal statement:
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) :=
sorry
Problem: putnam_1992_a3
Tags: 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 formal statement:
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) :=
sorry
Problem: putnam_1992_a4
Tags: 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 formal statement:
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) :=
sorry
Problem: putnam_1992_a5
Tags: 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 formal statement:
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) :=
sorry
Problem: putnam_1992_b1
Tags: 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 formal statement:
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) :=
sorry
Problem: putnam_1992_b2
Tags: 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 formal statement:
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) :=
  sorry
Problem: putnam_1992_b3
Tags: geometry, analysis

For any pair $(x,y)$ of real numbers, a sequence $(a_n(x,y))_{n \geq 0}$ is defined as follows:
\begin{align*}
a_0(x,y)&=x, \\
a_{n+1}(x,y)&=\frac{(a_n(x,y))^2+y^2}{2},\text{ for $n \geq 0$.}
\end{align*}
Find the area of the region $\{(x,y) \mid (a_n(x,y))_{n \geq 0}\text{ converges}\}$.

Solution: Show that the area is $4+\pi$.

Lean 4 formal statement:
noncomputable abbrev putnam_1992_b3_solution : ℝ := sorry
theorem putnam_1992_b3
  (a : (Fin 2 → ℝ) → (ℕ → ℝ))
  (ha : ∀ p, (a p) 0 = p 0 ∧ (∀ n, (a p) (n + 1) = (((a p) n) ^ 2 + (p 1) ^ 2) / 2)) :
  putnam_1992_b3_solution = (MeasureTheory.volume {p | ∃ L, Tendsto (a p) atTop (𝓝 L)}).toReal :=
sorry
Problem: putnam_1992_b4
Tags: algebra

Let $p(x)$ be a nonzero polynomial of degree less than $1992$ having no nonconstant factor in common with $x^3 - x$. Let
\[
\frac{d^{1992}}{dx^{1992}} \left( \frac{p(x)}{x^3 - x} \right) = \frac{f(x)}{g(x)}
\]
for polynomials $f(x)$ and $g(x)$. Find the smallest possible degree of $f(x)$.

Solution: Show that the minimum degree is $3984$.

Lean 4 formal statement:
abbrev putnam_1992_b4_solution : ℕ := sorry
theorem putnam_1992_b4
  (IsValid : Polynomial ℝ → Prop)
  (pair : Polynomial ℝ → Polynomial ℝ → Prop)
  (IsValid_def : ∀ p, IsValid p ↔ p ≠ 0 ∧ p.degree < 1992 ∧ IsCoprime p (X ^ 3 - X))
  (hpair : ∀ p f, pair p f ↔ ∃ g : Polynomial ℝ, iteratedDeriv 1992 (fun x ↦ p.eval x / (x ^ 3 - x)) = fun x ↦ f.eval x / g.eval x) :
  IsLeast {k : ℕ | ∃ p f, IsValid p ∧ pair p f ∧ k = f.degree} putnam_1992_b4_solution :=
sorry
Problem: putnam_1992_b5
Tags: linear_algebra, analysis

Let $D_n$ denote the value of the $(n-1) \times (n-1)$ determinant
\[
\left[
\begin{array}{cccccc}
3 & 1 & 1 & 1 & \cdots & 1 \\
1 & 4 & 1 & 1 & \cdots & 1 \\
1 & 1 & 5 & 1 & \cdots & 1 \\
1 & 1 & 1 & 6 & \cdots & 1 \\
\vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\
1 & 1 & 1 & 1 & \cdots & n+1
\end{array}
\right].
\]
Is the set $\left\{ \frac{D_n}{n!} \right\}_{n \geq 2}$ bounded?

Solution: Prove that the set is not bounded.

Lean 4 formal statement:
abbrev putnam_1992_b5_solution : Prop := sorry
theorem putnam_1992_b5
  (D : ℕ → ℚ)
  (hD : ∀ n, D n = Matrix.det (fun i j : Fin (n - 1) ↦ ite (i = j) ((i : ℕ) + 3 : ℚ) 1)) :
  putnam_1992_b5_solution ↔ (Bornology.IsBounded {x | ∃ n ≥ 2, D n / factorial n = x}) :=
sorry
Problem: putnam_1992_b6
Tags: linear_algebra

Let $M$ be a set of real $n \times n$ matrices such that
\begin{itemize}
\item[(i)] $I \in M$, where $I$ is the $n \times n$ identity matrix;
\item[(ii)] if $A \in M$ and $B \in M$, then either $AB \in M$ or $-AB \in M$, but not both;
\item[(iii)] if $A \in M$ and $B \in M$, then either $AB = BA$ or $AB = -BA$;
\item[(iv)] if $A \in M$ and $A \neq I$, there is at least one $B \in M$ such that $AB = -BA$.
\end{itemize}
Prove that $M$ contains at most $n^2$ matrices.

