ChristianZ97/PutnamBench-lean4 huggingface dataset:
Problem: putnam_2011_a2
Tags: analysis

Let $a_1,a_2,\dots$ and $b_1,b_2,\dots$ be sequences of positive real numbers such that $a_1 = b_1 = 1$ and $b_n = b_{n-1} a_n - 2$ for$n=2,3,\dots$. Assume that the sequence $(b_j)$ is bounded. Prove tha \[ S = \sum_{n=1}^\infty \frac{1}{a_1...a_n} \] converges, and evaluate $S$.

Solution: Show that the solution is $S = 3/2$.

Lean 4 formal statement:
noncomputable abbrev putnam_2011_a2_solution : ℝ := sorry
theorem putnam_2011_a2
(a b : ℕ → ℝ)
(habn : ∀ n : ℕ, a n > 0 ∧ b n > 0)
(hab1 : a 0 = 1 ∧ b 0 = 1)
(hb : ∀ n ≥ 1, b n = b (n-1) * a n - 2)
(hbnd : ∃ B : ℝ, ∀ n : ℕ, |b n| ≤ B)
: Tendsto (fun n => ∑ i : Fin n, 1/(∏ j : Fin (i + 1), (a j))) atTop (𝓝 putnam_2011_a2_solution) :=
sorry
Problem: putnam_2011_a3
Tags: analysis

Find a real number $c$ and a positive number $L$ for which $\lim_{r \to \infty} \frac{r^c \int_0^{\pi/2} x^r\sin x\,dx}{\int_0^{\pi/2} x^r\cos x\,dx}=L$.

Solution: Show that $(c,L)=(-1,2/\pi)$ works.

Lean 4 formal statement:
noncomputable abbrev putnam_2011_a3_solution : ℝ × ℝ := sorry
theorem putnam_2011_a3
: putnam_2011_a3_solution.2 > 0 ∧ Tendsto (fun r : ℝ => (r ^ putnam_2011_a3_solution.1 * ∫ x in Set.Ioo 0 (Real.pi / 2), x ^ r * Real.sin x) / (∫ x in Set.Ioo 0 (Real.pi / 2), x ^ r * Real.cos x)) atTop (𝓝 putnam_2011_a3_solution.2) :=
sorry
Problem: putnam_2011_a4
Tags: linear_algebra

For which positive integers $n$ is there an $n \times n$ matrix with integer entries such that every dot product of a row with itself is even, while every dot product of two different rows is odd?

Solution: Show that the answer is $n$ odd.

Lean 4 formal statement:
abbrev putnam_2011_a4_solution : Set ℕ := sorry
theorem putnam_2011_a4
    (nmat : ℕ → Prop)
    (hnmat : ∀ n, nmat n ↔
      ∃ A : Matrix (Fin n) (Fin n) ℤ,
        (∀ r, Even ((A r) ⬝ᵥ (A r))) ∧
        Pairwise fun r1 r2 ↦ Odd ((A r1) ⬝ᵥ (A r2)))
    (n : ℕ) (npos : 0 < n) :
    nmat n ↔ n ∈ putnam_2011_a4_solution :=
  sorry
Problem: putnam_2011_a5
Tags: analysis

Let $F:\mathbb{R}^2 \to \mathbb{R}$ and $g:\mathbb{R} \to \mathbb{R}$ be twice continuously differentiable functions with the following properties:
\begin{itemize}
\item $F(u,u)=0$ for every $u \in \mathbb{R}$;
\item for every $x \in \mathbb{R}$, $g(x)>0$ and $x^2g(x) \leq 1$;
\item for every $(u,v) \in \mathbb{R}^2$, the vector $\nabla F(u,v)$ is either $\mathbf{0}$ or parallel to the vector $\langle g(u),-g(v) \rangle$.
\end{itemize}
Prove that there exists a constant $C$ such that for every $n \geq 2$ and any $x_1,\dots,x_{n+1} \in \mathbb{R}$, we have $\min_{i \neq j} |F(x_i,x_j)| \leq \frac{C}{n}$.

Solution: None.

