ChristianZ97/PutnamBench-lean4 huggingface dataset:
Problem: putnam_2008_a3
Tags: number_theory

Start with a finite sequence $a_1, a_2, \dots, a_n$ of positive integers. If possible, choose two indices $j < k$ such that $a_j$ does not divide $a_k$, and replace $a_j$ and $a_k$ by $\mathrm{gcd}(a_j, a_k)$ and $\mathrm{lcm}(a_j, a_k)$, respectively. Prove that if this process is repeated, it must eventually stop and the final sequence does not depend on the choices made.

Solution: None.

Lean 4 formal statement:
theorem putnam_2008_a3
      (n : ℕ) (npos : n > 0)
      (a : Fin n → ℕ)
      (apos : ∀ i : Fin n, a i > 0)
      (cont : (Fin n → ℕ) → Prop)
      (hcont : ∀ s, cont s ↔ ∃ j k : Fin n, j < k ∧ ¬(s j ∣ s k))
      (init trans : (ℕ → Fin n → ℕ) → Prop)
      (hinit : ∀ P, init P ↔ P 0 = a)
      (htrans : ∀ P, trans P ↔ ∀ t : ℕ, cont (P t) →
            ∃ j k : Fin n, j < k ∧ ¬(P t j ∣ P t k) ∧ P (t + 1) j = Nat.gcd (P t j) (P t k) ∧ P (t + 1) k = Nat.lcm (P t j) (P t k) ∧
            ∀ i : Fin n, i ≠ j → i ≠ k → P (t + 1) i = P t i) :
      (∃ f, ∀ P, init P → trans P → ∃ t : ℕ, ¬cont (P t) ∧ P t = f) :=
sorry
Problem: putnam_2008_a4
Tags: algebra

Define $f : \mathbb{R} \to \mathbb{R} by $f(x) = x$ if $x \leq e$ and $f(x) = x * f(\ln(x))$ if $x > e$. Does $\sum_{n=1}^{\infty} 1/(f(n))$ converge?

Solution: Show that the sum does not converge.

Lean 4 formal statement:
abbrev putnam_2008_a4_solution : Prop := sorry
theorem putnam_2008_a4
(f : ℝ → ℝ)
(hf : f = fun x => if x ≤ Real.exp 1 then x else x * (f (Real.log x)))
: (∃ r : ℝ, Tendsto (fun N : ℕ => ∑ n ∈ Finset.range N, 1/(f (n + 1))) atTop (𝓝 r)) ↔ putnam_2008_a4_solution :=
sorry
Problem: putnam_2008_a5
Tags: algebra, geometry

Let $n \geq 3$ be an integer. Let $f(x)$ and $g(x)$ be polynomials with real coefficients such that the points $(f(1), g(1)), (f(2), g(2)), \dots, (f(n), g(n))$ in $\mathbb{R}^2$ are the vertices of a regular $n$-gon in counterclockwise order. Prove that at least one of $f(x)$ and $g(x)$ has degree greater than or equal to $n-1$.

Solution: None.

Lean 4 formal statement:
theorem putnam_2008_a5
(n : ℕ)
(nge3 : n ≥ 3)
(f g : Polynomial ℝ)
(hfg : ∃ O z : ℂ, z ≠ 0 ∧ ∀ k : ℕ, k ∈ Icc 1 n → (f.eval (k : ℝ)) + Complex.I * (g.eval (k : ℝ)) = O + z * Complex.exp (Complex.I * 2 * Real.pi * k / n))
: (f.natDegree ≥ n - 1 ∨ g.natDegree ≥ n - 1) :=
sorry
Problem: putnam_2008_a6
Tags: abstract_algebra

Prove that there exists a constant $c>0$ such that in every nontrivial finite group $G$ there exists a sequence of length at most $c \log |G|$ with the property that each element of $G$ equals the product of some subsequence. (The elements of $G$ in the sequence are not required to be distinct. A \emph{subsequence} of a sequence is obtained by selecting some of the terms, not necessarily consecutive, without reordering them; for example, $4, 4, 2$ is a subsequence of $2, 4, 6, 4, 2$, but $2, 2, 4$ is not.)

Solution: None.