Solution: None.

Lean 4 formal statement:
theorem putnam_1992_b6
  (n : ℕ)
  (npos : 0 < n)
  (M : Set (Matrix (Fin n) (Fin n) ℝ))
  (h1 : 1 ∈ M)
  (h2 : ∀ A ∈ M, ∀ B ∈ M, Xor' (A * B ∈ M) (-A * B ∈ M))
  (h3 : ∀ A ∈ M, ∀ B ∈ M, (A * B = B * A) ∨ (A * B = -B * A))
  (h4 : ∀ A ∈ M, A ≠ 1 → ∃ B ∈ M, A * B = -B * A) :
  M.encard ≤ n ^ 2 :=
sorry
Problem: putnam_1993_a1
Tags: analysis, algebra

The horizontal line $y=c$ intersects the curve $y=2x-3x^3$ in the first quadrant as in the figure. Find $c$ so that the areas of the two shaded regions are equal. [Figure not included. The first region is bounded by the $y$-axis, the line $y=c$ and the curve; the other lies under the curve and above the line $y=c$ between their two points of intersection.]

Solution: Show that the area of the two regions are equal when $c=4/9$.

Lean 4 formal statement:
noncomputable abbrev putnam_1993_a1_solution : ℝ := sorry
theorem putnam_1993_a1
: 0 < putnam_1993_a1_solution ∧ putnam_1993_a1_solution < (4 * Real.sqrt 2) / 9 ∧ (∫ x in Set.Ioo 0 ((Real.sqrt 2) / 3), max (putnam_1993_a1_solution - (2 * x - 3 * x ^ 3)) 0) = (∫ x in Set.Ioo 0 ((Real.sqrt 6) / 3), max ((2 * x - 3 * x ^ 3) - putnam_1993_a1_solution) 0) :=
sorry
Problem: putnam_1993_a2
Tags: algebra

Let $(x_n)_{n \geq 0}$ be a sequence of nonzero real numbers such that $x_n^2-x_{n-1}x_{n+1}=1$ for $n=1,2,3,\dots$. Prove there exists a real number $a$ such that $x_{n+1}=ax_n-x_{n-1}$ for all $n \geq 1$.

Solution: None.

Lean 4 formal statement:
theorem putnam_1993_a2
(x : ℕ → ℝ)
(xnonzero : ∀ n : ℕ, x n ≠ 0)
(hx : ∀ n ≥ 1, (x n) ^ 2 - x (n - 1) * x (n + 1) = 1)
: ∃ a : ℝ, ∀ n ≥ 1, x (n + 1) = a * x n - x (n - 1) :=
sorry
Problem: putnam_1993_a3
Tags: algebra

Let $\mathcal{P}_n$ be the set of subsets of $\{1,2,\dots,n\}$. Let $c(n,m)$ be the number of functions $f:\mathcal{P}_n \to \{1,2,\dots,m\}$ such that $f(A \cap B)=\min\{f(A),f(B)\}$. Prove that $c(n,m)=\sum_{j=1}^m j^n$.

Solution: None.

Lean 4 formal statement:
theorem putnam_1993_a3
(c : ℕ → ℕ → ℕ)
(hc : ∀ n ≥ 1, ∀ m ≥ 1, c n m = {f : Finset (Fin n) → Fin m | ∀ A B : Finset (Fin n), f (A ∩ B) = min (f A) (f B)}.encard)
: ∀ n ≥ 1, ∀ m ≥ 1, c n m = ∑' j : Set.Icc 1 m, (j : ℤ) ^ n :=
sorry
Problem: putnam_1993_a4
Tags: algebra

Let $x_1,x_2,\dots,x_{19}$ be positive integers each of which is less than or equal to $93$. Let $y_1,y_2,\dots,y_{93}$ be positive integers each of which is less than or equal to $19$. Prove that there exists a (nonempty) sum of some $x_i$'s equal to a sum of some $y_j$'s.

Solution: None.