Lean 4 formal statement:
theorem putnam_2011_a5
(F : (Fin 2 → ℝ) → ℝ)
(g : ℝ → ℝ)
(vec : ℝ → ℝ → (Fin 2 → ℝ))
(Fgrad : (Fin 2 → ℝ) → (Fin 2 → ℝ))
(parallel : (Fin 2 → ℝ) → (Fin 2 → ℝ) → Prop)
(hparallel : parallel = (fun u v : Fin 2 → ℝ => ∃ c : ℝ, u = c • v))
(Fgdiff : ContDiff ℝ 2 F ∧ ContDiff ℝ 2 g)
(prop1 : ∀ uu : Fin 2 → ℝ, uu 0 = uu 1 → F uu = 0)
(prop2 : ∀ x : ℝ, g x > 0 ∧ x ^ 2 * g x ≤ 1)
(hvec : ∀ x y : ℝ, (vec x y) 0 = x ∧ (vec x y) 1 = y)
(hFgrad : ∀ uv : Fin 2 → ℝ, Fgrad uv 0 = deriv (fun x : ℝ => F (vec x (uv 1))) (uv 0) ∧ Fgrad uv 1 = deriv (fun y : ℝ => F (vec (uv 0) y)) (uv 1))
(prop3 : ∀ uv : Fin 2 → ℝ, Fgrad uv = 0 ∨ parallel (Fgrad uv) (vec (g (uv 0)) (-g (uv 1))))
: ∃ C : ℝ, ∀ n ≥ 2, ∀ x : Fin (n + 1) → ℝ, sInf {Fxx : ℝ | ∃ i j : Fin (n + 1), i ≠ j ∧ Fxx = |F (vec (x i) (x j))|} ≤ C / n :=
sorry
Problem: putnam_2011_a6
Tags: abstract_algebra, linear_algebra

Let $G$ be an abelian group with $n$ elements, and let $\{g_1=e,g_2,\dots,g_k\} \subsetneq G$ be a (not necessarily minimal) set of distinct generators of $G$. A special die, which randomly selects one of the elements $g_1,g_2,\dots,g_k$ with equal probability, is rolled $m$ times and the selected elements are multiplied to produce an element $g \in G$. Prove that there exists a real number $b \in (0,1)$ such that $\lim_{m \to \infty} \frac{1}{b^{2m}} \sum_{x \in G} (\text{Prob}(g=x)-\frac{1}{n})^2$ is positive and finite.

Solution: None.

Lean 4 formal statement:
theorem putnam_2011_a6
(G : Type*) [CommGroup G] [Fintype G]
(n : ℕ)
(gset : Set G)
(k : ℕ)
(mgprob : ℕ → G → ℝ)
(hmgprob : mgprob = (fun (m : ℕ) (x : G) => ({f : Fin m → gset | (∏ i : Fin m, (f i).1) = x}.ncard : ℝ) / k ^ m))
(hn : n = (Fintype.elems : Finset G).card)
(hk : gset.encard = k)
(hgsetprop : k < n)
(hgset1 : 1 ∈ gset)
(hgsetgen : Subgroup.closure gset = ⊤)
: ∃ b ∈ Set.Ioo (0 : ℝ) 1, ∃ C > 0, Tendsto (fun m : ℕ => (1 / b ^ (2 * m)) * (∑ x : G, (mgprob m x - 1 / n) ^ 2)) atTop (𝓝 C) :=
sorry
Problem: putnam_2011_b1
Tags: algebra

Let $h$ and $k$ be positive integers. Prove that for every $\epsilon>0$, there are positive integers $m$ and $n$ such that $\epsilon<|h\sqrt{m}-k\sqrt{n}|<2\epsilon$.

Solution: None.

Lean 4 formal statement:
theorem putnam_2011_b1
(h k : ℤ)
(hkpos : h > 0 ∧ k > 0)
: ∀ ε > 0, ∃ m n : ℤ, m > 0 ∧ n > 0 ∧ ε < |h * Real.sqrt m - k * Real.sqrt n| ∧ |h * Real.sqrt m - k * Real.sqrt n| < 2 * ε :=
sorry
Problem: putnam_2011_b2
Tags: number_theory

Let $S$ be the set of all ordered triples $(p,q,r)$ of prime numbers for which at least one rational number $x$ satisfies $px^2+qx+r=0$. Which primes appear in seven or more elements of $S$?

Solution: Show that only the primes $2$ and $5$ appear seven or more times.

Lean 4 formal statement:
abbrev putnam_2011_b2_solution : Set ℕ := sorry
theorem putnam_2011_b2
  (S : Set (Fin 3 → ℕ))
  (t : ℕ)
  (hS : S = {s : Fin 3 → ℕ | (s 0).Prime ∧ (s 1).Prime ∧ (s 2).Prime ∧ ∃ x : ℚ, (s 0) * x ^ 2 + (s 1) * x + (s 2) = 0}) :
  (t.Prime ∧ ({s ∈ S | ∃ i : Fin 3, s i = t}.encard ≥ 7))
  ↔ t ∈ putnam_2011_b2_solution :=
sorry
Problem: putnam_2011_b3
Tags: analysis

Let $f$ and $g$ be (real-valued) functions defined on an open interval containing $0$, with $g$ nonzero and continuous at $0$. If $fg$ and $f/g$ are differentiable at $0$, must $f$ be differentiable at $0$?

Solution: Prove that $f$ is differentiable.

