purewhite42/putnambench_solving huggingface dataset:
Problem: 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}

Answer: fun n c ↦ c * n ! / (3 ^ n * ∏ m in Finset.Icc (1 : ℤ) n, (1 - 2 ^ (-m)))

Lean 4 formalization:
open Nat Filter Topology
Independent variables:
Hypotheses:
t: ℝ → ℝ
name: f
t: Differentiable ℝ f
name: hfdiff
t: ∀ x > 0, deriv f x = -3 * f x + 6 * f (2 * x)
name: hfderiv
t: ∀ x ≥ 0, |f x| ≤ Real.exp (- √x)
name: hdecay
t: ℕ → ℝ
name: μ
t: ∀ n, μ n = ∫ x in Set.Ioi 0, x ^ n * f x
name: μ_def
Conclusion:
((∀ n, μ n = answer 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))
Formal answer (type: ℕ → ℝ → ℝ):
answer = fun (n : ℕ) (c : ℝ) =>
  ((c * (↑(n ! : ℕ) : ℝ) : ℝ) /
      (((3 : ℝ) ^ n : ℝ) * ∏ m ∈ (Finset.Icc (1 : ℤ) (↑n : ℤ) : Finset ℤ), ((1 : ℝ) - ((2 : ℝ) ^ (-m : ℤ) : ℝ) : ℝ) :
        ℝ) :
    ℝ)
Problem: Can a countably infinite set have an uncountable collection of non-empty subsets such that the intersection of any two of them is finite?

Answer: True

Lean 4 formalization:
open Nat Filter Topology Set
Independent variables:
Hypotheses:
Conclusion:
(answer ↔
    (∃ S : Type,
      Countable S ∧ Infinite S ∧
      ∃ C : Set (Set S),
        ¬Countable C ∧
        (∀ R ∈ C, R ≠ ∅) ∧
        (∀ A ∈ C, ∀ B ∈ C, A ≠ B → (A ∩ B).Finite)
    ))
Formal answer (type: Prop):
answer = True
Problem: 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.

Answer: fun n : ℕ => (n)! + 2 ^ n

Lean 4 formalization:
open Filter Topology Nat
Independent variables:
Hypotheses:
t: ℕ → ℤ
name: T
t: T 0 = 2 ∧ T 1 = 3 ∧ T 2 = 6
name: hT012
t: ∀ n, T (n + 3) = (n + 7) * T (n + 2) - 4 * (n + 3) * T (n + 1) + (4 * n + 4) * T n
name: hTn
Conclusion:
(T = answer)
Formal answer (type: ℕ → ℤ):
answer = fun (n : ℕ) => ((↑(n ! : ℕ) : ℤ) + ((2 : ℤ) ^ n : ℤ) : ℤ)
Problem: Is $\sqrt{2}$ the limit of a sequence of numbers of the form $\sqrt[3]{n}-\sqrt[3]{m}$ ($n,m=0,1,2,\dots$)?

Answer: True

Lean 4 formalization:
open Filter Topology Nat
Independent variables:
Hypotheses:
t: ℝ → Prop
name: numform
t: ∀ x : ℝ, numform x ↔ ∃ n m : ℕ, x = n ^ ((1 : ℝ) / 3) - m ^ ((1 : ℝ) / 3)
name: hnumform
Conclusion:
(answer ↔ (∃ s : ℕ → ℝ, (∀ i : ℕ, numform (s i)) ∧ Tendsto s atTop (𝓝 (Real.sqrt 2))))
Formal answer (type: Prop):
answer = True
Problem: 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?

Answer: 3

Lean 4 formalization:
open Filter Topology Nat
Independent variables:
Hypotheses:
Conclusion:
(sInf {n : ℕ | ∃ S : Set (EuclideanSpace ℝ (Fin 2)), S.encard = n ∧ ∀ Q : EuclideanSpace ℝ (Fin 2), ∃ P ∈ S, Irrational (dist P Q)} = answer)
Formal answer (type: ℕ):
answer = (3 : ℕ)
Problem: If $\mathbf{A}$ and $\mathbf{B}$ are square matrices of the same size such that $\mathbf{ABAB}=\mathbf{0}$, does it follow that $\mathbf{BABA}=\mathbf{0}$?

Answer: False

Lean 4 formalization:
open Filter Topology Nat
Independent variables:
Hypotheses:
Conclusion:
(answer ↔
    (∀ n ≥ 1, ∀ A B : Matrix (Fin n) (Fin n) ℝ,
      A * B * A * B = 0 → B * A * B * A = 0))
Formal answer (type: Prop):
answer = False
Problem: 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.

