ChristianZ97/PutnamBench-lean4 huggingface dataset:
Problem: putnam_1972_b4
Tags: algebra

Let $n \geq 2$ be an integer. Show that there exists a polynomial $P(x,y,z)$ with integral coefficients such that $x \equiv P(x^n, x^{n+1}, x + x^{n+2})$.

Solution: None.

Lean 4 formal statement:
theorem putnam_1972_b4
(n : ℕ)
(hn : n > 1)
(vars : ℤ → ℤ → ℤ → (Fin 3 → ℤ))
(hvars : vars = fun a b c ↦ fun i ↦ ite (i = 0) a (ite (i = 1) b c))
: ∃ P : MvPolynomial (Fin 3) ℤ, ∀ x : ℤ, x = MvPolynomial.eval (vars (x^n) (x^(n+1)) (x + x^(n+2))) P :=
sorry
Problem: putnam_1972_b5
Tags: geometry

Let $ABCD$ be a skew (non-planar) quadrilateral. Prove that if $\angle ABC = \angle CDA$ and $\angle BCD = \angle DAB$, then $AB = CD$ and $AD = BC$.

Solution: None.

Lean 4 formal statement:
theorem putnam_1972_b5
(A B C D : EuclideanSpace ℝ (Fin 3))
(hnonplanar : ¬Coplanar ℝ {A, B, C, D})
(hangles : ∠ A B C = ∠ C D A ∧ ∠ B C D = ∠ D A B)
: dist A B = dist C D ∧ dist B C = dist D A :=
sorry
Problem: putnam_1972_b6
Tags: analysis

Let $n_1 < n_2 < \dots < n_k$ be a set of positive integers. Prove that the polynomail $1 + z^{n_1} + z^{n_2} + \dots + z^{n_k}$ has not roots inside the circle $|z| < (\frac{\sqrt{5}-1}{2}$.

Solution: None.

Lean 4 formal statement:
theorem putnam_1972_b6
(k : ℕ)
(hk : k ≥ 1)
(n : Fin k → ℤ)
(hn : ∀ i : Fin k, n i > 0)
(hn' : ∀ i j : Fin k, i < j → n i < n j)
(zpoly : ℂ → ℂ)
(hzpoly : zpoly = fun z => 1 + ∑ i : Fin k, z^(n i))
: ∀ z : ℂ, z ∈ ball 0 ((Real.sqrt 5 - 1)/2) → zpoly z ≠ 0 :=
sorry
Problem: putnam_1973_a1
Tags: geometry

Let $\triangle ABC$ be any triangle in the Euclidean plane, and let points $X$, $Y$, and $Z$ lie on sides $\overline{BC}$, $\overline{CA}$, and $\overline{AB}$ respectively. If $BX \le XC$, $CY \le YA$, and $AZ \le ZB$, prove that $[\triangle XYZ] \ge \frac{1}{4} [\triangle ABC]$.  Regardless of this constraint on $X$, $Y$, and $Z$, prove that one of $[\triangle AZY]$, $[\triangle BXZ]$, or $[\triangle CYX]$ is less than or equal to $[\triangle XYZ]$. (Here, $[\triangle]$ denotes the area of triangle $\triangle$.)

Solution: None.

Lean 4 formal statement:
theorem putnam_1973_a1
(A B C X Y Z : EuclideanSpace ℝ (Fin 2))
(hnoncol : ¬Collinear ℝ {A, B, C})
(hX : X ∈ segment ℝ B C)
(hY : Y ∈ segment ℝ C A)
(hZ : Z ∈ segment ℝ A B)
: ((dist B X ≤ dist X C ∧ dist C Y ≤ dist Y A ∧ dist A Z ≤ dist Z B) →
volume (convexHull ℝ {X, Y, Z}) ≥ (1/4) * volume (convexHull ℝ {A, B, C})) ∧
sInf {volume (convexHull ℝ {A, Z, Y}), volume (convexHull ℝ {B, X, Z}), volume (convexHull ℝ {C, Y, X})} ≤ volume (convexHull ℝ {X, Y, Z}) :=
sorry
Problem: putnam_1973_a2
Tags: analysis

Consider an infinite series whose $n$th term is given by $\pm \frac{1}{n}$, where the actual values of the $\pm$ signs repeat in blocks of $8$ (so the $\frac{1}{9}$ term has the same sign as the $\frac{1}{1}$ term, and so on). Call such a sequence balanced if each block contains four $+$ and four $-$ signs. Prove that being balanced is a sufficient condition for the sequence to converge. Is being balanced also necessary for the sequence to converge?

Solution: Show that the condition is necessary.

