ChristianZ97/PutnamBench-lean4 huggingface dataset:
Problem: putnam_1993_b4
Tags: analysis

The function $K(x,y)$ is positive and continuous for $0 \leq x \leq 1,0 \leq y \leq 1$, and the functions $f(x)$ and $g(x)$ are positive and continuous for $0 \leq x \leq 1$. Suppose that for all $x$, $0 \leq x \leq 1$, $\int_0^1 f(y)K(x,y)\,dy=g(x)$ and $\int_0^1 g(y)K(x,y)\,dy=f(x)$. Show that $f(x)=g(x)$ for $0 \leq x \leq 1$.

Solution: None.

Lean 4 formal statement:
theorem putnam_1993_b4
(K : ℝ × ℝ → ℝ)
(f g : ℝ → ℝ)
(Kpos : ∀ x y : Set.Icc (0 : ℝ) 1, K (x, y) > 0)
(Kcont : ContinuousOn K {(x, y) : ℝ × ℝ | x ∈ Set.Icc 0 1 ∧ y ∈ Set.Icc 0 1})
(fgpos : ∀ x : Set.Icc (0 : ℝ) 1, f x > 0 ∧ g x > 0)
(fgcont : ContinuousOn f (Set.Icc 0 1) ∧ ContinuousOn g (Set.Icc 0 1))
(fgint : ∀ x : Set.Icc (0 : ℝ) 1, (∫ y in Set.Ioo 0 1, f y * K (x, y)) = g x ∧ (∫ y in Set.Ioo 0 1, g y * K (x, y)) = f x)
: ∀ x : Set.Icc (0 : ℝ) 1, f x = g x :=
sorry
Problem: putnam_1993_b5
Tags: geometry, number_theory, linear_algebra

Show there do not exist four points in the Euclidean plane such that the pairwise distances between the points are all odd integers.

Solution: None.

Lean 4 formal statement:
theorem putnam_1993_b5:
  ¬∃ p : Fin 4 → (EuclideanSpace ℝ (Fin 2)),
    ∀ i j, i ≠ j →
      (∃ n : ℤ, dist (p i) (p j) = n ∧ Odd n) :=
  sorry
Problem: putnam_1993_b6
Tags: algebra

Let $S$ be a set of three, not necessarily distinct, positive integers. Show that one can transform $S$ into a set containing $0$ by a finite number of applications of the following rule: Select two of the three integers, say $x$ and $y$, where $x \leq y$ and replace them with $2x$ and $y-x$.

Solution: None.

Lean 4 formal statement:
theorem putnam_1993_b6
(S : Fin 3 → ℕ)
(f : Fin 3 → Fin 3 → (Fin 3 → ℕ) → (Fin 3 → ℕ))
(Spos : ∀ i : Fin 3, S i > 0)
(hf : ∀ i j k : Fin 3, (i ≠ j ∧ i ≠ k ∧ j ≠ k) → ∀ S' : Fin 3 → ℕ, if S' i ≤ S' j then ((f i j S') i = 2 * S' i ∧ (f i j S') j = S' j - S' i ∧ (f i j S') k = S' k) else (f i j S' = S'))
: ∃ (Ss : ℕ → (Fin 3 → ℕ)) (N : ℕ), Ss 0 = S ∧ (∃ i : Fin 3, Ss N i = 0) ∧ (∀ n : Fin N, ∃ i j : Fin 3, i ≠ j ∧ f i j (Ss n) = Ss ((n : ℕ) + 1)) :=
sorry
Problem: putnam_1994_a1
Tags: analysis

Suppose that a sequence $a_1,a_2,a_3,\dots$ satisfies $0< a_n \leq a_{2n}+a_{2n+1}$ for all $n \geq 1$. Prove that the series $\sum_{n=1}^\infty a_n$ diverges.

Solution: None.

Lean 4 formal statement:
theorem putnam_1994_a1
    (a : ℕ → ℝ)
    (ha : ∀ n ≥ 1, 0 < a n ∧ a n ≤ a (2 * n) + a (2 * n + 1)) :
    Tendsto (fun N : ℕ => ∑ n : Set.Icc 1 N, a n) atTop atTop :=
  sorry
Problem: putnam_1994_a3
Tags: geometry

Show that if the points of an isosceles right triangle of side length $1$ are each colored with one of four colors, then there must be two points of the same color which are at least a distance $2-\sqrt{2}$ apart.