Answer: 17711

Lean 4 formalization:
open Filter Topology Nat
Independent variables:
Hypotheses:
Conclusion:
(((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 =
    answer)
Formal answer (type: ℕ):
answer = (17711 : ℕ)
Problem: Find all real-valued continuously differentiable functions $f$ on the real line such that for all $x$, $(f(x))^2=\int_0^x [(f(t))^2+(f'(t))^2]\,dt+1990$.

Answer: $\{\sqrt{1990} e^x, -\sqrt{1990} e^x\}$

Lean 4 formalization:
open Filter Topology Nat
Independent variables:
Hypotheses:
t: (ℝ → ℝ) → Prop
name: P
t: ∀ f, P f ↔ ∀ x,
      (f x) ^ 2 = (∫ t in (0 : ℝ)..x, (f t) ^ 2 + (deriv f t) ^ 2) + 1990
name: P_def
t: ℝ → ℝ
name: f
Conclusion:
(f ∈ answer ↔ (ContDiff ℝ 1 f ∧ P f))
Formal answer (type: Set (ℝ → ℝ)):
answer =
  (Insert.insert (fun (x : ℝ) => ((√(1990 : ℝ) : ℝ) * (Real.exp x : ℝ) : ℝ))
      ({fun (x : ℝ) => ((-(√(1990 : ℝ) : ℝ) : ℝ) * (Real.exp x : ℝ) : ℝ)} : Set (ℝ → ℝ)) :
    Set (ℝ → ℝ))
Problem: 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}

Answer: True

Lean 4 formalization:
open Filter Topology Nat
Independent variables:
Hypotheses:
Conclusion:
(∀ (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))) ↔ answer)
Formal answer (type: Prop):
answer = True
Problem: 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?

Answer: True

Lean 4 formalization:
open Filter Polynomial Topology Nat
Independent variables:
Hypotheses:
Conclusion:
((∃ a : ℕ → ℝ, (∀ i, a i ≠ 0) ∧
      (∀ n ≥ 1, (∑ i in Finset.Iic n, a i • X ^ i : Polynomial ℝ).roots.toFinset.card = n)) ↔
    answer)
Formal answer (type: Prop):
answer = True
Problem: 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?

Answer: False

Lean 4 formalization:
open Filter Topology
Independent variables:
Hypotheses:
t: ℕ
name: n
t: 1 ≤ n
name: hn
Conclusion:
(answer ↔ (∃ 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))))
Formal answer (type: Prop):
answer = False
Problem: 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 $p(x)$.

Answer: the set of all real polynomials of degree 2 with two distinct real roots

Lean 4 formalization:
open Filter Topology
Independent variables:
t: Polynomial ℝ
name: p
Hypotheses:
t: ℕ
name: n
t: n = p.degree
name: hn
t: n ≥ 2
name: hge
Conclusion:
(p ∈ answer ↔
    (∃ 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)))
Formal answer (type: Set (Polynomial ℝ)):
answer =
  ({p : Polynomial ℝ |
      (p.degree : WithBot ℕ) = (2 : WithBot ℕ) ∧
        ∃ (r1 : ℝ) (r2 : ℝ), r1 ≠ r2 ∧ (Polynomial.eval r1 p : ℝ) = (0 : ℝ) ∧ (Polynomial.eval r2 p : ℝ) = (0 : ℝ)} :
    Set (Polynomial ℝ))
Problem: 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}

Answer: True

Lean 4 formalization:
open Filter Metric Topology
Independent variables:
Hypotheses:
Conclusion:
((∃ (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)) ↔
    answer)
Formal answer (type: Prop):
answer = True
Problem: Find the maximum value of $\int_0^y \sqrt{x^4+(y-y^2)^2}\,dx$ for $0 \leq y \leq 1$.

Answer: $\frac{1}{3}$

Lean 4 formalization:
open Filter Topology
Independent variables:
Hypotheses:
t: Set.Icc (0 : ℝ) 1 → ℝ
name: f
t: ∀ y : Set.Icc 0 1, f y = ∫ x in Set.Ioo 0 y, Real.sqrt (x ^ 4 + (y - y ^ 2) ^ 2)
name: hf
Conclusion:
(IsGreatest (f '' (Set.Icc 0 1)) answer)
Formal answer (type: ℝ):
answer = (1 / 3 : ℝ)
Problem: 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?