Lean 4 formal statement:
abbrev putnam_1973_a2_solution : Prop := sorry
theorem putnam_1973_a2
(L : List ℝ)
(hL : L.length = 8 ∧ ∀ i : Fin L.length, L[i] = 1 ∨ L[i] = -1)
(pluses : ℕ)
(hpluses : pluses = {i : Fin L.length | L[i] = 1}.ncard)
(S : ℕ → ℝ)
(hS : S = fun n : ℕ ↦ ∑ i ∈ Finset.Icc 1 n, L[i % 8]/i)
: (pluses = 4 → ∃ l : ℝ, Tendsto S atTop (𝓝 l)) ∧ (putnam_1973_a2_solution ↔ ((∃ l : ℝ, Tendsto S atTop (𝓝 l)) → pluses = 4)) :=
sorry
Problem: putnam_1973_a3
Tags: number_theory

Let $n$ be a fixed positive integer and let $b(n)$ be the minimum value of $k + \frac{n}{k}$ as $k$ is allowed to range through all positive integers. Prove that $b(n)$ and $\sqrt{4n + 1}$ have the same integer part.

Solution: None.

Lean 4 formal statement:
theorem putnam_1973_a3
(b : ℤ → ℝ)
(hb : b = fun (n : ℤ) => sInf {k + (n : ℝ)/(k : ℝ) | k > (0 : ℤ)})
: ∀ n : ℤ, n > 0 → floor (b n) = floor (Real.sqrt (4 * n + 1)) :=
sorry
Problem: putnam_1973_a4
Tags: analysis

How many zeros does the function $f(x) = 2^x - 1 - x^2$ have on the real line?

Solution: Show that the solution is 3.

Lean 4 formal statement:
abbrev putnam_1973_a4_solution : ℕ := sorry
theorem putnam_1973_a4
(f : ℝ → ℝ)
(hf : f = fun x => 2^x - 1 - x^2)
: putnam_1973_a4_solution = {x : ℝ | f x = 0}.ncard :=
sorry
Problem: putnam_1973_a6
Tags: geometry

Prove that it is impossible for seven distinct straight lines to be situated in the Euclidean plane so as to have at least six points where exactly three of these lines intersect and at least four points where exactly two of these lines interest.

Solution: None.

Lean 4 formal statement:
theorem putnam_1973_a6
(nint : ℕ → (Fin 7 → (ℝ × ℝ)) → ℕ)
(h_nint : nint = fun n lines => {p : ℝ × ℝ | ∃! S : Set (Fin 7), S.ncard = n ∧ p ∈ ⋂ i ∈ S, {pts | pts.2 = (lines i).1 * pts.1 + (lines i).2}}.ncard)
: ¬ ∃ lines : Fin 7 → (ℝ × ℝ), (∀ i j : Fin 7, i ≠ j → lines i ≠ lines j) ∧ nint 3 lines ≥ 6 ∧ nint 2 lines ≥ 4 :=
sorry
Problem: putnam_1973_b1
Tags: algebra

Let $a_1, \dots, a_{2n + 1}$ be a set of integers such that, if any one of them is removed, the remaining ones can be divided into two sets of $n$ integers with equal sums. Prove $a_1 = a_2 = \dots = a_{2n+1}$.

Solution: None.

Lean 4 formal statement:
theorem putnam_1973_b1
(n : ℕ)
(a : Finset.Icc 1 (2 * n + 1) → ℤ)
(h_remove : ∀ S : Finset (Finset.Icc 1 (2 * n + 1)), S.card = 2*n → ∃ T, T ⊆ S ∧ T.card = n ∧ ∑ i ∈ T, a i = ∑ i ∈ (S \ T), a i)
: ∀ i j : Finset.Icc 1 (2 * n + 1), a i = a j :=
sorry
Problem: putnam_1973_b2
Tags: number_theory

Let $z = x+iy$ be a complex number with $x$ and $y$ rational and with $\| z \| = 1$. Show thaat the number $\| z^{2n} - 1 \|$ is rational for every integer $n$.

Solution: None.

Lean 4 formal statement:
theorem putnam_1973_b2
(z : ℂ)
(hzrat : ∃ q1 q2 : ℚ, z.re = q1 ∧ z.im = q2)
(hznorm : ‖z‖ = 1)
: ∀ n : ℤ, ∃ q : ℚ, ‖z^(2*n) - 1‖ = q :=
sorry
Problem: putnam_1973_b3
Tags: number_theory, algebra

Let $p > 1$ be an integer with the property that $x^2 - x + p$ is prime for all $x$ in the range $0 < x < p$. Show there exists exactly one triple of integers $a,b,c$ satisfying $b^2 - 4ac = 1 - 4p$, $0 < a \leq c$, and $-a \leq b < a$.