Lean 4 formal statement:
theorem putnam_2008_a6
: ∃ c : ℝ, c > 0 ∧ ∀ G [Group G] (fing : Fintype G), fing.card > 1 → ∃ s : List G, s.length ≤ c * Real.log (fing.card : ℝ) ∧ ∀ g : G, ∃ t : List G, t.Sublist s ∧ t.prod = g :=
sorry
Problem: putnam_2008_b1
Tags: geometry, number_theory

What is the maximum number of rational points that can lie on a circle in $\mathbb{R}^2$ whose center is not a rational point? (A \emph{rational point} is a point both of whose coordinates are rational numbers.)

Solution: Show that the maximum number is $2$.

Lean 4 formal statement:
abbrev putnam_2008_b1_solution : ℕ := sorry
theorem putnam_2008_b1 :
  IsGreatest {n : ℕ | ∃ (c : EuclideanSpace ℝ (Fin 2)) (r : ℝ), ¬ is_rational_point c ∧ (Set.ncard {p : EuclideanSpace ℝ (Fin 2) | p ∈ real_circle c r ∧ is_rational_point p} = n)} putnam_2008_b1_solution :=
sorry
Problem: putnam_2008_b2
Tags: analysis

Let $F_0(x)=\ln x$. For $n \geq 0$ and $x>0$, let $F_{n+1}(x)=\int_0^x F_n(t)\,dt$. Evaluate $\lim_{n \to \infty} \frac{n!F_n(1)}{\ln n}$.

Solution: Show that the desired limit is $-1$.

Lean 4 formal statement:
abbrev putnam_2008_b2_solution : ℝ := sorry
theorem putnam_2008_b2
(F : ℕ → ℝ → ℝ)
(hF0 : ∀ x : ℝ, F 0 x = Real.log x)
(hFn : ∀ n : ℕ, ∀ x > 0, F (n + 1) x = ∫ t in Set.Ioo 0 x, F n t)
: Tendsto (fun n : ℕ => ((n)! * F n 1) / Real.log n) atTop (𝓝 putnam_2008_b2_solution) :=
sorry
Problem: putnam_2008_b3
Tags: geometry

What is the largest possible radius of a circle contained in a $4$-dimensional hypercube of side length $1$?

Solution: Show that the answer is $\frac{\sqrt 2}{2}$.

Lean 4 formal statement:
noncomputable abbrev putnam_2008_b3_solution : ℝ := sorry
theorem putnam_2008_b3
    (H : Set (EuclideanSpace ℝ (Fin 4)))
    (H_def : H = {P : EuclideanSpace ℝ (Fin 4) | ∀ i : Fin 4, |P i| ≤ 1 / 2})
    (contains : ℝ → Prop)
    (contains_def : ∀ r, contains r ↔
      ∃ᵉ (A : AffineSubspace ℝ (EuclideanSpace ℝ (Fin 4))) (C ∈ A),
        Module.finrank ℝ A.direction = 2 ∧
        sphere C r ∩ A ⊆ H) :
    IsGreatest contains putnam_2008_b3_solution :=
  sorry
Problem: putnam_2008_b4
Tags: algebra, number_theory

Let $p$ be a prime number. Let $h(x)$ be a polynomial with integer coefficients such that $h(0), h(1), \dots, h(p^2-1)$ are distinct modulo $p^2$. Show that $h(0), h(1), \dots, h(p^3-1)$ are distinct modulo $p^3$.

Solution: None.

Lean 4 formal statement:
theorem putnam_2008_b4
(p : ℕ)
(hp : Nat.Prime p)
(h : Polynomial ℤ)
(hh : ∀ i j : Finset.range (p ^ 2), h.eval i ≡ h.eval j [ZMOD p ^ 2] → i = j)
: (∀ i j : Finset.range (p ^ 3), h.eval i ≡ h.eval j [ZMOD p ^ 3] → i = j) :=
sorry
Problem: putnam_2008_b5
Tags: analysis

Find all continuously differentiable functions f : \mathbb{R} \to \mathbb{R} such that for every rational number $q$, the number $f(q)$ is rational and has the same denominator as $q$.

Solution: Show that the solution is the set of all functions of the form n + x, n - x where n is any integer.