Lean 4 formal statement:
abbrev putnam_2011_b3_solution : Prop := sorry
theorem putnam_2011_b3 :
  ((∀ f g : ℝ → ℝ,
    g 0 ≠ 0 → ContinuousAt g 0 →
    DifferentiableAt ℝ (fun x ↦ f x * g x) 0 →
    DifferentiableAt ℝ (fun x ↦ f x / g x) 0 →
    (DifferentiableAt ℝ f 0))
  ↔ putnam_2011_b3_solution) :=
sorry
Problem: putnam_2011_b4
Tags: linear_algebra

In a tournament, $2011$ players meet $2011$ times to play a multiplayer game. Every game is played by all $2011$ players together and ends with each of the players either winning or losing. The standings are kept in two $2011 \times 2011$ matrices, $T=(T_{hk})$ and $W=(W_{hk})$. Initially, $T=W=0$. After every game, for every $(h,k)$ (including for $h=k$), if players $h$ and $k$ tied (that is, both won or both lost), the entry $T_{hk}$ is increased by $1$, while if player $h$ won and player $k$ lost, the entry $W_{hk}$ is increased by $1$ and $W_{kh}$ is decreased by $1$. Prove that at the end of the tournament, $\det(T+iW)$ is a non-negative integer divisible by $2^{2010}$.

Solution: None.

Lean 4 formal statement:
theorem putnam_2011_b4
(games : Fin 2011 → Fin 2011 → Bool)
(T W : Matrix (Fin 2011) (Fin 2011) ℂ)
(hT : T = (fun p1 p2 => ({g | games g p1 = games g p2}.ncard : ℂ)))
(hW : W = (fun p1 p2 => ({g | games g p1 ∧ !games g p2}.ncard - {g | !games g p1 ∧ games g p2}.ncard : ℂ)))
: ∃ n : ℕ, (T + Complex.I • W).det = n ∧ (2 ^ 2010) ∣ n :=
sorry
Problem: putnam_2011_b5
Tags: analysis

Let $a_1, a_2, \dots$ be real numbers. Suppose that there is a constant $A$ such that for all $n$,
\[
\int_{-\infty}^\infty \left( \sum_{i=1}^n \frac{1}{1 + (x-a_i)^2} \right)^2\,dx \leq An.
\]
Prove there is a constant $B>0$ such that for all $n$,
\[
\sum_{i,j=1}^n (1 + (a_i - a_j)^2) \geq Bn^3.
\]

Solution: None.

Lean 4 formal statement:
theorem putnam_2011_b5
  (a : ℕ → ℝ)
  (h : ∃ A : ℝ, ∀ n : ℕ, ∫ x : ℝ, ((∑ i : Finset.range n, 1 / (1 + (x - a i) ^ 2)) ^ 2) ≤ A * n) :
  ∃ B : ℝ, B > 0 ∧
    ∀ n : ℕ, ∑' i : Finset.range n, ∑' j : Finset.range n, (1 + (a i - a j) ^ 2) ≥ B * n ^ 3 :=
sorry
Problem: putnam_2011_b6
Tags: number_theory

Let $p$ be an odd prime. Show that for at least $(p+1)/2$ values of $n$ in $\{0,1,2,\dots,p-1\}$,
\[
\sum_{k=0}^{p-1} k! n^k \qquad \mbox{is not divisible by $p$.}
\]

Solution: None.

Lean 4 formal statement:
theorem putnam_2011_b6
(p : ℕ)
(hp : Odd p ∧ Nat.Prime p)
: {n ∈ Finset.range p | ¬ p ∣ ∑ k : Finset.range p, Nat.factorial k * n^(k : ℕ)}.card ≥ (p + 1)/2 :=
sorry
Problem: putnam_2012_a1
Tags: geometry, algebra

Let $d_1,d_2,\dots,d_{12}$ be real numbers in the open interval $(1,12)$. Show that there exist distinct indices $i,j,k$ such that $d_i,d_j,d_k$ are the side lengths of an acute triangle.

Solution: None.

Lean 4 formal statement:
theorem putnam_2012_a1
(d : Fin 12 → ℝ)
(hd : ∀ i, d i ∈ Set.Ioo 1 12)
: ∃ i j k, i ≠ j ∧ i ≠ k ∧ j ≠ k ∧ d k ≥ d i ∧ d k ≥ d j ∧ (d i) ^ 2 + (d j) ^ 2 > (d k) ^ 2 :=
sorry
Problem: putnam_2012_a2
Tags: abstract_algebra

Let $*$ be a commutative and associative binary operation on a set $S$. Assume that for every $x$ and $y$ in $S$, there exists $z$ in $S$ such that $x*z=y$. (This $z$ may depend on $x$ and $y$.) Show that if $a,b,c$ are in $S$ and $a*c=b*c$, then $a=b$.

Solution: None.