Solution: None.

Lean 4 formal statement:
theorem putnam_1973_b3
(p : ℕ)
(pgt1 : p > 1)
(hprime : ∀ x ∈ Set.Ico 0 p, Nat.Prime (x^2 - x + p))
: ∃! triple : ℤ × ℤ × ℤ, let (a,b,c) := triple; b^2 - 4*a*c = 1 - 4*p ∧ 0 < a ∧ a ≤ c ∧ -a ≤ b ∧ b < a :=
sorry
Problem: putnam_1973_b4
Tags: analysis

Suppose $f$ is a function on $[0,1]$ with continuous derivative satisfying $0 < f'(x) \leq 1$ and $f 0 = 0$. Prove that $\left[\int_0^1 f(x) dx\right]]^2 \geq \int_0^1 (f(x))^3 dx$, and find an example where equality holds.

Solution: Show that one such example where equality holds is the identity function.

Lean 4 formal statement:
abbrev putnam_1973_b4_solution : ℝ → ℝ := sorry
theorem putnam_1973_b4
(f : ℝ → ℝ)
(hprop : (ℝ → ℝ) → Prop)
(hprop_def : hprop = fun g => ContDiff ℝ 1 g ∧ (∀ x : ℝ, 0 < deriv g x ∧ deriv g x ≤ 1) ∧ g 0 = 0)
(hf : hprop f)
: (∫ x in Icc 0 1, f x)^2 ≥ ∫ x in Icc 0 1, (f x)^3 ∧ (hprop putnam_1973_b4_solution ∧ (∫ x in Icc 0 1, putnam_1973_b4_solution x)^2 = ∫ x in Icc 0 1, (putnam_1973_b4_solution x)^3) :=
sorry
Problem: putnam_1974_a1
Tags: number_theory

Call a set of positive integers 'conspiratorial' if no three of them are pairwise relatively prime. What is the largest number of elements in any conspiratorial subset of the integers 1 through 16?

Solution: Show that the answer is 11.

Lean 4 formal statement:
abbrev putnam_1974_a1_solution : ℕ := sorry
theorem putnam_1974_a1
    (conspiratorial : Set ℤ → Prop)
    (hconspiratorial : ∀ S, conspiratorial S ↔ ∀ a ∈ S, ∀ b ∈ S, ∀ c ∈ S, (a > 0 ∧ b > 0 ∧ c > 0) ∧ ((a ≠ b ∧ b ≠ c ∧ a ≠ c) → (Int.gcd a b > 1 ∨ Int.gcd b c > 1 ∨ Int.gcd a c > 1))) :
    IsGreatest {k | ∃ S, S ⊆ Icc 1 16 ∧ conspiratorial S ∧ S.encard = k} putnam_1974_a1_solution :=
  sorry
Problem: putnam_1974_a3
Tags: number_theory

A well-known theorem asserts that a prime $p > 2$ can be written as the sum of two perfect squres if and only if $p \equiv 1 \bmod 4$. Find which primes $p > 2$ can be written in each of the following forms, using (not necessarily positive) integers $x$ and $y$: (a) $x^2 + 16y^2$, (b) $4x^2 + 4xy + 5y^2$.

Solution: Show that that the answer to (a) is the set of primes which are $1 \bmod 8$, and the solution to (b) is the set of primes which are $5 \bmod 8$.

Lean 4 formal statement:
abbrev putnam_1974_a3_solution : (Set ℕ) × (Set ℕ) := sorry
theorem putnam_1974_a3
(assmption : ∀ p : ℕ, p.Prime ∧ p > 2 → ((∃ m n : ℤ, p = m^2 + n^2) ↔ p ≡ 1 [MOD 4]))
: ∀ p : ℕ, ((p.Prime ∧ p > 2 ∧ (∃ x y : ℤ, p = x^2 + 16*y^2)) ↔ p ∈ putnam_1974_a3_solution.1) ∧ ((p.Prime ∧ p > 2 ∧ (∃ x y : ℤ, p = 4*x^2 + 4*x*y + 5*y^2)) ↔ p ∈ putnam_1974_a3_solution.2) :=
sorry
Problem: putnam_1974_a4
Tags: algebra

Evaluate in closed form: $\frac{1}{2^{n-1}} \sum_{k < n/2} (n-2k)*{n \choose k}$.

Solution: Show that the solution is $\frac{n}{2^{n-1}} * {(n-1) \choose \left[ (n-1)/2 \right]}$.