Answer: the set of positive integers $A$ such that $A$ is a perfect square

Lean 4 formalization:
open Filter Topology
Independent variables:
t: ℤ
name: A
Hypotheses:
t: ℤ → ℤ
name: m
t: ℤ → ℤ
name: S
t: ℕ → ℤ
name: a
t: ∀ n, 0 ≤ n → (m n) ^ 2 ≤ n ∧ (∀ m' : ℤ, m' ^ 2 ≤ n → m' ≤ m n)
name: hm
t: ∀ n, 0 ≤ n → S n = n - (m n) ^ 2
name: hS
t: a 0 = A ∧ (∀ k, a (k + 1) = a k + S (a k))
name: ha
t: A > 0
name: hA
Conclusion:
((∃ (K : ℕ) (c : ℤ), ∀ k ≥ K, a k = c) ↔ A ∈ answer)
Formal answer (type: Set ℤ):
answer = ({A : ℤ | ∃ x > (0 : ℤ), A = (x ^ (2 : ℕ) : ℤ)} : Set ℤ)
Problem: 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 $\{x^2:x \in \mathbb{Z}_p\} \cap \{y^2+1:y \in \mathbb{Z}_p\}$?

Answer: $\lceil p / 4 \rceil$

Lean 4 formalization:
open Filter Topology
Independent variables:
t: ℕ
name: p
Hypotheses:
t: Odd p
name: podd
t: Prime p
name: pprime
Conclusion:
(({z : ZMod p | ∃ x : ZMod p, z = x ^ 2} ∩ {z : ZMod p | ∃ y : ZMod p, z = y ^ 2 + 1}).encard = answer p)
Formal answer (type: ℕ → ℕ):
answer = fun (p : ℕ) => (⌈((↑p : ℝ) / (4 : ℝ) : ℝ)⌉₊ : ℕ)
Problem: 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: $\sinh u=(e^u-e^{-u})/2$.)

Answer: (fun a b : ℝ => |Real.log (a / b)|)

Lean 4 formalization:
open Filter Topology
Independent variables:
t: ℝ
name: a
t: ℝ
name: b
Hypotheses:
t: a > 0 ∧ b > 0
name: abpos
Conclusion:
(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))} (answer a b))
Formal answer (type: ℝ → ℝ → ℝ):
answer = fun (a b : ℝ) => (|(Real.log (a / b : ℝ) : ℝ)| : ℝ)
Problem: Define $C(\alpha)$ to be the coefficient of $x^{1992}$ in the power series about $x=0$ of $(1 + x)^\alpha$. Evaluate
\[
\int_0^1 \left( C(-y-1) \sum_{k=1}^{1992} \frac{1}{y+k} \right)\,dy.
\]

Answer: 1992

Lean 4 formalization:
open Topology Filter
Independent variables:
Hypotheses:
t: ℝ → ℝ
name: C
t: C = fun α ↦ taylorCoeffWithin (fun x ↦ (1 + x) ^ α) 1992 Set.univ 0
name: hC
Conclusion:
(∫ y in (0)..1, C (-y - 1) * ∑ k in Finset.Icc (1 : ℕ) 1992, 1 / (y + k) = answer)
Formal answer (type: ℝ):
answer = (1992 : ℝ)
Problem: For a given positive integer $m$, find all triples $(n, x, y)$ of positive integers, with $n$ relatively prime to $m$, which satisfy
\[
(x^2 + y^2)^m = (xy)^n.
\]

Answer: $\emptyset$ if $m$ is odd, otherwise $\{(m + 1, 2^{m/2}, 2^{m/2})\}$

Lean 4 formalization:
open Topology Filter Nat
Independent variables:
t: ℕ
name: m
Hypotheses:
t: m > 0
name: mpos
t: Set (ℕ × ℕ × ℕ)
name: S
t: ∀ 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
name: hS
Conclusion:
(S = answer m)
Formal answer (type: ℕ → Set (ℕ × ℕ × ℕ)):
answer = fun (m : ℕ) =>
  (if Odd m then (∅ : Set (ℕ × ℕ × ℕ))
    else
      ({(((m + (1 : ℕ) : ℕ), ((((2 : ℕ) ^ (m / (2 : ℕ) : ℕ) : ℕ), ((2 : ℕ) ^ (m / (2 : ℕ) : ℕ) : ℕ)) : ℕ × ℕ)) :
            ℕ × ℕ × ℕ)} :
        Set (ℕ × ℕ × ℕ)) :
    Set (ℕ × ℕ × ℕ))
Problem: 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 $f^{(k)}(0), k = 1, 2, 3, \dots$.