Solution: None.

Lean 4 formal statement:
theorem putnam_1994_a3
(T : Set (EuclideanSpace ℝ (Fin 2)))
(hT : T = convexHull ℝ {!₂[0,0], !₂[1,0], !₂[0,1]})
(Tcolors : T → Fin 4)
: ∃ p q : T, Tcolors p = Tcolors q ∧ dist p.1 q.1 ≥ 2 - Real.sqrt 2 :=
sorry
Problem: putnam_1994_a4
Tags: linear_algebra

Let $A$ and $B$ be $2 \times 2$ matrices with integer entries such that $A$, $A+B$, $A+2B$, $A+3B$, and $A+4B$ are all invertible matrices whose inverses have integer entries. Show that $A+5B$ is invertible and that its inverse has integer entries.

Solution: None.

Lean 4 formal statement:
theorem putnam_1994_a4
  (A B : Matrix (Fin 2) (Fin 2) ℤ)
  (ABinv : Nonempty (Invertible A) ∧
    Nonempty (Invertible (A + B)) ∧
    Nonempty (Invertible (A + 2 * B)) ∧
    Nonempty (Invertible (A + 3 * B)) ∧
    Nonempty (Invertible (A + 4 * B)))
  : Nonempty (Invertible (A + 5 * B)) :=
sorry
Problem: putnam_1994_a5
Tags: analysis

Let $(r_n)_{n \geq 0}$ be a sequence of positive real numbers such that $\lim_{n \to \infty} r_n=0$. Let $S$ be the set of numbers representable as a sum $r_{i_1}+r_{i_2}+\cdots+r_{i_{1994}}$, with $i_1< i_2<\cdots< i_{1994}$. Show that every nonempty interval $(a,b)$ contains a nonempty subinterval $(c,d)$ that does not intersect $S$.

Solution: None.

Lean 4 formal statement:
theorem putnam_1994_a5
  (r : ℕ → ℝ)
  (S : Set ℝ)
  (rpos : ∀ n, r n > 0)
  (rlim : Tendsto r atTop (𝓝 0))
  (hS : S = {x | ∃ i : Fin 1994 → ℕ, (∀ j k : Fin 1994, j < k → i j < i k) ∧ (x = ∑ j : Fin 1994, r (i j))}) :
  ∀ a b : ℝ, a < b → (∃ c d : ℝ, a ≤ c ∧ c < d ∧ d ≤ b ∧ (Set.Ioo c d) ∩ S = ∅) :=
sorry
Problem: putnam_1994_a6
Tags: algebra

Let $f_1,\dots,f_{10}$ be bijections of the set of integers such that for each integer $n$, there is some composition $f_{i_1} \circ f_{i_2} \circ \cdots \circ f_{i_m}$ of these functions (allowing repetitions) which maps 0 to $n$. Consider the set of $1024$ functions $\mathcal{F}=\{f_1^{e_1} \circ f_2^{e_2} \circ \cdots \circ f_{10}^{e_{10}}\}$, $e_i=0$ or $1$ for $1 \leq i \leq 10$. ($f_i^0$ is the identity function and $f_i^1=f_i$.) Show that if $A$ is any nonempty finite set of integers, then at most $512$ of the functions in $\mathcal{F}$ map $A$ to itself.

Solution: None.

Lean 4 formal statement:
theorem putnam_1994_a6
(f : Fin 10 → Equiv.Perm ℤ)
(mijcomp : ℕ → (ℕ → Fin 10) → ℕ → (ℤ → ℤ))
(F : Finset (ℤ → ℤ))
(hf: ∀ n : ℤ, ∃ m : ℕ, ∃ i : ℕ → Fin 10, m ≥ 1 ∧ (mijcomp m i 0) 0 = n)
(hmijcomp : ∀ m ≥ 1, ∀ (i : ℕ → Fin 10) (j : Fin m), mijcomp m i j = if (j = m - 1) then (f (i j) : ℤ → ℤ) else (f (i j) ∘ mijcomp m i (j + 1)))
(hF : F = {g : ℤ → ℤ | ∃ e : Fin 10 → Fin 2, g = (f 0)^[e 0] ∘ (f 1)^[e 1] ∘ (f 2)^[e 2] ∘ (f 3)^[e 3] ∘ (f 4)^[e 4] ∘ (f 5)^[e 5] ∘ (f 6)^[e 6] ∘ (f 7)^[e 7] ∘ (f 8)^[e 8] ∘ (f 9)^[e 9]})
: ∀ A : Finset ℤ, A.Nonempty → {g ∈ F | g '' A = A}.card ≤ 512 :=
sorry
Problem: putnam_1994_b1
Tags: algebra

Find all positive integers $n$ that are within $250$ of exactly $15$ perfect squares.