Lean 4 formal statement:
noncomputable abbrev putnam_1974_a4_solution : ℕ → ℚ := sorry
theorem putnam_1974_a4
    (n : ℕ)
    (hn : 0 < n) :
    (1 : ℚ) / (2 ^ (n - 1)) * ∑ k ∈ Finset.Icc 0 ⌊n / 2⌋₊, (n - 2 * k) * (n.choose k) =
    putnam_1974_a4_solution n :=
  sorry
Problem: putnam_1974_a6
Tags: algebra

Given $n$, let $k(n)$ be the minimal degree of any monic integral polynomial $f$ such that the value of $f(x)$ is divisible by $n$ for every integer $x$. Find the value of $k(1000000)$.

Solution: Show that the answer is 25.

Lean 4 formal statement:
abbrev putnam_1974_a6_solution : ℕ := sorry
theorem putnam_1974_a6
(hdivnallx : Polynomial ℤ → Prop)
(hdivnallx_def : hdivnallx = fun f => Monic f ∧ (∀ x : ℤ, (10^6 : ℤ) ∣ f.eval x))
: sInf {d : ℕ | ∃ f : Polynomial ℤ, hdivnallx f ∧ d = f.natDegree} = putnam_1974_a6_solution :=
sorry
Problem: putnam_1974_b1
Tags: algebra, geometry

Prove that the optimal configuration of 5 (not necessarily distinct) points $p_1, \dots, p_5$ on the unit circle which maximizes the sum of the ten distances \[\Sigma_{i < j}, d(p_i, p_j) \] is the one which evenly spaces the points like a regular pentagon.

Solution: None.

Lean 4 formal statement:
abbrev putnam_1974_b1_solution : (Fin 5 → EuclideanSpace ℝ (Fin 2)) → Prop := sorry
theorem putnam_1974_b1
    (d : (Fin 5 → EuclideanSpace ℝ (Fin 2)) → ℝ)
    (d_def : ∀ p, d p = ∑ ⟨i, j⟩ : Fin 5 × Fin 5, if i < j then dist (p i) (p j) else 0)
    (p : Fin 5 → EuclideanSpace ℝ (Fin 2))
    (hp : ∀ i, ‖p i‖ = 1) :
    d p = sSup {d q | (q) (hq : ∀ i, ‖q i‖ = 1)} ↔ putnam_1974_b1_solution p :=
  sorry
Problem: putnam_1974_b2
Tags: analysis

Let $y(x)$ be a continuously differentiable real-valued function of a real vairable $x$. Show that if $(y')^2 + y^3 \to 0$ as $x \to +\infty$, then $y(x)$ and $y'(x) \to 0$ as $x \to +\infty$.

Solution: None.

Lean 4 formal statement:
theorem putnam_1974_b2
(y : ℝ → ℝ)
(ycontdiff : ContDiff ℝ 1 y)
(limy : Tendsto (fun x : ℝ => (deriv y x)^2 + (y x)^3) atTop (𝓝 0))
: Tendsto y atTop (𝓝 0) ∧ Tendsto (deriv y) atTop (𝓝 0) :=
sorry
Problem: putnam_1974_b3
Tags: number_theory

Prove that if $\alpha$ is a real number such that $\cos (\pi \alpha) = 1/3$, the $\alpha$ is irrational.

Solution: None.

Lean 4 formal statement:
theorem putnam_1974_b3
(α : ℝ)
(ha : Real.cos (Real.pi * α) = (1 : ℝ)/3)
: Irrational α :=
sorry
Problem: putnam_1974_b4
Tags: analysis

Let $f : \mathbb{R} \to \mathbb{R}$ be continuous in each variable seperately. Show that there exists a sequence of continuous functions $g_n : \mathbb{R}^2 \to \mathbb{R}$ such that $f(x,y) = \lim_{n \to \infty} g_n(x,y)$ for all $(x,y) \in \mathbb{R}^2$.

Solution: None.

Lean 4 formal statement:
theorem putnam_1974_b4
(f : ℝ × ℝ → ℝ)
(hfcontx : ∀ y : ℝ, Continuous (fun x => f ⟨x, y⟩))
(hfconty : ∀ x : ℝ, Continuous (fun y => f ⟨x, y⟩))
: ∃ g : ℕ → (ℝ × ℝ → ℝ), (∀ n : ℕ, Continuous (g n)) ∧ (∀ x y : ℝ, Tendsto (fun n => g n ⟨x, y⟩) atTop (𝓝 (f ⟨x, y⟩))) :=
sorry
Problem: putnam_1974_b5
Tags: analysis

Show that $1 + (n/1!) + (n^2/2!) + \dots + (n^n/n!) > e^n/2$ for every integer $n \geq 0$.

Solution: None.