Lean 4 formal statement:
theorem putnam_2012_a2
(S : Type*) [CommSemigroup S]
(a b c : S)
(hS : ∀ x y : S, ∃ z : S, x * z = y)
(habc : a * c = b * c)
: a = b :=
sorry
Problem: putnam_2012_a3
Tags: analysis, algebra

Let $f: [-1, 1] \to \mathbb{R}$ be a continuous function such that
\begin{itemize}
\item[(i)]
$f(x) = \frac{2-x^2}{2} f \left( \frac{x^2}{2-x^2} \right)$ for every $x$ in $[-1, 1]$,
\item[(ii)]
$f(0) = 1$, and
\item[(iii)]
$\lim_{x \to 1^-} \frac{f(x)}{\sqrt{1-x}}$ exists and is finite.
\end{itemize}
Prove that $f$ is unique, and express $f(x)$ in closed form.

Solution: $f(x) = \sqrt{1-x^2}$ for all $x \in [-1,1]$.

Lean 4 formal statement:
noncomputable abbrev putnam_2012_a3_solution : ℝ → ℝ := sorry
theorem putnam_2012_a3
(S : Set ℝ)
(hS : S = Set.Icc (-1 : ℝ) 1)
(fsat : (ℝ → ℝ) → Prop)
(hfsat : fsat = fun f : ℝ → ℝ => ContinuousOn f S ∧
(∀ x ∈ S, f x = ((2 - x^2)/2)*f (x^2/(2 - x^2))) ∧ f 0 = 1 ∧
(∃ y : ℝ, leftLim (fun x : ℝ => (f x)/Real.sqrt (1 - x)) 1 = y))
: fsat putnam_2012_a3_solution ∧ ∀ f : ℝ → ℝ, fsat f → ∀ x ∈ S, f x = putnam_2012_a3_solution x :=
sorry
Problem: putnam_2012_a4
Tags: algebra

Let $q$ and $r$ be integers with $q>0$, and let $A$ and $B$ be intervals on the real line. Let $T$ be the set of all $b+mq$ where $b$ and $m$ are integers with $b$ in $B$, and let $S$ be the set of all integers $a$ in $A$ such that $ra$ is in $T$. Show that if the product of the lengths of $A$ and $B$ is less than $q$, then $S$ is the intersection of $A$ with some arithmetic progression.

Solution: None.

Lean 4 formal statement:
theorem putnam_2012_a4
    (IsFiniteAP : Set ℤ → Prop)
    (IsFiniteAP_def : ∀ s,
      IsFiniteAP s ↔ ∃ n : ℕ, ∃ a d : ℤ, 0 < d ∧ s = {a + i * d | i : Fin n})
    (q r : ℤ)
    (A B : Fin 2 → ℝ)
    (T : Set ℤ)
    (S : Set ℤ)
    (qpos : q > 0)
    (ABlt : A 0 < A 1 ∧ B 0 < B 1)
    (hT : T = {x : ℤ | ∃ b m : ℤ, (b : ℝ) ∈ Set.Icc (B 0) (B 1) ∧ x = b + m * q})
    (hS : S = {a : ℤ | (a : ℝ) ∈ Set.Icc (A 0) (A 1) ∧ r * a ∈ T})
    (prod_lt : (A 1 - A 0) * (B 1 - B 0) < q) :
    IsFiniteAP {x | x ∈ S ∧ (x : ℝ) ∈ Set.Icc (A 0) (A 1)} :=
  sorry
Problem: putnam_2012_a5
Tags: linear_algebra

Let $\FF_p$ denote the field of integers modulo a prime $p$, and let $n$ be a positive integer. Let $v$ be a fixed vector in $\FF_p^n$, let $M$ be an $n \times n$ matrix with entries of $\FF_p$, and define $G: \FF_p^n \to \FF_p^n$ by $G(x) = v + Mx$. Let $G^{(k)}$ denote the $k$-fold composition of $G$ with itself, that is, $G^{(1)}(x) = G(x)$ and $G^{(k+1)}(x) = G(G^{(k)}(x))$. Determine all pairs $p, n$ for which there exist $v$ and $M$ such that the $p^n$ vectors $G^{(k)}(0)$, $k=1,2,\dots,p^n$ are distinct.

Solution: Show that the solution is the pairs $(p,n)$ with $n = 1$ as well as the single pair $(2,2)$.

Lean 4 formal statement:
abbrev putnam_2012_a5_solution : Set (ℕ × ℕ) := sorry
theorem putnam_2012_a5
  (n p : ℕ)
  (hn : n > 0)
  (hp : Nat.Prime p)
  {F : Type*} [Field F] [Fintype F]
  (hK : Fintype.card F = p)
  (G : Matrix (Fin n) (Fin n) F → (Fin n → F) → (Fin n → F) → (Fin n → F))
  (hG : ∀ M v x, G M v x = v + mulVec M x) :
  (n, p) ∈ putnam_2012_a5_solution ↔
    ∃ᵉ (M : Matrix (Fin n) (Fin n) F) (v : (Fin n → F)),
      ¬(∃ i j : Finset.range (p^n), i ≠ j ∧ (G M v)^[i + 1] 0 = (G M v)^[j + 1] 0) :=
  sorry
Problem: putnam_2012_a6
Tags: analysis

Let $f(x,y)$ be a continuous, real-valued function on $\mathbb{R}^2$. Suppose that, for every rectangular region $R$ of area $1$, the double integral of $f(x,y)$ over $R$ equals $0$.
Must $f(x,y)$ be identically $0$?