Solution: Show that an integer $n$ is within $250$ of exactly $15$ perfect squares if and only if either $315 \leq n \leq 325$ or $332 \leq n \leq 350$.

Lean 4 formal statement:
abbrev putnam_1994_b1_solution : Set ℤ := sorry
theorem putnam_1994_b1
    (n : ℤ) :
    n ∈ putnam_1994_b1_solution ↔
    (0 < n ∧ {m : ℕ | |n - m ^ 2| ≤ 250}.encard = 15) :=
  sorry
Problem: putnam_1994_b2
Tags: geometry, algebra

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?

Solution: Show that there exists such a line if and only if $c<243/8$.

Lean 4 formal statement:
abbrev putnam_1994_b2_solution : Set ℝ := sorry
theorem putnam_1994_b2
  (c : ℝ) :
  (∃ m b : ℝ,
    {x : ℝ | m * x + b = x ^ 4 + 9 * x ^ 3 + c * x ^ 2 + 9 * x + 4}.encard = 4)
  ↔ c ∈ putnam_1994_b2_solution :=
sorry
Problem: putnam_1994_b3
Tags: analysis

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 $x>N$.

Solution: Show that the desired set is $(-\infty,1)$.

Lean 4 formal statement:
abbrev putnam_1994_b3_solution : Set ℝ := sorry
theorem putnam_1994_b3 :
    {k | ∀ f (hf : (∀ x, 0 < f x ∧ f x < deriv f x) ∧ Differentiable ℝ f),
      ∃ N, ∀ x > N, Real.exp (k * x) < f x} = putnam_1994_b3_solution :=
  sorry
Problem: putnam_1994_b4
Tags: linear_algebra, number_theory, analysis

For $n \geq 1$, let $d_n$ be the greatest common divisor of the entries of $A^n-I$, where $A=\begin{pmatrix} 3 & 2 \\ 4 & 3 \end{pmatrix}$ and $I=\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$. Show that $\lim_{n \to \infty} d_n=\infty$.

Solution: None.

Lean 4 formal statement:
theorem putnam_1994_b4
(matgcd : Matrix (Fin 2) (Fin 2) ℤ → ℤ)
(A : Matrix (Fin 2) (Fin 2) ℤ)
(d : ℕ → ℤ)
(hmatgcd : ∀ M, matgcd M = Int.gcd (Int.gcd (Int.gcd (M 0 0) (M 0 1)) (M 1 0)) (M 1 1))
(hA : A 0 0 = 3 ∧ A 0 1 = 2 ∧ A 1 0 = 4 ∧ A 1 1 = 3)
(hd : ∀ n ≥ 1, d n = matgcd (A ^ n - 1))
: Tendsto d atTop atTop :=
sorry
Problem: putnam_1994_b5
Tags: algebra

For any real number $\alpha$, define the function $f_\alpha(x)=\lfloor \alpha x \rfloor$. Let $n$ be a positive integer. Show that there exists an $\alpha$ such that for $1 \leq k \leq n$, $f_\alpha^k(n^2)=n^2-k=f_{\alpha^k}(n^2)$.

Solution: None.

Lean 4 formal statement:
theorem putnam_1994_b5
  (n : ℕ) (hn : n > 0)
  (f : ℝ → ℤ → ℤ)
  (hf : ∀ (α : ℝ) (x : ℤ), f α x = Int.floor (α * x)) :
  ∃ α : ℝ,
    ∀ k ∈ Set.Icc 1 n,
      ((f α)^[k] (n ^ 2) = n ^ 2 - k) ∧
      (f (α ^ k) (n ^ 2) = n ^ 2 - k) :=
sorry
Problem: putnam_1994_b6
Tags: number_theory

For any integer $a$, set $n_a=101a-100 \cdot 2^a$. Show that for $0 \leq a,b,c,d \leq 99$, $n_a+n_b \equiv n_c+n_d \pmod{10100}$ implies $\{a,b\}=\{c,d\}$.