Lean 4 formal statement:
abbrev putnam_2008_b5_solution : Set (ℝ → ℝ) := sorry
theorem putnam_2008_b5
(fqsat : (ℝ → ℝ) → ℚ → Prop)
(hfqsat : ∀ f q, fqsat f q ↔ ContDiff ℝ 1 f ∧ (∃ p : ℚ, p = f q ∧ p.den = q.den))
: ∀ f : (ℝ → ℝ), (∀ q : ℚ, fqsat f q) ↔ f ∈ putnam_2008_b5_solution :=
sorry
Problem: putnam_2008_b6
Tags: number_theory

Let $n$ and $k$ be positive integers. Say that a permutation $\sigma$ of $\{1,2,\dots,n\} is $k-limited$ if \|\sigma(i) - i\| \leq k$ for all $i$. Prove that the number of $k-limited$ permutations $\{1,2,\dots,n\}$ is odd if and only if $n \equiv 0$ or $1 (mod 2k+1)$.

Solution: None.

Lean 4 formal statement:
theorem putnam_2008_b6
(n k : ℕ)
(hnk : n > 0 ∧ k > 0)
: Odd (Set.ncard {s : Equiv.Perm (Fin n) | klimited k n s}) ↔ (n ≡ 0 [MOD 2*k+1] ∨ n ≡ 1 [MOD 2*k+1]) :=
sorry
Problem: putnam_2009_a1
Tags: geometry, algebra

Let $f$ be a real-valued function on the plane such that for every square $ABCD$ in the plane, $f(A)+f(B)+f(C)+f(D)=0$. Does it follow that $f(P)=0$ for all points $P$ in the plane?

Solution: Prove that $f$ is identically $0$.

Lean 4 formal statement:
abbrev putnam_2009_a1_solution : Prop := sorry
theorem putnam_2009_a1
: ((∀ f : (ℝ × ℝ) → ℝ, (∀ O v : ℝ × ℝ, v ≠ (0, 0) → f (O.1, O.2) + f (O.1 + v.1, O.2 + v.2) + f (O.1 + v.1 - v.2, O.2 + v.2 + v.1) + f (O.1 - v.2, O.2 + v.1) = 0) → ∀ P : ℝ × ℝ, f P = 0) ↔ putnam_2009_a1_solution) :=
sorry
Problem: putnam_2009_a2
Tags: analysis

Functions $f,g,h$ are differentiable on some open interval around $0$
and satisfy the equations and initial conditions
\begin{gather*}
f' = 2f^2gh+\frac{1}{gh},\quad f(0)=1, \\
g'=fg^2h+\frac{4}{fh}, \quad g(0)=1, \\
h'=3fgh^2+\frac{1}{fg}, \quad h(0)=1.
\end{gather*}
Find an explicit formula for $f(x)$, valid in some open interval around $0$.

Solution: Prove that the formula is
\[
f(x) = 2^{-1/12} \left(\frac{\sin(6x+\pi/4)}{\cos^2(6x+\pi/4)}\right)^{1/6}.
\]

Lean 4 formal statement:
noncomputable abbrev putnam_2009_a2_solution : ℝ → ℝ := sorry
theorem putnam_2009_a2
(f g h : ℝ → ℝ)
(a b : ℝ)
(hab : 0 ∈ Ioo a b)
(hdiff : DifferentiableOn ℝ f (Ioo a b) ∧ DifferentiableOn ℝ g (Ioo a b) ∧ DifferentiableOn ℝ h (Ioo a b))
(hf : (∀ x ∈ Ioo a b, deriv f x = 2 * (f x)^2 * (g x) * (h x) + 1 / ((g x) * (h x))) ∧ f 0 = 1)
(hg : (∀ x ∈ Ioo a b, deriv g x = (f x) * (g x)^2 * (h x) + 4 / ((f x) * (h x))) ∧ g 0 = 1)
(hh : (∀ x ∈ Ioo a b, deriv h x = 3 * (f x) * (g x) * (h x)^2 + 1 / ((f x) * (g x))) ∧ h 0 = 1)
: (∃ c d : ℝ, 0 ∈ Ioo c d ∧ ∀ x ∈ Ioo c d, f x = putnam_2009_a2_solution x) :=
sorry
Problem: putnam_2009_a3
Tags: linear_algebra, analysis

Let $d_n$ be the determinant of the $n \times n$ matrix whose entries, from left to right and then from top to bottom, are $\cos 1, \cos 2, \dots, \cos n^2$. (For example,\[ d_3 = \left|\begin{matrix} \cos 1 & \cos 2 & \cos 3 \\ \cos 4 & \cos 5 & \cos 6 \\  \cos 7 & \cos 8 & \cos 9 \end{matrix} \right|. \]The argument of $\cos$ is always in radians, not degrees.) Evaluate $\lim_{n\to\infty} d_n$.