Answer: fun k ↦ ite (Even k) ((-1) ^ (k / 2) * factorial k) 0

Lean 4 formalization:
open Topology Filter Nat Function
Independent variables:
Hypotheses:
t: ℝ → ℝ
name: f
t: ContDiff ℝ ⊤ f
name: hfdiff
t: ∀ n : ℕ, n > 0 → f (1 / n) = n ^ 2 / (n ^ 2 + 1)
name: hf
Conclusion:
(∀ k : ℕ, k > 0 → iteratedDeriv k f 0 = answer k)
Formal answer (type: ℕ → ℝ):
answer = fun (k : ℕ) => (if Even k then (((-1 : ℝ) ^ (k / (2 : ℕ) : ℕ) : ℝ) * (↑(k ! : ℕ) : ℝ) : ℝ) else (0 : ℝ) : ℝ)
Problem: 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 $A_S$?

Answer: fun n ↦ 2 * n - 3

Lean 4 formalization:
open Topology Filter Nat Function
Independent variables:
t: ℕ
name: n
Hypotheses:
t: n ≥ 2
name: hn
t: Finset ℝ → Set ℝ
name: A
t: A = fun S ↦ {x | ∃ a ∈ S, ∃ b ∈ S, a ≠ b ∧ (a + b) / 2 = x}
name: hA
Conclusion:
(IsLeast {k : ℤ | ∃ S : Finset ℝ, S.card = n ∧ k = (A S).ncard} (answer n))
Formal answer (type: ℕ → ℤ):
answer = fun (n : ℕ) => (((2 : ℤ) * (↑n : ℤ) : ℤ) - (3 : ℤ) : ℤ)
Problem: 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 $\{(x,y) \mid (a_n(x,y))_{n \geq 0}\text{ converges}\}$.

Answer: 4 + π

Lean 4 formalization:
open Topology Filter Nat Function Polynomial
Independent variables:
Hypotheses:
t: (Fin 2 → ℝ) → (ℕ → ℝ)
name: a
t: ∀ p, (a p) 0 = p 0 ∧ (∀ n, (a p) (n + 1) = (((a p) n) ^ 2 + (p 1) ^ 2) / 2)
name: ha
Conclusion:
(answer = (MeasureTheory.volume {p | ∃ L, Tendsto (a p) atTop (𝓝 L)}).toReal)
Formal answer (type: ℝ):
answer = ((4 : ℝ) + (Real.pi : ℝ) : ℝ)
Problem: 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 $f(x)$.

Answer: 3984

Lean 4 formalization:
open Topology Filter Nat Function Polynomial
Independent variables:
Hypotheses:
t: Polynomial ℝ → Prop
name: IsValid
t: Polynomial ℝ → Polynomial ℝ → Prop
name: pair
t: ∀ p, IsValid p ↔ p ≠ 0 ∧ p.degree < 1992 ∧ IsCoprime p (X ^ 3 - X)
name: IsValid_def
t: ∀ 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
name: hpair
Conclusion:
(IsLeast {k : ℕ | ∃ p f, IsValid p ∧ pair p f ∧ k = f.degree} answer)
Formal answer (type: ℕ):
answer = (3984 : ℕ)
Problem: 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?

Answer: False

Lean 4 formalization:
open Topology Filter Nat Function Polynomial
Independent variables:
Hypotheses:
t: ℕ → ℚ
name: D
t: ∀ n, D n = Matrix.det (fun i j : Fin (n - 1) ↦ ite (i = j) ((i : ℕ) + 3 : ℚ) 1)
name: hD
Conclusion:
(answer ↔ (Bornology.IsBounded {x | ∃ n ≥ 2, D n / factorial n = x}))
Formal answer (type: Prop):
answer = False
Problem: 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.]

Answer: 4/9

Lean 4 formalization:
Independent variables:
Hypotheses:
Conclusion:
(0 < answer ∧ answer < (4 * Real.sqrt 2) / 9 ∧
    (∫ x in Set.Ioo 0 ((Real.sqrt 2) / 3), max (answer - (2 * x - 3 * x ^ 3)) 0) =
    (∫ x in Set.Ioo 0 ((Real.sqrt 6) / 3), max ((2 * x - 3 * x ^ 3) - answer) 0))
Formal answer (type: ℝ):
answer = (4 / 9 : ℝ)
Problem: Find the smallest positive integer $n$ such that for every integer $m$ with $0<m<1993$, there exists an integer $k$ for which $\frac{m}{1993}<\frac{k}{n}<\frac{m+1}{1994}$.