Solution: None.

Lean 4 formal statement:
theorem putnam_1994_b6
(n : ℕ → ℤ)
(hn : ∀ a : ℕ, n a = 101 * a - 100 * 2 ^ a)
: ∀ a b c d : Set.Icc 0 99, (n a + n b ≡ n c + n d [ZMOD 10100]) → (({a, b} : Set (Set.Icc 0 99)) = {c, d}) :=
sorry
Problem: putnam_1995_a1
Tags: algebra

Let $S$ be a set of real numbers which is closed under multiplication (that is, if $a$ and $b$ are in $S$, then so is $ab$). Let $T$ and $U$ be disjoint subsets of $S$ whose union is $S$. Given that the product of any {\em three} (not necessarily distinct) elements of $T$ is in $T$ and that the product of any three elements of $U$ is in $U$, show that at least one of the two subsets $T,U$ is closed under multiplication.

Solution: None.

Lean 4 formal statement:
theorem putnam_1995_a1
(S : Set ℝ)
(hS : ∀ a ∈ S, ∀ b ∈ S, a * b ∈ S)
(T U : Set ℝ)
(hsub : T ⊆ S ∧ U ⊆ S)
(hunion : T ∪ U = S)
(hdisj : T ∩ U = ∅)
(hT3 : ∀ a ∈ T, ∀ b ∈ T, ∀ c ∈ T, a * b * c ∈ T)
(hU3 : ∀ a ∈ U, ∀ b ∈ U, ∀ c ∈ U, a * b * c ∈ U)
: (∀ a ∈ T, ∀ b ∈ T, a * b ∈ T) ∨ (∀ a ∈ U, ∀ b ∈ U, a * b ∈ U) :=
sorry
Problem: putnam_1995_a2
Tags: analysis

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?

Solution: Show that the solution is those pairs $(a,b)$ where $a = b$.

Lean 4 formal statement:
abbrev putnam_1995_a2_solution : Set (ℝ × ℝ) := sorry
theorem putnam_1995_a2
(habconv : (ℝ × ℝ) → Prop)
(habconv_def : 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))
: ∀ ab : ℝ × ℝ, ab.1 > 0 ∧ ab.2 > 0 → (habconv ab ↔ ab ∈ putnam_1995_a2_solution) :=
sorry
Problem: putnam_1995_a3
Tags: number_theory

The number $d_{1}d_{2}\dots d_{9}$ has nine (not necessarily distinct) decimal digits. The number $e_{1}e_{2}\dots e_{9}$ is such that each of the nine 9-digit numbers formed by replacing just one of the digits $d_{i}$ is $d_{1}d_{2}\dots d_{9}$ by the corresponding digit $e_{i}$ ($1 \leq i \leq 9$) is divisible by 7. The number $f_{1}f_{2}\dots f_{9}$ is related to $e_{1}e_{2}\dots e_{9}$ is the same way: that is, each of the nine numbers formed by replacing one of the $e_{i}$ by the corresponding $f_{i}$ is divisible by 7. Show that, for each $i$, $d_{i}-f_{i}$ is divisible by 7. [For example, if $d_{1}d_{2}\dots d_{9} = 199501996$, then $e_{6}$ may be 2 or 9, since $199502996$ and $199509996$ are multiples of 7.]

Solution: None.

Lean 4 formal statement:
theorem putnam_1995_a3
(relation : (Fin 9 → ℤ) → (Fin 9 → ℤ) → Prop)
(digits_to_num : (Fin 9 → ℤ) → ℤ)
(hdigits_to_num : digits_to_num = fun dig => ∑ i : Fin 9, (dig i) * 10^i.1)
(hrelation : ∀ d e : (Fin 9 → ℤ), relation d e ↔ (∀ i : Fin 9, d i < 10 ∧ d i ≥ 0 ∧ e i < 10 ∧ e i ≥ 0) ∧ (∀ i : Fin 9, 7 ∣ (digits_to_num (fun j : Fin 9 => if j = i then e j else d j))))
: ∀ d e f : (Fin 9 → ℤ), ((relation d e) ∧ (relation e f)) → (∀ i : Fin 9, 7 ∣ d i - f i) :=
sorry
Problem: putnam_1995_a4
Tags: combinatorics

Suppose we have a necklace of $n$ beads. Each bead is labeled with an integer and the sum of all these labels is $n-1$. Prove that we can cut the necklace to form a string whose consecutive labels $x_{1},x\_{2},\dots,x_{n}$ satisfy \[\sum_{i=1}^{k} x_{i} \leq k-1 \qquad \mbox{for} \quad k=1,2,\dots,n.\]

Solution: None.