Solution: Show that the limit is 0.

Lean 4 formal statement:
abbrev putnam_2009_a3_solution : ℝ := sorry
theorem putnam_2009_a3
(cos_matrix : (n : ℕ) → Matrix (Fin n) (Fin n) ℝ)
(hM : ∀ n : ℕ, ∀ i j : Fin n, (cos_matrix n) i j = Real.cos (1 + n * i + j))
: Tendsto (fun n => (cos_matrix n).det) atTop (𝓝 putnam_2009_a3_solution) :=
sorry
Problem: putnam_2009_a4
Tags: number_theory

Let $S$ be a set of rational numbers such that
\begin{enumerate}
\item[(a)] $0 \in S$;
\item[(b)] If $x \in S$ then $x+1\in S$ and $x-1\in S$; and
\item[(c)] If $x\in S$ and $x\not\in\{0,1\}$, then $\frac{1}{x(x-1)}\in S$.
\end{enumerate}
Must $S$ contain all rational numbers?

Solution: Prove that $S$ need not contain all rationals.

Lean 4 formal statement:
abbrev putnam_2009_a4_solution : Prop := sorry
theorem putnam_2009_a4
: ((∀ S : Set ℚ, 0 ∈ S → (∀ x ∈ S, x + 1 ∈ S ∧ x - 1 ∈ S) → (∀ x ∈ S, x ∉ ({0, 1} : Set ℚ) → 1 / (x * (x - 1)) ∈ S) → ∀ r : ℚ, r ∈ S) ↔ putnam_2009_a4_solution) :=
sorry
Problem: putnam_2009_a5
Tags: abstract_algebra

Is there a finite abelian group $G$ such that the product of the orders of all its elements is 2^{2009}?

Solution: Show that the answer is no such finite abelian group exists.

Lean 4 formal statement:
abbrev putnam_2009_a5_solution : Prop := sorry
theorem putnam_2009_a5
: (∃ (G : Type*) (_ : CommGroup G) (_ : Fintype G), ∏ g : G, orderOf g = 2^2009) ↔ putnam_2009_a5_solution :=
sorry
Problem: putnam_2009_b1
Tags: number_theory

Show that every positive rational number can be written as a quotient of products of factorails of (not necessarily distinct) primes. For example, 10/9 = (2! * 5!)/(3! * 3! * 3!).

Solution: None.

Lean 4 formal statement:
theorem putnam_2009_b1
(isquotprodprimefact : ℚ → Prop)
(hisquotprodprimefact : ∀ q, isquotprodprimefact q ↔
  (∃ (k m : ℕ) (a : Fin k → ℕ) (b : Fin m → ℕ),
    (∀ i : Fin k, Nat.Prime (a i)) ∧ (∀ j : Fin m, Nat.Prime (b j))
    ∧ (q = (∏ i : Fin k, Nat.factorial (a i))/(∏ j : Fin m, Nat.factorial (b j)))))
: ∀ q : ℚ, q > 0 → isquotprodprimefact q :=
sorry
Problem: putnam_2009_b2
Tags: analysis, algebra

A game involves jumping to the right on the real number line. If $a$ and $b$ are real numbers and $b > a$, the cost of jumping from $a$ to $b$ is $b^3-ab^2$. For what real numbers $c$ can one travel from $0$ to $1$ in a finite number of jumps with total cost exactly $c$?

Solution: Prove that the possible costs are $1/3 < c \leq 1.$

Lean 4 formal statement:
abbrev putnam_2009_b2_solution : Set ℝ := sorry
theorem putnam_2009_b2
: ({c : ℝ | ∃ s : ℕ → ℝ, s 0 = 0 ∧ StrictMono s ∧ (∃ n : ℕ, s n = 1 ∧ ((∑ i ∈ Finset.range n, ((s (i + 1)) ^ 3 - (s i) * (s (i + 1)) ^ 2)) = c))} = putnam_2009_b2_solution) :=
sorry
Problem: putnam_2009_b3
Tags: number_theory

Call a subset $S$ of $\{1, 2, \dots, n\}$ \emph{mediocre} if it has the following property: Whenever $a$ and $b$ are elements of $S$ whose average is an integer, that average is also an element of $S$. Let $A(n)$ be the number of mediocre subsets of $\{1,2,\dots,n\}$. [For instance, every subset of $\{1,2,3\}$ except $\{1,3\}$ is mediocre, so $A(3) = 7$.] Find all positive integers $n$ such that $A(n+2) - 2A(n+1) + A(n) = 1$.