Lean 4 formal statement:
theorem putnam_1974_b5
: ∀ n ≥ 0, ∑ i ∈ Finset.Icc (0 : ℕ) n, (n^i : ℝ)/(Nat.factorial i) > (Real.exp n)/2 :=
sorry
Problem: putnam_1974_b6
Tags: set_theory

For a set with $1000$ elements, how many subsets are there whose candinality is respectively $\equiv 0 \bmod 3, \equiv 1 \bmod 3, \equiv 2 \bmod 3$?

Solution: Show that there answer is that there are $(2^1000-1)/3$ subsets of cardinality $\equiv 0 \bmod 3$ and $\equiv 1 \bmod 3$, and $1 + (2^1000-1)/3$ subsets of cardinality $\equiv 2 \bmod 3$.

Lean 4 formal statement:
abbrev putnam_1974_b6_solution : (ℕ × ℕ × ℕ) := sorry
theorem putnam_1974_b6
(n : ℤ)
(hn : n = 1000)
(count0 count1 count2 : ℕ)
(hcount0 : count0 = {S | S ⊆ Finset.Icc 1 n ∧ S.card ≡ 0 [MOD 3]}.ncard)
(hcount1 : count1 = {S | S ⊆ Finset.Icc 1 n ∧ S.card ≡ 1 [MOD 3]}.ncard)
(hcount2 : count2 = {S | S ⊆ Finset.Icc 1 n ∧ S.card ≡ 2 [MOD 3]}.ncard)
: (count0, count1, count2) = putnam_1974_b6_solution :=
sorry
Problem: putnam_1975_a1
Tags: algebra, number_theory

If an integer $n$ can be written as the sum of two triangular numbers (that is, $n = \frac{a^2 + a}{2} + \frac{b^2 + b}{2}$ for some integers $a$ and $b$), express $4n + 1$ as the sum of the squares of two integers $x$ and $y$, giving $x$ and $y$ in terms of $a$ and $b$. Also, show that if $4n + 1 = x^2 + y^2$ for some integers $x$ and $y$, then $n$ can be written as the sum of two triangular numbers.

Solution: $x = a + b + 1$ and $y = a - b$ (or vice versa).

Lean 4 formal statement:
abbrev putnam_1975_a1_solution : ((ℤ × ℤ) → ℤ) × ((ℤ × ℤ) → ℤ) := sorry
theorem putnam_1975_a1
(nab nxy : (ℤ × ℤ × ℤ) → Prop)
(hnab : nab = fun (n, a, b) => n = (a^2 + (a : ℚ))/2 + (b^2 + (b : ℚ))/2)
(hnxy : nxy = fun (n, x, y) => 4*n + 1 = x^2 + y^2)
: (∀ n a b : ℤ, nab (n, a, b) → nxy (n, putnam_1975_a1_solution.1 (a, b), putnam_1975_a1_solution.2 (a, b))) ∧
∀ n : ℤ, (∃ x y : ℤ, nxy (n, x, y)) → ∃ a b : ℤ, nab (n, a, b) :=
sorry
Problem: putnam_1975_a2
Tags: algebra

For which ordered pairs $(b, c)$ of real numbers do both roots of $z^2 + bz + c$ lie strictly inside the unit disk (i.e., $\{|z| < 1\}$) in the complex plane?

Solution: The desired region is the strict interior of the triangle with vertices $(0, -1)$, $(2, 1)$, and $(-2, 1)$.

Lean 4 formal statement:
abbrev putnam_1975_a2_solution : (ℝ × ℝ) → Prop := sorry
theorem putnam_1975_a2
: ∀ b c : ℝ, (∀ z : ℂ, (X^2 + (C (b : ℂ))*X + (C (c : ℂ)) : Polynomial ℂ).eval z = 0 → ‖z‖ < 1) ↔ putnam_1975_a2_solution (b, c) :=
sorry
Problem: putnam_1975_a3
Tags: algebra

If $a$, $b$, and $c$ are real numbers satisfying $0 < a < b < c$, at what points in the set $$\{(x, y, z) \in \mathbb{R}^3 : x^b + y^b + z^b = 1, x \ge 0, y \ge 0, z \ge 0\}$$ does $f(x, y, z) = x^a + y^b + z^c$ attain its maximum and minimum?

Solution: $f$ attains its maximum at $\left(x_0, (1 - x_0^b)^{\frac{1}{b}}, 0\right)$ and its minimum at $\left(0, (1 - z_0^b)^{\frac{1}{b}}, z_0\right)$, where $x_0 = \left(\frac{a}{b}\right)^{\frac{1}{b-a}}$ and $z_0 = \left(\frac{b}{c}\right)^{\frac{1}{c-b}}$.