Lean 4 formal statement:
theorem putnam_1995_a4
(n : ℕ)
(hn : n > 0)
(necklace : Fin n → ℤ)
(hnecklacesum : ∑ i : Fin n, necklace i = n - 1)
: ∃ cut, ∀ k, ∑ i : {j : Fin n | j.1 ≤ k}, necklace (cut + i) ≤ k :=
sorry
Problem: putnam_1995_a5
Tags: linear_algebra, analysis

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?

Solution: Show that the answer is yes, the functions must be linearly dependent.

Lean 4 formal statement:
abbrev putnam_1995_a5_solution : Prop := sorry
theorem putnam_1995_a5 :
  putnam_1995_a5_solution ↔
  (∀ (n : ℕ) (x : Fin n → (ℝ → ℝ)) (a : Fin n → Fin n → ℝ),
    (0 < n) →
    (∀ i, Differentiable ℝ (x i)) →
    (∀ i j, a i j > 0) →
    (∀ t i, (deriv (x i)) t = ∑ j : Fin n, (a i j) * ((x j) t)) →
    (∀ i, Tendsto (x i) atTop (𝓝 0)) →
    ¬(∀ b : Fin n → ℝ, (∀ t : ℝ, ∑ i : Fin n, (b i) * ((x i) t) = 0) →
      (∀ i, b i = 0))) :=
  sorry
Problem: putnam_1995_a6
Tags: algebra

Suppose that each of $n$ people writes down the numbers $1,2,3$ in random order in one column of a $3 \times n$ matrix, with all orders equally likely and with the orders for different columns independent of each other. Let the row sums $a,b,c$ of the resulting matrix be rearranged (if necessary) so that $a \leq b \leq c$. Show that for some $n \geq 1995$, it is at least four times as likely that both $b=a+1$ and $c=a+2$ as that $a=b=c$.

Solution: None.

Lean 4 formal statement:
theorem putnam_1995_a6
(SM : (n : ℕ) → Set (Matrix (Fin 3) (Fin n) ℤ))
(Srowsums : (n : ℕ) → Matrix (Fin 3) (Fin n) ℤ → Set ℤ)
(sumsprog : (Set ℤ) → Prop)
(hSM : ∀ n, SM n = {M : Matrix (Fin 3) (Fin n) ℤ | ∀ j : Fin n, {M i j | i : Fin 3} = {1, 2, 3}})
(hSrowsums : ∀ n M, Srowsums n M = {∑ j : Fin n, M i j | i : Fin 3})
(hsumsprog : ∀ sums, sumsprog sums ↔ sums.encard = 3 ∧ (∃ a b c : ℤ, {a, b, c} = sums ∧ b = a + 1 ∧ c = a + 2))
: ∃ n ≥ 1995, {M : (SM n) | sumsprog (Srowsums n M)}.encard ≥ 4 * {M : (SM n) | (Srowsums n M).encard = 1}.encard :=
sorry
Problem: putnam_1995_b1
Tags: combinatorics

For a partition $\pi$ of $\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$, let $\pi(x)$ be the number of elements in the part containing $x$. Prove that for any two partitions $\pi$ and $\pi'$, there are two distinct numbers $x$ and $y$ in $\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$ such that $\pi(x) = \pi(y)$ and $\pi'(x) = \pi'(y)$. [A {\em partition} of a set $S$ is a collection of disjoint subsets (parts) whose union is $S$.]

Solution: None.

Lean 4 formal statement:
theorem putnam_1995_b1
  (Pi : Finpartition (Finset.range 9) → (Finset.range 9) → ℕ)
  (Pi_def : ∀ partition k, Pi partition k = (Exists.choose (Finpartition.exists_mem partition k.2)).card) :
  ∀ Pt1 Pt2,
    ∃ x y : Finset.range 9,
      x ≠ y ∧
      Pi Pt1 x = Pi Pt1 y ∧
      Pi Pt2 x = Pi Pt2 y :=
sorry
Problem: putnam_1995_b3
Tags: linear_algebra

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.)