Solution: Show that the answer is $n = 2^k - 1$ for some integer $k$.

Lean 4 formal statement:
abbrev putnam_2009_b3_solution : Set ℤ := sorry
theorem putnam_2009_b3
(mediocre : ℤ → Set ℤ → Prop)
(hmediocre : ∀ n S, mediocre n S ↔ (S ⊆ Icc 1 n) ∧ ∀ a ∈ S, ∀ b ∈ S, 2 ∣ a + b → (a + b) / 2 ∈ S)
(A : ℤ → ℤ)
(hA : A = fun n ↦ ({S : Set ℤ | mediocre n S}.ncard : ℤ))
: ({n : ℤ | n > 0 ∧ A (n + 2) - 2 * A (n + 1) + A n = 1} = putnam_2009_b3_solution) :=
sorry
Problem: putnam_2009_b4
Tags: algebra, linear_algebra

Say that a polynomial with real coefficients in two variables, $x,y$, is \emph{balanced} if the average value of the polynomial on each circle centered at the origin is $0$. The balanced polynomials of degree at most $2009$ form a vector space $V$ over $\mathbb{R}$. Find the dimension of $V$.

Solution: Prove that the dimension of $V$ is $2020050$.

Lean 4 formal statement:
abbrev putnam_2009_b4_solution : ℕ := sorry
theorem putnam_2009_b4
    (IsBalanced : MvPolynomial (Fin 2) ℝ → Prop)
    (IsBalanced_def : ∀ P, IsBalanced P ↔ ∀ r > 0,
      (∫ t in (0 : ℝ)..(2 * π), eval ![r * cos t, r * sin t] P) / (2 * π * r) = 0)
    (V : Submodule ℝ (MvPolynomial (Fin 2) ℝ))
    (V_def : ∀ P, P ∈ V ↔ IsBalanced P ∧ P.totalDegree ≤ 2009) :
    Module.rank ℝ V = putnam_2009_b4_solution :=
  sorry
Problem: putnam_2009_b5
Tags: analysis

Let $f: (1, \infty) \to \mathbb{R}$ be a differentiable function such that
\[
 f'(x) = \frac{x^2 - f(x)^2}{x^2 (f(x)^2 + 1)}
\qquad \mbox{for all $x>1$.}
\]
Prove that $\lim_{x \to \infty} f(x) = \infty$.

Solution: None.

Lean 4 formal statement:
theorem putnam_2009_b5
(f : ℝ → ℝ)
(hfdiff : DifferentiableOn ℝ f (Ioi 1))
(hf : ∀ x > 1, deriv f x = (x ^ 2 - (f x) ^ 2) / ((x ^ 2) * ((f x) ^ 2 + 1)))
: (Tendsto f atTop atTop) :=
sorry
Problem: putnam_2009_b6
Tags: number_theory

Prove that for every positive integer $n$, there is a sequence of integers $a_0, a_1, \dots, a_{2009}$ with $a_0 = 0$ and $a_{2009} = n$ such that each term after $a_0$ is either an earlier term plus $2^k$ for some nonnegative integer $k$, or of the form $b\,\mathrm{mod}\,c$ for some earlier positive terms $b$ and $c$. [Here $b\,\mathrm{mod}\,c$ denotes the remainder when $b$ is divided by $c$, so $0 \leq (b\,\mathrm{mod}\,c) < c$.]

Solution: None.

Lean 4 formal statement:
theorem putnam_2009_b6
  (n : ℕ) (npos : n > 0) :
  (∃ a : ℕ → ℤ,
    a 0 = 0 ∧ a 2009 = n ∧
    ∀ i : Icc 1 2009,
      ((∃ j k : ℕ, j < i ∧ a i = a j + 2 ^ k) ∨
      ∃ b c : ℕ, b < i ∧ c < i ∧ a b > 0 ∧ a c > 0 ∧ a i = (a b) % (a c))) :=
sorry
Problem: putnam_2010_a1
Tags: algebra

Given a positive integer $n$, what is the largest $k$ such that the numbers $1,2,\dots,n$ can be put into $k$ boxes so that the sum of the numbers in each box is the same? [When $n=8$, the example $\{1,2,3,6\},\{4,8\},\{5,7\}$ shows that the largest $k$ is \emph{at least} $3$.]