Answer: 3987

Lean 4 formalization:
Independent variables:
Hypotheses:
Conclusion:
(IsLeast
    {n : ℕ | 0 < n ∧
      ∀ m ∈ Set.Ioo (0 : ℤ) (1993), ∃ k : ℤ,
      ((m : ℝ) / 1993 < (k : ℝ) / n) ∧ ((k : ℝ) / n < ((m : ℝ) + 1) / 1994) }
    answer)
Formal answer (type: ℕ):
answer = (3987 : ℕ)
Problem: 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.

Answer: (5/4, -1/4)

Lean 4 formalization:
Independent variables:
Hypotheses:
Conclusion:
((MeasureTheory.volume
    {p : Fin 2 → ℝ | 0 < p 0 ∧ p 0 < 1 ∧ 0 < p 1 ∧ p 1 < 1 ∧ Even (round (p 0 / p 1))}
  ).toReal = answer.1 + answer.2 * Real.pi)
Formal answer (type: ℚ × ℚ):
answer = (((5 / 4 : ℚ), (-1 / 4 : ℚ)) : ℚ × ℚ)
Problem: Find all positive integers $n$ that are within $250$ of exactly $15$ perfect squares.

Answer: the set of integers $n$ such that $315 \leq n \leq 325$ or $332 \leq n \leq 350$

Lean 4 formalization:
open Filter Topology
Independent variables:
t: ℤ
name: n
Hypotheses:
Conclusion:
(n ∈ answer ↔ (0 < n ∧ {m : ℕ | |n - m ^ 2| ≤ 250}.encard = 15))
Formal answer (type: Set ℤ):
answer = ({n : ℤ | (315 : ℤ) ≤ n ∧ n ≤ (325 : ℤ) ∨ (332 : ℤ) ≤ n ∧ n ≤ (350 : ℤ)} : Set ℤ)
Problem: For which real numbers $c$ is there a straight line that intersects the curve $x^4+9x^3+cx^2+9x+4$ in four distinct points?

Answer: {c : ℝ | c < 243 / 8}

Lean 4 formalization:
open Filter Topology
Independent variables:
t: ℝ
name: c
Hypotheses:
Conclusion:
(answer = {c : ℝ | (∃ m b : ℝ,
    {x : ℝ | m * x + b = x ^ 4 + 9 * x ^ 3 + c * x ^ 2 + 9 * x + 4}.encard = 4)})
Formal answer (type: Set ℝ):
answer = ({c : ℝ | c < (243 / 8 : ℝ)} : Set ℝ)
Problem: Find the set of all real numbers $k$ with the following property: For any positive, differentiable function $f$ that satisfies $f'(x)>f(x)$ for all $x$, there is some number $N$ such that $f(x)>e^{kx}$ for all $x>N$.

Answer: the set of all real numbers less than 1

Lean 4 formalization:
open Filter Topology
Independent variables:
Hypotheses:
Conclusion:
({k | ∀ f (hf : (∀ x, 0 < f x ∧ f x < deriv f x) ∧ Differentiable ℝ f),
      ∃ N, ∀ x > N, Real.exp (k * x) < f x} = answer)
Formal answer (type: Set ℝ):
answer = (Set.Iio (1 : ℝ) : Set ℝ)
Problem: For what pairs $(a,b)$ of positive real numbers does the improper integral \[ \int_{b}^{\infty} \left( \sqrt{\sqrt{x+a}-\sqrt{x}} - \sqrt{\sqrt{x}-\sqrt{x-b}} \right)\,dx \] converge?