Solution: Show that the solution is $45$ if $n = 1$, $45^2*10$ if $n = 2$, and $0$ if $n$ is greater than 2.

Lean 4 formal statement:
abbrev putnam_1995_b3_solution : ℕ → ℤ := sorry
theorem putnam_1995_b3
(n : ℕ)
(hn : n > 0)
(digits_set : Set (ℕ → ℕ))
(hdigits_set : digits_set = {f | f 0 ≠ 0 ∧ (∀ i : Fin (n ^ 2), f i ≤ 9) ∧ (∀ i ≥ n ^ 2, f i = 0)})
(digits_to_matrix : (ℕ → ℕ) → Matrix (Fin n) (Fin n) ℤ)
(hdigits_to_matrix : digits_to_matrix = fun f => (fun i j => f (i.1 * n + j.1)))
: ∑' f : digits_set, (digits_to_matrix f).det = putnam_1995_b3_solution n :=
sorry
Problem: putnam_1995_b4
Tags: algebra

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.

Solution: Show that the solution is $(3 + 1*\sqrt{5})/2.

Lean 4 formal statement:
abbrev putnam_1995_b4_solution : ℤ × ℤ × ℤ × ℤ := sorry
theorem putnam_1995_b4
    (contfrac : ℝ)
    (hcontfrac : contfrac = 2207 - 1 / contfrac)
    (hcontfrac' : 1 < contfrac) :
    let ⟨a, b, c, d⟩ := putnam_1995_b4_solution
    contfrac ^ ((1 : ℝ) / 8) = (a + b * sqrt c) / d :=
  sorry
Problem: putnam_1995_b6
Tags: algebra, number_theory

For a positive real number $\alpha$, define \[ S(\alpha) = \{ \lfloor n\alpha \rfloor : n = 1,2,3,\dots \}. \] Prove that $\{1,2,3,\dots\}$ cannot be expressed as the disjoint union of three sets $S(\alpha), S(\beta)$ and $S(\gamma)$. [As usual, $\lfloor x \rfloor$ is the greatest integer $\leq x$.]

Solution: None.

Lean 4 formal statement:
theorem putnam_1995_b6
  (S : ℝ → Set ℕ)
  (hS : S = fun (α : ℝ) => {x : ℕ | ∃ n : ℕ, n ≥ 1 ∧ x = floor (n * α)}) :
  ¬ ∃ α β γ,
    α > 0 ∧ β > 0 ∧ γ > 0 ∧
    (S α) ∩ (S β) = ∅ ∧ (S β) ∩ (S γ) = ∅ ∧ (S α) ∩ (S γ) = ∅ ∧
    Set.Ici 1 = (S α) ∪ (S β) ∪ (S γ) :=
sorry
Problem: putnam_1996_a2
Tags: geometry

Let $C_1$ and $C_2$ be circles whose centers are $10$ units apart, and whose radii are $1$ and $3$. Find, with proof, the locus of all points $M$ for which there exist points $X$ on $C_1$ and $Y$ on $C_2$ such that $M$ is the midpoint of the line segment $XY$.

Solution: Let $O_1$ and $O_2$ be the centers of $C_1$ and $C_2$, respectively. Then show that the desired locus is an annulus centered at the midpoint $O$ of $O_1O_2$, with inner radius $1$ and outer radius $2$.

Lean 4 formal statement:
abbrev putnam_1996_a2_solution : (EuclideanSpace ℝ (Fin 2)) → (EuclideanSpace ℝ (Fin 2)) → Set (EuclideanSpace ℝ (Fin 2)) := sorry
theorem putnam_1996_a2
(O1 O2 : EuclideanSpace ℝ (Fin 2))
(C1 C2 : Set (EuclideanSpace ℝ (Fin 2)))
(hC1 : C1 = sphere O1 1)
(hC2 : C2 = sphere O2 3)
(hO1O2 : dist O1 O2 = 10)
: {M : EuclideanSpace ℝ (Fin 2) | ∃ X Y, X ∈ C1 ∧ Y ∈ C2 ∧ M = midpoint ℝ X Y} = putnam_1996_a2_solution O1 O2 :=
sorry
Problem: putnam_1996_a3
Tags: combinatorics

Suppose that each of 20 students has made a choice of anywhere from 0 to 6 courses from a total of 6 courses offered. Prove or disprove: there are 5 students and 2 courses such that all 5 have chosen both courses or all 5 have chosen neither course.