Solution: Show that the largest such $k$ is $\lceil \frac{n}{2} \rceil$.

Lean 4 formal statement:
noncomputable abbrev putnam_2010_a1_solution : ℕ → ℕ := sorry
theorem putnam_2010_a1
    (n : ℕ)
    (kboxes : ℕ → Prop)
    (npos : n > 0)
    (hkboxes : ∀ k : ℕ, kboxes k =
      (∃ boxes : Finset.Icc 1 n → Fin k, ∀ i j : Fin k,
        ∑ x ∈ Finset.univ.filter (boxes · = i), (x : ℕ) =
        ∑ x ∈ Finset.univ.filter (boxes · = j), (x : ℕ))) :
    IsGreatest kboxes (putnam_2010_a1_solution n) :=
  sorry
Problem: putnam_2010_a2
Tags: analysis

Find all differentiable functions $f:\mathbb{R} \to \mathbb{R}$ such that
\[
f'(x) = \frac{f(x+n)-f(x)}{n}
\]
for all real numbers $x$ and all positive integers $n$.

Solution: The solution consists of all functions of the form $f(x) = cx+d$ for some real numbers $c,d$.

Lean 4 formal statement:
abbrev putnam_2010_a2_solution : Set (ℝ → ℝ) := sorry
theorem putnam_2010_a2
: {f : ℝ → ℝ | Differentiable ℝ f ∧
∀ x : ℝ, ∀ n : ℤ, n > 0 → deriv f x = (f (x + n) - f x)/n} = putnam_2010_a2_solution :=
sorry
Problem: putnam_2010_a3
Tags: analysis

Suppose that the function $h : \mathbb{R}^2 → \mathbb{R}$ has continuous partial derivatives and satisfies the equation $h(x, y) = a \frac{\partial h}{\partial x}(x, y) +b \frac{\partial h}{\partial y}(x, y)$ for some constants $a, b$. Prove that if there is a constant $M$ such that $|h(x, y)| ≤ M$ for all $(x, y) ∈ \mathbb{R}^2$, then $h$ is identically zero.

Solution: None.

Lean 4 formal statement:
theorem putnam_2010_a3
    (h : ℝ × ℝ → ℝ)
    (a b M : ℝ)
    (H : ContDiff ℝ 1 h)
    (H' : ∀ x, h x = a * (fderiv ℝ h x (1, 0)) + b * (fderiv ℝ h x (0, 1)))
    (H'' : ∀ x, |h x| ≤ M) : h = 0 :=
  sorry
Problem: putnam_2010_a4
Tags: number_theory

Prove that for each positive integer $n$, the number $10^{10^{10^n}} + 10^{10^n} + 10^n - 1$ is not prime.

Solution: None.

Lean 4 formal statement:
theorem putnam_2010_a4
: ∀ n : ℕ, n > 0 → ¬Nat.Prime (10^10^10^n + 10^10^n + 10^n - 1) :=
sorry
Problem: putnam_2010_a5
Tags: abstract_algebra, algebra

Let $G$ be a group, with operation $*$. Suppose that \begin{enumerate} \item[(i)] $G$ is a subset of $\mathbb{R}^3$ (but $*$ need not be related to addition of vectors); \item[(ii)] For each $\mathbf{a},\mathbf{b} \in G$, either $\mathbf{a}\times \mathbf{b} = \mathbf{a}*\mathbf{b}$ or $\mathbf{a}\times \mathbf{b} = 0$ (or both), where $\times$ is the usual cross product in $\mathbb{R}^3$. \end{enumerate} Prove that $\mathbf{a} \times \mathbf{b} = 0$ for all $\mathbf{a}, \mathbf{b} \in G$.

Solution: None.

Lean 4 formal statement:
theorem putnam_2010_a5
    (G : Type*) [Group G]
    (i : G ↪ (Fin 3 → ℝ))
    (h : ∀ a b, (i a) ⨯₃ (i b) = i (a * b) ∨ (i a) ⨯₃ (i b) = 0)
    (a b : G) :
    (i a) ⨯₃ (i b) = 0 :=
  sorry
Problem: putnam_2010_a6
Tags: analysis

Let $f:[0,\infty)\to \mathbb{R}$ be a strictly decreasing continuous function
such that $\lim_{x\to\infty} f(x) = 0$. Prove that
$\int_0^\infty \frac{f(x)-f(x+1)}{f(x)}\,dx$ diverges.