Answer: {x | let ⟨a,b⟩ := x; a = b}

Lean 4 formalization:
open Filter Topology Real
Independent variables:
Hypotheses:
t: (ℝ × ℝ) → Prop
name: habconv
t: habconv = fun ⟨a,b⟩ => ∃ limit : ℝ, Tendsto (fun t : ℝ => ∫ x in (Set.Icc b t), (sqrt (sqrt (x + a) - sqrt x) - sqrt (sqrt x - sqrt (x - b)))) atTop (𝓝 limit)
name: habconv_def
Conclusion:
(∀ ab : ℝ × ℝ, ab.1 > 0 ∧ ab.2 > 0 → (habconv ab ↔ ab ∈ answer))
Formal answer (type: Set (ℝ × ℝ)):
answer = ({((a, b) : ℝ × ℝ) : ℝ × ℝ | a = b} : Set (ℝ × ℝ))
Problem: Let $x_{1},x_{2},\dots,x_{n}$ be differentiable (real-valued) functions of a single variable $f$ which satisfy \begin{align*} \frac{dx_{1}}{dt} &= a_{11}x_{1} + a_{12}x_{2} + \cdots + a_{1n}x_{n} \ \frac{dx_{2}}{dt} &= a_{21}x_{1} + a_{22}x_{2} + \cdots + a_{2n}x_{n} \ \vdots && \vdots \ \frac{dx_{n}}{dt} &= a_{n1}x_{1} + a_{n2}x_{2} + \cdots + a_{nn}x_{n} \end{align*} for some constants $a_{ij}>0$. Suppose that for all $i$, $x_{i}(t) \to 0$ as $t \to \infty$. Are the functions $x_{1},x_{2},\dots,x_{n}$ necessarily linearly dependent?

Answer: True

Lean 4 formalization:
open Filter Topology Real
Independent variables:
Hypotheses:
t: ℕ
name: n
t: Fin n → (ℝ → ℝ)
name: x
t: Fin n → Fin n → ℝ
name: a
t: 0 < n
name: npos
t: ∀ i, Differentiable ℝ (x i)
name: hdiff
t: ∀ i j, a i j > 0
name: hpos
t: ∀ t i, (deriv (x i)) t = ∑ j : Fin n, (a i j) * ((x j) t)
name: hsys
t: ∀ i, Tendsto (x i) atTop (𝓝 0)
name: htendsto
Conclusion:
(answer ↔ ¬(∀ b : Fin n → ℝ, (∀ t : ℝ, ∑ i : Fin n, (b i) * ((x i) t) = 0) → (∀ i, b i = 0)))
Formal answer (type: Prop):
answer = True
Problem: To each positive integer with $n^{2}$ decimal digits, we associate the determinant of the matrix obtained by writing the digits in order across the rows. For example, for $n=2$, to the integer 8617 we associate $\det \left( \begin{array}{cc} 8 & 6 \ 1 & 7 \end{array} \right) = 50$. Find, as a function of $n$, the sum of all the determinants associated with $n^{2}$-digit integers. (Leading digits are assumed to be nonzero; for example, for $n=2$, there are 9000 determinants.)

Answer: 45 if n = 1, 10 * 45² if n = 2, and 0 otherwise

Lean 4 formalization:
open Filter Topology Real Nat
Independent variables:
t: ℕ
name: n
Hypotheses:
t: n > 0
name: hn
t: Set (ℕ → ℕ)
name: digits_set
t: digits_set = {f | f 0 ≠ 0 ∧ (∀ i : Fin (n ^ 2), f i ≤ 9) ∧ (∀ i ≥ n ^ 2, f i = 0)}
name: hdigits_set
t: (ℕ → ℕ) → Matrix (Fin n) (Fin n) ℤ
name: digits_to_matrix
t: digits_to_matrix = fun f => (fun i j => f (i.1 * n + j.1))
name: hdigits_to_matrix
Conclusion:
(∑' f : digits_set, (digits_to_matrix f).det = answer n)
Formal answer (type: ℕ → ℤ):
answer = fun (n : ℕ) =>
  (if n = (1 : ℕ) then (45 : ℤ) else (if n = (2 : ℕ) then ((10 : ℤ) * ((45 : ℤ) ^ (2 : ℕ) : ℤ) : ℤ) else (0 : ℤ) : ℤ) :
    ℤ)
Problem: Evaluate \[ \sqrt[8]{2207 - \frac{1}{2207-\frac{1}{2207-\dots}}}. \] Express your answer in the form $\frac{a+b\sqrt{c}}{d}$, where $a,b,c,d$ are integers.

Answer: ⟨3,1,5,2⟩

Lean 4 formalization:
open Filter Topology Real Nat
Independent variables:
Hypotheses:
t: ℝ
name: contfrac
t: contfrac = 2207 - 1 / contfrac
name: hcontfrac
t: 1 < contfrac
name: hcontfrac'
Conclusion:
(let ⟨a, b, c, d⟩ := answer;
    contfrac ^ ((1 : ℝ) / 8) = (a + b * sqrt c) / d)
Formal answer (type: ℤ × ℤ × ℤ × ℤ):
answer = (((3 : ℤ), (((1 : ℤ), (((5 : ℤ), (2 : ℤ)) : ℤ × ℤ)) : ℤ × ℤ × ℤ)) : ℤ × ℤ × ℤ × ℤ)