Solution: Show that the solution is that the statement is false.

Lean 4 formal statement:
abbrev putnam_1996_a3_solution : Prop := sorry
theorem putnam_1996_a3 :
    (∀ choices : Fin 20 → Set (Fin 6),
      ∃ (students : Finset (Fin 20)) (courses : Finset (Fin 6)),
        students.card = 5 ∧
        courses.card = 2 ∧
        (↑courses ⊆ ⋂ s ∈ students, choices s ∨ ↑courses ⊆ ⋂ s ∈ students, (choices s)ᶜ))
    ↔ putnam_1996_a3_solution :=
  sorry
Problem: putnam_1996_a4
Tags: algebra

Let $S$ be a set of ordered triples $(a, b, c)$ of distinct elements of a finite set $A$. Suppose that \begin{enumerate} \item $(a,b,c) \in S$ if and only if $(b,c,a) \in S$; \item $(a,b,c) \in S$ if and only if $(c,b,a) \notin S$; \item $(a,b,c)$ and $(c,d,a)$ are both in $S$ if and only if $(b,c,d)$ and $(d,a,b)$ are both in $S$. \end{enumerate} Prove that there exists a one-to-one function $g$ from $A$ to $\R$ such that $g(a) < g(b) < g(c)$ implies $(a,b,c) \in S$.

Solution: None.

Lean 4 formal statement:
theorem putnam_1996_a4
(A : Type*)
[Finite A]
(S : Set (A × A × A))
(hSdistinct : ∀ a b c : A, ⟨a, b, c⟩ ∈ S → a ≠ b ∧ b ≠ c ∧ a ≠ c)
(hS1 : ∀ a b c : A, [a,b,c].Pairwise (· ≠ ·) → (⟨a, b, c⟩ ∈ S ↔ ⟨b, c, a⟩ ∈ S))
(hS2 : ∀ a b c : A, [a,b,c].Pairwise (· ≠ ·) → (⟨a, b, c⟩ ∈ S ↔ ⟨c, b, a⟩ ∉ S))
(hS3 : ∀ a b c d : A, [a,b,c,d].Pairwise (· ≠ ·) → ((⟨a, b, c⟩ ∈ S ∧ ⟨c, d, a⟩ ∈ S) ↔ (⟨b,c,d⟩ ∈ S ∧ ⟨d,a,b⟩ ∈ S)))
: ∃ g : A → ℝ, Injective g ∧ (∀ a b c : A, g a < g b ∧ g b < g c → ⟨a,b,c⟩ ∈ S) :=
sorry
Problem: putnam_1996_a5
Tags: number_theory

If $p$ is a prime number greater than 3 and $k = \lfloor 2p/3 \rfloor$, prove that the sum \[\binom p1 + \binom p2 + \cdots + \binom pk \] of binomial coefficients is divisible by $p^2$.

Solution: None.

Lean 4 formal statement:
theorem putnam_1996_a5
(p : ℕ)
(hpprime : Prime p)
(hpge3 : p > 3)
(k : ℕ)
(hk : k = Nat.floor (2*p/(3 : ℚ)))
: p^2 ∣ ∑ i ∈ Finset.Icc 1 k, Nat.choose p i :=
sorry
Problem: putnam_1996_a6
Tags: analysis, algebra

Let $c>0$ be a constant. Give a complete description, with proof, of the set of all continuous functions $f:\mathbb{R} \to \mathbb{R}$ such that $f(x)=f(x^2+c)$ for all $x \in \mathbb{R}$.

Solution: Show that if $c \leq 1/4$ then $f$ must be constant, and if $c>1/4$ then $f$ can be defined on $[0,c]$ as any continuous function with equal values on the endpoints, then extended to $x>c$ by the relation $f(x)=f(x^2+c)$, then extended further to $x<0$ by the relation $f(x)=f(-x)$.