Solution: None.

Lean 4 formal statement:
theorem putnam_2010_a6
(f : ℝ → ℝ)
(hf : (StrictAntiOn f (Ici 0) ∧ ContinuousOn f (Ici 0) ∧ Tendsto f atTop (𝓝 0)))
: ¬∃ y : ℝ, Tendsto (fun z : ℝ => ∫ x in Ico 0 z, ((f x) - f (x + 1))/(f x)) atTop (𝓝 y) :=
sorry
Problem: putnam_2010_b1
Tags: analysis

Is there an infinite sequence of real numbers $a_1, a_2, a_3, \dots$ such that \[ a_1^m + a_2^m + a_3^m + \cdots = m \] for every positive integer $m$?

Solution: Show that the solution is no such infinite sequence exists.

Lean 4 formal statement:
abbrev putnam_2010_b1_solution : Prop := sorry
theorem putnam_2010_b1
: (∃ a : ℕ → ℝ, ∀ m : ℕ, m > 0 → ∑' i : ℕ, (a i)^m = m) ↔ putnam_2010_b1_solution :=
sorry
Problem: putnam_2010_b2
Tags: geometry

Given that $A$, $B$, and $C$ are noncollinear points in the plane with integer coordinates such that the distances $AB$, $AC$, and $BC$ are integers, what is the smallest possible value of $AB$?

Solution: Show that the smallest distance is $3$.

Lean 4 formal statement:
abbrev putnam_2010_b2_solution : ℕ := sorry
theorem putnam_2010_b2
  (ABCintcoords ABCintdists ABCall: EuclideanSpace ℝ (Fin 2) → EuclideanSpace ℝ (Fin 2) → EuclideanSpace ℝ (Fin 2) → Prop)
  (hABCintcoords : ∀ A B C, ABCintcoords A B C ↔ (∀ i : Fin 2, A i = round (A i) ∧ B i = round (B i) ∧ C i = round (C i)))
  (hABCintdists : ∀ A B C, ABCintdists A B C ↔ (dist A B = round (dist A B) ∧ dist A C = round (dist A C) ∧ dist B C = round (dist B C)))
  (hABCall : ∀ A B C, ABCall A B C ↔ (¬Collinear ℝ {A, B, C} ∧ ABCintcoords A B C ∧ ABCintdists A B C)) :
  IsLeast {y | ∃ A B C, ABCall A B C ∧ y = dist A B} putnam_2010_b2_solution :=
sorry
Problem: putnam_2010_b3
Tags: analysis

There are $2010$ boxes labeled $B_1, B_2, \dots, B_{2010}$, and $2010n$ balls have been distributed among them, for some positive integer $n$. You may redistribute the balls by a sequence of moves, each of which consists of choosing an $i$ and moving \emph{exactly} $i$ balls from box $B_i$ into any one other box. For which values of $n$ is it possible to reach the distribution with exactly $n$ balls in each box, regardless of the initial distribution of balls?

Solution: Prove that it is possible if and only if $n \geq 1005$.

Lean 4 formal statement:
abbrev putnam_2010_b3_solution : Set ℕ := sorry
theorem putnam_2010_b3
      (n : ℕ) (hn : n > 0)
      (trans : (ℕ → Fin 2010 → ℕ) → ℕ → Prop)
      (htrans : ∀ P T, trans P T ↔ ∀ t : ℕ, t < T → ∃ i j,
            i ≠ j ∧
            P t i ≥ i.1 + 1 ∧ P (t + 1) i = P t i - (i.1 + 1) ∧ P (t + 1) j = P t j + (i.1 + 1) ∧
            ∀ k : Fin 2010, k ≠ i → k ≠ j → P (t + 1) k = P t k) :
      (∀ B, ∑ i, B i = 2010 * n → ∃ᵉ (P) (T), P 0 = B ∧ trans P T ∧ ∀ i, P T i = n)
      ↔ n ∈ putnam_2010_b3_solution :=
sorry
Problem: putnam_2010_b4
Tags: algebra

Find all pairs of polynomials $p(x)$ and $q(x)$ with real coefficients for which $p(x)q(x+1)-p(x+1)q(x)=1$.

Solution: Show that the pairs $(p,q)$ satisfying the given equation are those of the form $p(x)=ax+b,q(x)=cx+d$ for $a,b,c,d \in \mathbb{R}$ such that $bc-ad=1$.