Solution: Prove that $f(x,y)$ must be identically $0$.

Lean 4 formal statement:
abbrev putnam_2012_a6_solution : Prop := sorry
theorem putnam_2012_a6
(p : ((ℝ × ℝ) → ℝ) → Prop)
(hp : ∀ f, p f ↔
    Continuous f ∧
    ∀ x1 x2 y1 y2 : ℝ, x2 > x1 → y2 > y1
      → (x2 - x1) * (y2 - y1) = 1 → ∫ x in x1..x2, ∫ y in y1..y2, f (x, y) = 0)
: ((∀ f x y, p f → f (x, y) = 0) ↔ putnam_2012_a6_solution) :=
sorry
Problem: putnam_2012_b1
Tags: algebra

Let $S$ be a class of functions from $[0, \infty)$ to $[0, \infty)$ that satisfies:
\begin{itemize}
\item[(i)]
The functions $f_1(x) = e^x - 1$ and $f_2(x) = \ln(x+1)$ are in $S$;
\item[(ii)]
If $f(x)$ and $g(x)$ are in $S$, the functions $f(x) + g(x)$ and $f(g(x))$ are in $S$;
\item[(iii)]
If $f(x)$ and $g(x)$ are in $S$ and $f(x) \geq g(x)$ for all $x \geq 0$, then the function
$f(x) - g(x)$ is in $S$.
\end{itemize}
Prove that if $f(x)$ and $g(x)$ are in $S$, then the function $f(x) g(x)$ is also in $S$.

Solution: None.

Lean 4 formal statement:
theorem putnam_2012_b1
(S : Set (Set.Ici (0 : ℝ) → ℝ))
(rngS : ∀ f ∈ S, ∀ x : Set.Ici (0 : ℝ), f x ∈ Set.Ici (0 : ℝ))
(f1 : Set.Ici (0 : ℝ) → ℝ)
(hf1 : f1 = fun (x : Set.Ici (0 : ℝ)) ↦ exp x - 1)
(f2 : Set.Ici (0 : ℝ) → ℝ)
(hf2 : f2 = fun (x : Set.Ici (0 : ℝ)) ↦ Real.log (x + 1))
(hf1mem : f1 ∈ S)
(hf2mem : f2 ∈ S)
(hsum : ∀ f ∈ S, ∀ g ∈ S, (fun x ↦ (f x) + (g x)) ∈ S)
(hcomp : ∀ f ∈ S, ∀ g ∈ S, ∀ gnneg : Set.Ici (0 : ℝ) → Set.Ici (0 : ℝ), ((∀ x : Set.Ici (0 : ℝ), g x = gnneg x) → (fun x ↦ f (gnneg x)) ∈ S))
(hdiff : ∀ f ∈ S, ∀ g ∈ S, (∀ x : Set.Ici (0 : ℝ), f x ≥ g x) → (fun x ↦ (f x) - (g x)) ∈ S)
: (∀ f ∈ S, ∀ g ∈ S, (fun x ↦ (f x) * (g x)) ∈ S) :=
sorry
Problem: putnam_2012_b3
Tags: combinatorics

A round-robin tournament of $2n$ teams lasted for $2n-1$ days, as follows. On each day, every team played one game against another team, with one team winning and one team losing in each of the $n$ games. Over the course of the tournament, each team played every other team exactly once. Can one necessarily choose one winning team from each day without choosing any team more than once?

Solution: Show that the answer is yes.

Lean 4 formal statement:
abbrev putnam_2012_b3_solution : Prop := sorry
theorem putnam_2012_b3
(nmatchupsgames nmatchupsall : (n : ℕ) → (Fin (2 * n - 1) → (Fin (2 * n) → Fin (2 * n))) → Prop)
(nmatchupswins nmatchupswinschoices: (n : ℕ) → (Fin (2 * n - 1) → (Fin (2 * n) → Fin (2 * n))) → (Fin (2 * n - 1) → (Fin (2 * n) → Bool)) → Prop)
(hnmatchupsall : ∀ n matchups, nmatchupsall n matchups ↔ ∀ t1 t2, t1 ≠ t2 → (∃ d, matchups d t1 = t2))
(hnmatchupsgames : ∀ n matchups, nmatchupsgames n matchups ↔ ∀ d, ∀ t, matchups d t ≠ t ∧ matchups d (matchups d t) = t)
(hnmatchupswins : ∀ n matchups wins, nmatchupswins n matchups wins ↔ ∀ d t, wins d t = !(wins d (matchups d t)))
(hnmatchupswinschoices : ∀ n matchups wins, nmatchupswinschoices n matchups wins ↔ ∃ choices, (∀ d, wins d (choices d)) ∧ Function.Injective choices) :
  (∀ n ≥ 1, ∀ matchups wins,
    (nmatchupsgames n matchups ∧ nmatchupsall n matchups ∧ nmatchupswins n matchups wins) →
    nmatchupswinschoices n matchups wins)
  ↔ putnam_2012_b3_solution :=
sorry
Problem: putnam_2012_b4
Tags: analysis