Lean 4 formal statement:
theorem putnam_1993_a4
  (x : Fin 19 → ℤ)
  (y : Fin 93 → ℤ)
  (hx : ∀ i, 0 < x i ∧ x i ≤ 93)
  (hy : ∀ j, 0 < y j ∧ y j ≤ 19) :
  ∃ (is : Finset (Fin 19)) (js : Finset (Fin 93)),
    is ≠ ∅ ∧ (∑ i : is, x i) = (∑ j : js, y j) :=
sorry
Problem: putnam_1993_a5
Tags: analysis

Show that $\int_{-100}^{-10} (\frac{x^2-x}{x^3-3x+1})^2\,dx+\int_{\frac{1}{101}}^{\frac{1}{11}} (\frac{x^2-x}{x^3-3x+1})^2\,dx+\int_{\frac{101}{100}}^{\frac{11}{10}} (\frac{x^2-x}{x^3-3x+1})^2\,dx$ is a rational number.

Solution: None.

Lean 4 formal statement:
theorem putnam_1993_a5
: ¬Irrational ((∫ x in Set.Ioo (-100) (-10), (((x ^ 2 - x) / (x ^ 3 - 3 * x + 1)) ^ 2)) + (∫ x in Set.Ioo (1 / 101) (1 / 11), (((x ^ 2 - x) / (x ^ 3 - 3 * x + 1)) ^ 2)) + (∫ x in Set.Ioo (101 / 100) (11 / 10), (((x ^ 2 - x) / (x ^ 3 - 3 * x + 1)) ^ 2))) :=
sorry
Problem: putnam_1993_a6
Tags: algebra

The infinite sequence of $2$'s and $3$'s $2,3,3,2,3,3,3,2,3,3,3,2,3,3,2,3,3,3,2,3,3,3,2,3,3,3,2,3,3,2,3,3,3,2,\dots$ has the property that, if one forms a second sequence that records the number of $3$'s between successive $2$'s, the result is identical to the given sequence. Show that there exists a real number $r$ such that, for any $n$, the $n$th term of the sequence is $2$ if and only if $n=1+\lfloor rm \rfloor$ for some nonnegative integer $m$. (Note: $\lfloor x \rfloor$ denotes the largest integer less than or equal to $x$.)

Solution: None.

Lean 4 formal statement:
theorem putnam_1993_a6
(seq : ℕ → ℤ)
(hseq23 : ∀ n, seq n = 2 ∨ seq n = 3)
(hseq2inds : ∀ n, seq n = 2 ↔ (∃ N : ℕ, n = ∑ i : Fin N, (seq i + 1)))
: ∃ r : ℝ, ∀ n, seq n = 2 ↔ (∃ m : ℤ, n + 1 = 1 + Int.floor (r * m)) :=
sorry
Problem: putnam_1993_b1
Tags: algebra

Find the smallest positive integer $n$ such that for every integer $m$ with $0< m<1993$, there exists an integer $k$ for which $\frac{m}{1993}<\frac{k}{n}<\frac{m+1}{1994}$.

Solution: Show that the smallest positive integer $n$ satisfying the condition is $n=3987$.

Lean 4 formal statement:
abbrev putnam_1993_b1_solution : ℕ := sorry
theorem putnam_1993_b1 :
    IsLeast
    {n : ℕ | 0 < n ∧
      ∀ m ∈ Set.Ioo (0 : ℤ) (1993), ∃ k : ℤ,
      (m / 1993 < (k : ℝ) / n) ∧ ((k : ℝ) / n < (m + 1) / 1994) }
    putnam_1993_b1_solution :=
  sorry
Problem: putnam_1993_b3
Tags: probability, number_theory, geometry

Two real numbers $x$ and $y$ are chosen at random in the interval $(0,1)$ with respect to the uniform distribution. What is the probability that the closest integer to $x/y$ is even? Express the answer in the form $r+s\pi$, where $r$ and $s$ are rational numbers.

Solution: Show that the limit is $(5-\pi)/4$. That is, $r=5/4$ and $s=-1/4$.

Lean 4 formal statement:
abbrev putnam_1993_b3_solution : ℚ × ℚ := sorry
theorem putnam_1993_b3 :
  let (r, s) := putnam_1993_b3_solution;
  (MeasureTheory.volume
    {p : Fin 2 → ℝ | 0 < p ∧ p < 1 ∧ Even (round (p 0 / p 1))}
  ).toReal
  = r + s * Real.pi :=
sorry