Lean 4 formal statement:
abbrev putnam_2010_b4_solution : Set (Polynomial ℝ × Polynomial ℝ) := sorry
theorem putnam_2010_b4
(p q : Polynomial ℝ)
: (∀ x : ℝ, p.eval x * q.eval (x + 1) - p.eval (x + 1) * q.eval x = 1) ↔ (p, q) ∈ putnam_2010_b4_solution :=
sorry
Problem: putnam_2010_b5
Tags: analysis

Is there a strictly increasing function $f: \mathbb{R} \to \mathbb{R}$ such that $f'(x) = f(f(x))$ for all $x$?

Solution: Show that the solution is no such function exists.

Lean 4 formal statement:
abbrev putnam_2010_b5_solution : Prop := sorry
theorem putnam_2010_b5 :
  (∃ f : ℝ → ℝ, StrictMono f ∧ Differentiable ℝ f ∧ (∀ x : ℝ, deriv f x = f (f x))) ↔ putnam_2010_b5_solution :=
sorry
Problem: putnam_2010_b6
Tags: linear_algebra

Let $A$ be an $n \times n$ matrix of real numbers for some $n \geq 1$. For each positive integer $k$, let $A^{[k]}$ be the matrix obtained by raising each entry to the $k$th power. Show that if $A^k=A^{[k]}$ for $k=1,2,\dots,n+1$, then $A^k=A^{[k]}$ for all $k \geq 1$.

Solution: None.

Lean 4 formal statement:
theorem putnam_2010_b6
(n : ℕ)
(npos : n ≥ 1)
(A : Matrix (Fin n) (Fin n) ℝ)
(Apow : ℕ → Matrix (Fin n) (Fin n) ℝ)
(hApow : ∀ k > 0, Apow k = (fun i j : Fin n => (A i j) ^ k))
: (∀ k ∈ Set.Icc 1 (n + 1), A ^ k = Apow k) → (∀ k ≥ 1, A ^ k = Apow k) :=
sorry
Problem: putnam_2011_a1
Tags: geometry, algebra

Define a \emph{growing spiral} in the plane to be a sequence of points with integer coordinates $P_0=(0,0),P_1,\dots,P_n$ such that $n \geq 2$ and:
\begin{itemize}
\item the directed line segments $P_0P_1,P_1P_2,\dots,P_{n-1}P_n$ are in the successive coordinate directions east (for $P_0P_1$), north, west, south, east, etc.;
\item the lengths of these line segments are positive and strictly increasing.
\end{itemize}
How many of the points $(x,y)$ with integer coordinates $0 \leq x \leq 2011,0 \leq y \leq 2011$ \emph{cannot} be the last point, $P_n$ of any growing spiral?

Solution: Show that there are $10053$ excluded points.

Lean 4 formal statement:
abbrev putnam_2011_a1_solution : ℕ := sorry
theorem putnam_2011_a1
  (IsSpiral : List (Fin 2 → ℤ) → Prop)
  (IsSpiral_def : ∀ P, IsSpiral P ↔ P.length ≥ 3 ∧ P[0]! = 0 ∧
  (∃ l : Fin (P.length - 1) → ℕ, (∀ i, 0 < l i) ∧ StrictMono l ∧ (∀ i : Fin (P.length - 1),
    (i.1 % 4 = 0 → (P[i] 0 + l i = P[i.1 + 1]! 0 ∧ P[i] 1 = P[i.1 + 1]! 1)) ∧
    (i.1 % 4 = 1 → (P[i] 0 = P[i.1 + 1]! 0 ∧ P[i] 1 + l i = P[i.1 + 1]! 1)) ∧
    (i.1 % 4 = 2 → (P[i] 0 - l i = P[i.1 + 1]! 0 ∧ P[i] 1 = P[i.1 + 1]! 1)) ∧
    (i.1 % 4 = 3 → (P[i] 0 = P[i.1 + 1]! 0 ∧ P[i] 1 - l i = P[i.1 + 1]! 1))))) :
  {p | 0 ≤ p 0 ∧ p 0 ≤ 2011 ∧ 0 ≤ p 1 ∧ p 1 ≤ 2011 ∧ ¬∃ spiral, IsSpiral spiral ∧ spiral.getLast! = p}.encard = putnam_2011_a1_solution :=
sorry