Suppose that $a_0 = 1$ and that $a_{n+1} = a_n + e^{-a_n}$ for $n=0,1,2,\dots$. Does $a_n - \log n$
have a finite limit as $n \to \infty$? (Here $\log n = \log_e n = \ln n$.)

Solution: Prove that the sequence has a finite limit.

Lean 4 formal statement:
noncomputable abbrev putnam_2012_b4_solution : Prop := sorry
theorem putnam_2012_b4
(a : ℕ → ℝ)
(ha0 : a 0 = 1)
(han : ∀ n : ℕ, a (n + 1) = a n + exp (-a n))
: ((∃ L : ℝ, Tendsto (fun n ↦ a n - Real.log n) atTop (𝓝 L)) ↔ putnam_2012_b4_solution) :=
sorry
Problem: putnam_2012_b5
Tags: analysis

Prove that, for any two bounded functions $g_1, g_2: \RR \to [1, \infty)$, there exist functions $h_1, h_2: \RR \to \RR$ such that, for every $x \in \RR$, \[ \sup_{s \in \RR} (g_1(s)^x g_2(s))  = \max_{t \in \RR} (x h_1(t) + h_2(t)).\]

Solution: None.

Lean 4 formal statement:
theorem putnam_2012_b5
(g1 g2 : ℝ → ℝ)
(hgim : ∀ x : ℝ, g1 x ≥ 1 ∧ g2 x ≥ 1)
(hgbd : ∃ B1 B2 : ℝ, ∀ x : ℝ, g1 x ≤ B1 ∧ g2 x ≤ B2)
: ∃ h1 h2 : ℝ → ℝ, ∀ x : ℝ, sSup {((g1 s)^x * (g2 s)) | s : ℝ} = sSup {(x * (h1 t) + h2 t) | t : ℝ} :=
sorry
Problem: putnam_2012_b6
Tags: number_theory

Let $p$ be an odd prime number such that $p \equiv 2 \pmod{3}$. Define a permutation $\pi$ of the residue classes modulo $p$ by $\pi(x) \equiv x^3 \pmod{p}$. Show that $\pi$ is an even permutation if and only if $p \equiv 3 \pmod{4}$.

Solution: None.

Lean 4 formal statement:
theorem putnam_2012_b6
(p : ℕ)
(hpodd : Odd p)
(hpprime : Nat.Prime p)
(hpmod3 : p ≡ 2 [MOD 3])
(P : Equiv.Perm (Fin p))
(hP : ∀ i : Fin p, P i = (i * i * i))
: Equiv.Perm.signAux P = 1 ↔ (p ≡ 3 [MOD 4]) :=
sorry
Problem: putnam_2013_a2
Tags: number_theory, algebra

Let $S$ be the set of all positive integers that are \emph{not} perfect squares. For $n$ in $S$, consider choices of integers
$a_1, a_2, \dots, a_r$ such that $n < a_1<  a_2 < \cdots < a_r$
and $n \cdot a_1 \cdot a_2 \cdots a_r$ is a perfect square, and
let $f(n)$ be the minumum of $a_r$ over all such choices. For example,
$2 \cdot 3 \cdot 6$ is a perfect square, while $2 \cdot 3$, $2 \cdot 4$,
$2 \cdot 5$, $2 \cdot 3 \cdot 4$, $2 \cdot 3 \cdot 5$, $2 \cdot 4 \cdot 5$, and $2 \cdot 3 \cdot 4 \cdot 5$ are not, and so $f(2) = 6$.
Show that the function $f$ from $S$ to the integers is one-to-one.

Solution: None.