Lean 4 formal statement:
noncomputable abbrev putnam_1975_a3_solution : ((ℝ × ℝ × ℝ) → (ℝ × ℝ × ℝ)) × ((ℝ × ℝ × ℝ) → (ℝ × ℝ × ℝ)) := sorry
theorem putnam_1975_a3
(a b c : ℝ)
(hi : 0 < a ∧ a < b ∧ b < c)
(P : (ℝ × ℝ × ℝ) → Prop)
(f : (ℝ × ℝ × ℝ) → ℝ)
(hP : P = fun (x, y, z) => x ≥ 0 ∧ y ≥ 0 ∧ z ≥ 0 ∧ x^b + y^b + z^b = 1)
(hf : f = fun (x, y, z) => x^a + y^b + z^c)
: (P (putnam_1975_a3_solution.1 (a, b, c)) ∧ ∀ x y z : ℝ, P (x, y, z) →
f (x, y, z) ≤ f (putnam_1975_a3_solution.1 (a, b, c))) ∧
(P (putnam_1975_a3_solution.2 (a, b, c)) ∧ ∀ x y z : ℝ, P (x, y, z) →
f (x, y, z) ≥ f (putnam_1975_a3_solution.2 (a, b, c))) :=
sorry
Problem: putnam_1975_a4
Tags: algebra

Let $n = 2m$, where $m$ is an odd integer greater than 1. Let $\theta = e^{2\pi i/n}$. Expression $(1 - \theta)^{-1}$ explicitly as a polynomial in $\theta$ \[ a_k \theta^k  + a_{k-1}\theta^{k-1} + \dots + a_1\theta + a_0\], with integer coefficients $a_i$.

Solution: Show that the solution is the polynomial $0 + \theta + \theta^3 + \dots + \theta^{m-2}$, alternating consecutive coefficients between 0 and 1.

Lean 4 formal statement:
noncomputable abbrev putnam_1975_a4_solution : ℕ → Polynomial ℤ := sorry
theorem putnam_1975_a4
(m : ℕ)
(hm : Odd m ∧ m > 1)
(θ : ℂ)
(hθ : θ = cexp (2 * Real.pi * I / (2 * m)))
: 1/(1 - θ) = Polynomial.aeval θ (putnam_1975_a4_solution m) :=
sorry
Problem: putnam_1975_a5
Tags: analysis

On some interval $I$ of the real line, let $y_1(x), y_2(x)$ be linearly independent solutions of the differential equation \[y'' = f(x)y\], where $f(x)$ is a continuous real-valued function. Suppose that $y_1(x) > 0$ and $y_2(x) > 0$ on $I$. Show that there exists a positive constant $c$ such that, on $I$, the function \[z(x) = c \sqrt{y_1(x)y_2(x)}\] satisfies the equation \[z'' + 1/z^3 = f(x)z.\]

Solution: None.

Lean 4 formal statement:
theorem putnam_1975_a5
(I : Interval ℝ)
(f y1 y2 : ℝ → ℝ)
(hfcont : Continuous f)
(hycdiff : ContDiff ℝ 2 y1 ∧ ContDiff ℝ 2 y2)
(hy1sol : ∀ x : ℝ, iteratedDeriv 2 y1 x = (f x) * (y1 x))
(hy2sol : ∀ x : ℝ, iteratedDeriv 2 y2 x = (f x) * (y2 x))
(hylinindep : ∀ c1 c2 : ℝ, (∀ x : ℝ, c1 * y1 x + c2 * y2 x = 0) → (c1 = 0 ∧ c2 = 0))
(hyI : ∀ x ∈ I, y1 x > 0 ∧ y2 x > 0)
: ∃ c : ℝ, c > 0 ∧ (let z : ℝ → ℝ := fun x => c * Real.sqrt ((y1 x) * (y2 x)); ∀ x ∈ I, iteratedDeriv 2 z x + 1/(z x)^3 = (f x) * (z x)) :=
sorry
Problem: putnam_1975_b1
Tags: abstract_algebra, number_theory

Let $H$ be a subgroup of the additive group of ordered pairs of integers under componentwise addition. If $H$ is generated by the elements $(3, 8)$, $(4, -1)$, and $(5, 4)$, then $H$ is also generated by two elements $(1, b)$ and $(0, a)$ for some integer $b$ and positive integer $a$. Find $a$.

Solution: $a$ must equal $7$.