Lean 4 formal statement:
abbrev putnam_1996_a6_solution : ℝ → Set (ℝ → ℝ) := sorry
theorem putnam_1996_a6
(c : ℝ)
(f : ℝ → ℝ)
(cgt0 : c > 0)
: (Continuous f ∧ ∀ x : ℝ, f x = f (x ^ 2 + c)) ↔ f ∈ putnam_1996_a6_solution c :=
sorry
Problem: putnam_1996_b1
Tags: algebra

Define a \emph{selfish} set to be a set which has its own cardinality (number of elements) as an element. Find, with proof, the number of subsets of $\{1,2,\ldots,n\}$ which are \emph{minimal} selfish sets, that is, selfish sets none of whose proper subsets is selfish.

Solution: Show that the number of subsets is $F_n$, the $n$th Fibonacci number.

Lean 4 formal statement:
abbrev putnam_1996_b1_solution : ℕ → ℕ := sorry
theorem putnam_1996_b1
(selfish : Finset ℕ → Prop)
(n : ℕ)
(hselfish : ∀ s : Finset ℕ, selfish s = (s.card ∈ s))
(npos : n ≥ 1)
: {s : Finset ℕ | (s : Set ℕ) ⊆ Set.Icc 1 n ∧ selfish s ∧ (∀ ss : Finset ℕ, ss ⊂ s → ¬selfish ss)}.encard = putnam_1996_b1_solution n :=
sorry
Problem: putnam_1996_b2
Tags: analysis

Show that for every positive integer $n$, $(\frac{2n-1}{e})^{\frac{2n-1}{2}}<1 \cdot 3 \cdot 5 \cdots (2n-1)<(\frac{2n+1}{e})^{\frac{2n+1}{2}}$.

Solution: None.

Lean 4 formal statement:
theorem putnam_1996_b2
(n : ℕ)
(prododd : ℝ)
(npos : n > 0)
(hprododd : prododd = ∏ i ∈ Finset.range (2 * n), if Odd i then i else 1)
: ((2 * n - 1 : ℝ) / Real.exp 1) ^ ((2 * n - 1 : ℝ) / 2) < prododd ∧ prododd < ((2 * n + 1 : ℝ) / Real.exp 1) ^ ((2 * n + 1 : ℝ) / 2) :=
sorry
Problem: putnam_1996_b3
Tags: algebra

Given that $\{x_1,x_2,\ldots,x_n\}=\{1,2,\ldots,n\}$, find, with proof, the largest possible value, as a function of $n$ (with $n \geq 2$), of $x_1x_2+x_2x_3+\cdots+x_{n-1}x_n+x_nx_1$.

Solution: Show that the maximum is $(2n^3+3n^2-11n+18)/6$.

Lean 4 formal statement:
abbrev putnam_1996_b3_solution : ℕ → ℕ := sorry
theorem putnam_1996_b3
  (n : ℕ) (hn : n ≥ 2) :
  IsGreatest
  {k | ∃ x : ℕ → ℤ,
    (x '' (Finset.range n) = Set.Icc (1 : ℤ) n) ∧
    ∑ i : Fin n, x i * x ((i + 1) % n) = k}
  (putnam_1996_b3_solution n) :=
sorry
Problem: putnam_1996_b4
Tags: linear_algebra

For any square matrix $A$, we can define $\sin A$ by the usual power series: $\sin A=\sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!}A^{2n+1}$. Prove or disprove: there exists a $2 \times 2$ matrix $A$ with real entries such that $\sin A=\begin{pmatrix} 1 & 1996 \\ 0 & 1 \end{pmatrix}$.

Solution: Show that there does not exist such a matrix $A$.

Lean 4 formal statement:
abbrev putnam_1996_b4_solution : Prop := sorry
theorem putnam_1996_b4
(matsin : Matrix (Fin 2) (Fin 2) ℝ → Matrix (Fin 2) (Fin 2) ℝ)
(mat1996 : Matrix (Fin 2) (Fin 2) ℝ)
(hmatsin : ∀ A, matsin A = ∑' n : ℕ, ((-(1 : ℝ)) ^ n / (2 * n + 1)!) • A ^ (2 * n + 1))
(hmat1996 : mat1996 0 0 = 1 ∧ mat1996 0 1 = 1996 ∧ mat1996 1 0 = 0 ∧ mat1996 1 1 = 1)
: (∃ A, matsin A = mat1996) ↔ putnam_1996_b4_solution :=
sorry