Lean 4 formal statement:
theorem putnam_2013_a2
  (S : Set ℤ)
  (hS : S = {n : ℤ | n > 0 ∧ ¬∃ m : ℤ, m ^ 2 = n})
  (P : ℤ → List ℤ → Prop)
  (hP : ∀ n a, P n a ↔
    a.length > 0 ∧ n < a[0]! ∧
    (∃ m : ℤ, m ^ 2 = n * a.prod) ∧
    (∀ i : Fin (a.length - 1), a[i] < a[i+(1:ℕ)]))
  (T : ℤ → Set ℤ)
  (hT : T = fun n : ℤ => {m : ℤ | ∃ a : List ℤ, P n a ∧ a[a.length - 1]! = m})
  (f : ℤ → ℤ)
  (hf : ∀ n ∈ S, ((∃ r ∈ T n, f n = r) ∧ ∀ r ∈ T n, f n ≤ r)) :
  InjOn f S :=
sorry
Problem: putnam_2013_a3
Tags: analysis

Suppose that the real numbers \( a_0, a_1, \ldots, a_n \) and \( x \), with \( 0 < x < 1 \), satisfy $ \frac{a_0}{1-x} + \frac{a_1}{(1-x)^2} + \cdots + \frac{a_n}{(1-x)^{n+1}} = 0. $ Prove that there exists a real number \( y \) with \( 0 < y < 1 \) such that $ a_0 + a_1y + \cdots + a_ny^n = 0. $.

Solution: None.

Lean 4 formal statement:
theorem putnam_2013_a3
(n : ℕ)
(a : Set.Icc 0 n → ℝ)
(x : ℝ)
(hx : 0 < x ∧ x < 1)
(hsum : (∑ i : Set.Icc 0 n, a i / (1 - x ^ (i.1 + 1))) = 0)
: ∃ y : ℝ, 0 < y ∧ y < 1 ∧ (∑ i : Set.Icc 0 n, a i * y ^ i.1) = 0 :=
sorry
Problem: putnam_2013_a4
Tags: algebra

A finite collection of digits $0$ and $1$ is written around a circle. An \emph{arc} of length $L \geq 0$ consists of $L$ consecutive digits around the circle. For each arc $w$, let $Z(w)$ and $N(w)$ denote the number of $0$'s in $w$ and the number of $1$'s in $w$, respectively. Assume that $|Z(w)-Z(w')| \leq 1$ for any two arcs $w,w'$ of the same length. Suppose that some arcs $w_1,\dots,w_k$ have the property that $Z=\frac{1}{k}\sum_{j=1}^k Z(w_j)$ and $N=\frac{1}{k}\sum_{j=1}^k N(w_j)$ are both integers. Prove that there exists an arc $w$ with $Z(w)=Z$ and $N(w)=N$.

Solution: None.

Lean 4 formal statement:
theorem putnam_2013_a4
(n k : ℕ)
(circle : Fin n → Fin 2)
(Z N : Fin n × Fin (n + 1) → ℤ)
(ws : Fin k → Fin n × Fin (n + 1))
(Zsum Nsum : ℤ)
(npos : n ≥ 1) (kpos : k ≥ 1)
(hZ : ∀ w, Z w = ∑ l ∈ {x : Fin n | x.1 < w.2.1}, if (circle (w.1 + l) = 0) then 1 else 0)
(hN : ∀ w, N w = ∑ l ∈ {x : Fin n | x.1 < w.2.1}, if (circle (w.1 + l) = 1) then 1 else 0)
(Zle1 : ∀ w w', w.2 = w'.2 → |(Z w : ℤ) - Z w'| ≤ 1)
(hZsum : Zsum = ((1 : ℝ) / k) * ∑ j : Fin k, Z (ws j))
(hNsum : Nsum = ((1 : ℝ) / k) * ∑ j : Fin k, N (ws j)) :
∃ w, Z w = Zsum ∧ N w = Nsum :=
sorry
Problem: putnam_2013_a5
Tags: geometry, linear_algebra

For $m \geq 3$, a list of $\binom{m}{3}$ real numbers $a_{ijk}$ ($1 \leq i< j< k \leq m$) is said to be \emph{area definite} for $\mathbb{R}^n$ if the inequality $\sum_{1 \leq i< j< k \leq m} a_{ijk} \cdot \text{Area}(\Delta A_iA_jA_k) \geq 0$ holds for every choice of $m$ points $A_1,\dots,A_m$ in $\mathbb{R}^n$. For example, the list of four numbers $a_{123}=a_{124}=a_{134}=1$, $a_{234}=-1$ is area definite for $\mathbb{R}^2$. Prove that if a list of $\binom{m}{3}$ numbers is area definite for $\mathbb{R}^2$, then it is area definite for $\mathbb{R}^3$.

Solution: None.