Lean 4 formal statement:
abbrev putnam_1975_b1_solution : ℤ := sorry
theorem putnam_1975_b1
(H : Set (ℤ × ℤ))
(hH : H = {(x, y) : (ℤ × ℤ) | ∃ u v w : ℤ, (x, y) = (u*3 + v*4 + w*5, u*8 + v*(-1) + w*4)})
: (∃ b : ℤ, H = {(x, y) : (ℤ × ℤ) | ∃ u v : ℤ, (x, y) = (u, u*b + v*putnam_1975_b1_solution)}) ∧ putnam_1975_b1_solution > 0 :=
sorry
Problem: putnam_1975_b2
Tags: analysis, geometry

In three-dimensional Euclidean space, define a \emph{slab} to be the open set of points lying between two parallel planes. The distance between the planes is called the \emph{thickness} of the slab. Given an infinite sequence $S_1, S_2, \dots$ of slabs of thicknesses $d_1, d_2, \dots,$ respectively, such that $\Sigma_{i=1}^{\infty} d_i$ converges, prove that there is some point in the space which is not contained in any of the slabs.

Solution: None.

Lean 4 formal statement:
theorem putnam_1975_b2
(slab : (Fin 3 → ℝ) → ℝ → ℝ → Set (Fin 3 → ℝ))
(hslab : slab = fun normal offset thickness => {x : Fin 3 → ℝ | offset < normal ⬝ᵥ x ∧ normal ⬝ᵥ x < offset + thickness})
(normals : ℕ → (Fin 3 → ℝ))
(offsets : ℕ → ℝ)
(thicknesses : ℕ → ℝ)
(hnormalsunit : ∀ i : ℕ, ‖normals i‖ = 1)
(hthicknessespos : ∀ i : ℕ, thicknesses i > 0)
(hthicknessesconv : ∃ C : ℝ, Tendsto (fun i : ℕ => ∑ j ∈ Finset.range i, thicknesses j) atTop (𝓝 C))
: Set.univ ≠ ⋃ i : ℕ, slab (normals i) (offsets i) (thicknesses i) :=
sorry
Problem: putnam_1975_b3
Tags: analysis, algebra

Let $s_k (a_1, a_2, \dots, a_n)$ denote the $k$-th elementary symmetric function; that is, the sum of all $k$-fold products of the $a_i$. For example, $s_1 (a_1, \dots, a_n) = \sum_{i=1}^{n} a_i$, and $s_2 (a_1, a_2, a_3) = a_1a_2 + a_2a_3 + a_1a_3$. Find the supremum $M_k$ (which is never attained) of $$\frac{s_k (a_1, a_2, \dots, a_n)}{(s_1 (a_1, a_2, \dots, a_n))^k}$$ across all $n$-tuples $(a_1, a_2, \dots, a_n)$ of positive real numbers with $n \ge k$.

Solution: The supremum $M_k$ is $\frac{1}{k!}$.

Lean 4 formal statement:
noncomputable abbrev putnam_1975_b3_solution : ℕ → ℝ := sorry
theorem putnam_1975_b3
: ∀ k : ℕ, k > 0 → (∀ a : Multiset ℝ, (∀ i ∈ a, i > 0) ∧ card a ≥ k →
(esymm a k)/(esymm a 1)^k ≤ putnam_1975_b3_solution k) ∧
∀ M : ℝ, M < putnam_1975_b3_solution k → (∃ a : Multiset ℝ, (∀ i ∈ a, i > 0) ∧ card a ≥ k ∧
(esymm a k)/(esymm a 1)^k > M) :=
sorry
Problem: putnam_1975_b4
Tags: analysis

Let $C = \{(x, y) \in \mathbb{R}^2 : x^2 + y^2 = 1\}$ denote the unit circle. Does there exist $B \subseteq C$ for which $B$ is topologically closed and contains exactly one point from each pair of diametrically opposite points in $C$?

Solution: Such $B$ does not exist.

Lean 4 formal statement:
abbrev putnam_1975_b4_solution : Prop := sorry
theorem putnam_1975_b4
(P : ℝ × ℝ → Prop)
(hP : P = fun (x, y) => x^2 + y^2 = 1)
: (∃ B ⊆ setOf P, IsClosed B ∧ ∀ x y : ℝ, P (x, y) → Xor' ((x, y) ∈ B) ((-x, -y) ∈ B)) ↔ putnam_1975_b4_solution :=
sorry
Problem: putnam_1975_b5
Tags: analysis, algebra

Let $f_0(x) = e^x$ and $f_{n+1}(x) = xf'_n(x)$ for all $n \ge 0$. Prove that $$\sum_{n=0}^{\infty} \frac{f_n(1)}{n!} = e^e.$$

Solution: None.

Lean 4 formal statement:
theorem putnam_1975_b5
(e : ℝ)
(he : e = Real.exp 1)
(f : ℕ → ℝ → ℝ)
(h0 : ∀ x : ℝ, f 0 x = e^x)
(hf : ∀ n : ℕ, ∀ x : ℝ, f (n + 1) x = x * (deriv (f n) x))
: ∑' n : ℕ, (f n 1)/(Nat.factorial n) = e^e :=
sorry
Problem: putnam_1975_b6
Tags: analysis

Show that if $s_n = 1 + \frac{1}{2} + \frac{1}{3} + \dots + 1/n, then $n(n+1)^{1/n} < n + s_n$ whenever $n > 1$ and $(n-1)n^{-1/(n-1)} < n - s_n$ whenever $n > 2$.

Solution: None.

Lean 4 formal statement:
theorem putnam_1975_b6
(s : ℕ → ℝ)
(hs : s = fun (n : ℕ) => ∑ i ∈ Finset.Icc 1 n, 1/(i : ℝ))
: (∀ n : ℕ, n > 1 → n * (n+1 : ℝ)^(1/(n : ℝ)) < n + s n) ∧ (∀ n : ℕ, n > 2 → ((n : ℝ) - 1)*((n : ℝ)^(-1/(n-1 : ℝ))) < n - s n) :=
sorry
Problem: putnam_1976_a2
Tags: algebra

Let $P(x, y) = x^2y + xy^2$, $Q(x, y) = x^2 + xy + y^2$, $F_n(x, y) = (x + y)^n - x^n - y^n$, and $G_n(x, y) = (x + y)^n + x^n + y^n$. Prove that for all positive integers $n$, either $F_n$ or $G_n$ can be represented as a polynomial in $P$ and $Q$ with integer coefficients.

Solution: None.

Lean 4 formal statement:
theorem putnam_1976_a2
    (P Q : MvPolynomial (Fin 2) ℤ)
    (hP : P = X 0 ^ 2 * X 1 + X 0 * X 1 ^ 2)
    (hQ : Q = X 0 ^ 2 + X 0 * X 1 + X 1 ^ 2)
    (F G : ℕ → MvPolynomial (Fin 2) ℤ)
    (hF : ∀ n, F n = (X 0 + X 1) ^ n - X 0 ^ n - X 1 ^ n)
    (hG : ∀ n, G n = (X 0 + X 1) ^ n + X 0 ^ n + X 1 ^ n)
    (n : ℕ)
    (hn : 0 < n) :
    ∃ A : MvPolynomial (Fin 2) ℤ,
      F n = aeval ![P, Q] A ∨
      G n = aeval ![P, Q] A :=
  sorry
Problem: putnam_1976_a3
Tags: number_theory

Find all integer solutions $(p, r, q, s)$ of the equation $|p^r - q^s| = 1$, where $p$ and $q$ are prime and $r$ and $s$ are greater than $1$.

Solution: The only solutions are $(p, r, q, s) = (3, 2, 2, 3)$ and $(p, r, q, s) = (2, 3, 3, 2)$.

Lean 4 formal statement:
abbrev putnam_1976_a3_solution : Set (ℕ × ℕ × ℕ × ℕ) := sorry
theorem putnam_1976_a3
: {(p, r, q, s) : ℕ × ℕ × ℕ × ℕ | Nat.Prime p ∧ Nat.Prime q ∧ r > 1 ∧ s > 1 ∧ |(p^r : ℤ) - q^s| = 1} = putnam_1976_a3_solution :=
sorry
Problem: putnam_1976_a4
Tags: algebra

Let $r$ be a real root of $P(x) = x^3 + ax^2 + bx - 1$, where $a$ and $b$ are integers and $P$ is irreducible over the rationals. Suppose that $r + 1$ is a root of $x^3 + cx^2 + dx + 1$, where $c$ and $d$ are also integers. Express another root $s$ of $P$ as a function of $r$ that does not depend on the values of $a$, $b$, $c$, or $d$.

Solution: The possible answers are $s = -\frac{1}{r + 1}$ and $s = -\frac{r + 1}{r}$.

Lean 4 formal statement:
noncomputable abbrev putnam_1976_a4_solution : (ℝ → ℝ) × (ℝ → ℝ) := sorry
theorem putnam_1976_a4
(a b c d : ℤ)
(r : ℝ)
(P Q: Polynomial ℚ)
(hP : P = X^3 + (C (a : ℚ))*X^2 + (C (b : ℚ))*X - (C 1) ∧ aeval r P = 0 ∧ Irreducible P)
(hQ : Q =  X^3 + (C (c : ℚ))*X^2 + (C (d : ℚ))*X + (C 1) ∧ aeval (r + 1) Q = 0)
: ∃ s : ℝ, aeval s P = 0 ∧ (s = putnam_1976_a4_solution.1 r ∨ s = putnam_1976_a4_solution.2 r) :=
sorry