Lean 4 formal statement:
theorem putnam_2013_a5
(m : ℕ)
(area2 : (Fin 2 → ℝ) → (Fin 2 → ℝ) → (Fin 2 → ℝ) → ℝ)
(area3 : (Fin 3 → ℝ) → (Fin 3 → ℝ) → (Fin 3 → ℝ) → ℝ)
(areadef2 : (Fin m → Fin m → Fin m → ℝ) → Prop)
(areadef3 : (Fin m → Fin m → Fin m → ℝ) → Prop)
(mge3 : m ≥ 3)
(harea2 : ∀ a b c, area2 a b c = (volume (convexHull ℝ {a, b, c})).toReal)
(harea3 : ∀ a b c, area3 a b c = (μH[2] (convexHull ℝ {a, b, c})).toReal)
(hareadef2 : ∀ a, areadef2 a ↔ ∀ A : Fin m → (Fin 2 → ℝ), (∑ i : Fin m, ∑ j : Fin m, ∑ k : Fin m, if (i < j ∧ j < k) then (a i j k * area2 (A i) (A j) (A k)) else 0) ≥ 0)
(hareadef3 : ∀ a, areadef3 a ↔ ∀ A : Fin m → (Fin 3 → ℝ), (∑ i : Fin m, ∑ j : Fin m, ∑ k : Fin m, if (i < j ∧ j < k) then (a i j k * area3 (A i) (A j) (A k)) else 0) ≥ 0)
: ∀ a, areadef2 a → areadef3 a :=
sorry
Problem: putnam_2013_a6
Tags: algebra

Define a function $w:\mathbb{Z} \times \mathbb{Z} \to \mathbb{Z}$ as follows. For $|a|,|b| \leq 2$, let $w(a,b)$ be as in the table shown; otherwise, let $w(a,b)=0$.
\begin{center}
\begin{tabular}{|cc|r|r|r|r|r|}
\hline
\multicolumn{2}{|c|}{\multirow{2}{*}{$w(a,b)$}} & \multicolumn{5}{|c|}{$b$} \\
& & -2 & -1 & 0 & 1 & 2 \\
\hline
& -2 & -1 & -2 & 2 & -2 & -1 \\
& -1 & -2 & 4 & -4 & 4 & -2 \\
$a$ & 0 & 2 & -4 & 12 & -4 & 2 \\
& 1 & -2 & 4 & -4 & 4 & -2 \\
& 2 & -1 & -2 & 2 & -2 & -1 \\
\hline
\end{tabular}
\end{center}
For every finite subset $S$ of $\mathbb{Z} \times \mathbb{Z}$, define $A(S)=\sum_{(\mathbf{s},\mathbf{s}') \in S \times S} w(\mathbf{s}-\mathbf{s}')$. Prove that if $S$ is any finite nonempty subset of $\mathbb{Z} \times \mathbb{Z}$, then $A(S)>0$. (For example, if $S=\{(0,1),(0,2),(2,0),(3,1)\}$, then the terms in $A(S)$ are $12,12,12,12,4,4,0,0,0,0,-1,-1,-2,-2,-4,-4$.)

Solution: None.

Lean 4 formal statement:
theorem putnam_2013_a6
  (w : ℤ → ℤ → ℤ)
  (A : Finset (ℤ × ℤ) → ℤ)
  (hwn1 : w (-2) (-2) = -1 ∧ w 2 (-2) = -1 ∧ w (-2) 2 = -1 ∧ w 2 2 = -1)
  (hwn2 : w (-1) (-2) = -2 ∧ w 1 (-2) = -2 ∧ w (-2) (-1) = -2 ∧ w 2 (-1) = -2 ∧ w (-2) 1 = -2 ∧ w 2 1 = -2 ∧ w (-1) 2 = -2 ∧ w 1 2 = -2)
  (hw2 : w 0 (-2) = 2 ∧ w (-2) 0 = 2 ∧ w 2 0 = 2 ∧ w 0 2 = 2)
  (hw4 : w (-1) (-1) = 4 ∧ w 1 (-1) = 4 ∧ w (-1) 1 = 4 ∧ w 1 1 = 4)
  (hwn4 : w 0 (-1) = -4 ∧ w (-1) 0 = -4 ∧ w 1 0 = -4 ∧ w 0 1 = -4)
  (hw12 : w 0 0 = 12)
  (hw0 : ∀ a b : ℤ, (|a| > 2 ∨ |b| > 2) → w a b = 0)
  (hA : ∀ S, A S = ∑ s ∈ S, ∑ s' ∈ S, w (s - s').1 (s - s').2) :
  ∀ S : Finset (ℤ × ℤ), Nonempty S → A S > 0 :=
sorry
Problem: putnam_2013_b1
Tags: algebra

For positive integers $n$, let the numbers $c(n)$ be determined by the rules $c(1)=1$, $c(2n)=c(n)$, and $c(2n+1)=(-1)^nc(n)$. Find the value of $\sum_{n=1}^{2013} c(n)c(n+2)$.

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

Lean 4 formal statement:
abbrev putnam_2013_b1_solution : ℤ := sorry
theorem putnam_2013_b1
(c : ℕ → ℤ)
(hc1 : c 1 = 1)
(hceven : ∀ n : ℕ, n > 0 → c (2 * n) = c n)
(hcodd : ∀ n : ℕ, n > 0 → c (2 * n + 1) = (-1) ^ n * c n)
: (∑ n : Set.Icc 1 2013, c n * c (n.1 + 2)) = putnam_2013_b1_solution :=
sorry