ChristianZ97/putnambench-satp huggingface dataset:
Problem putnam_1972_a3 (analysis)

We call a function $f$ from $[0,1]$ to the reals to be supercontinuous on $[0,1]$ if the Cesaro-limit exists for the sequence $f(x_1), f(x_2), f(x_3), \dots$ whenever it does for the sequence $x_1, x_2, x_3 \dots$. Find all supercontinuous functions on $[0,1]$.

Solution: Show that the solution is the set of affine functions.

Lean 4 formalization:
import Mathlib
open EuclideanGeometry Filter Topology Set
abbrev putnam_1972_a3_solution : Set (ℝ → ℝ) := sorry
theorem putnam_1972_a3
    (climit_exists : (ℕ → ℝ) → Prop)
    (supercontinuous : (ℝ → ℝ) → Prop)
    (hclimit_exists : ∀ x, climit_exists x ↔ ∃ C : ℝ, Tendsto (fun n => (∑ i ∈ Finset.range n, (x i))/(n : ℝ)) atTop (𝓝 C))
    (hsupercontinuous : ∀ f, supercontinuous f ↔ ∀ (x : ℕ → ℝ), (∀ i : ℕ, x i ∈ Icc 0 1) → climit_exists x → climit_exists (fun i => f (x i))) :
    {f | supercontinuous f} = putnam_1972_a3_solution := by
Problem putnam_1972_a5 (number_theory)

Show that if $n$ is an integer greater than $1$, then $n$ does not divide $2^n - 1$.

Solution: None.

Lean 4 formalization:
import Mathlib
open EuclideanGeometry Filter Topology Set
theorem putnam_1972_a5
(n : ℕ)
(hn : n > 1)
: ¬((n : ℤ) ∣ 2^n - 1) := by
Problem putnam_1972_a6 (analysis)

Let $f$ be an integrable function in $0 \leq x \leq 1$ and suppose for all $0 \leq i \leq n-1, \int_0^1 x^i f(x) dx = 0$. Further suppose that $\int_0^1 x^n f(x) dx = 1$. Show that $|f(x)| \geq 2^n(n+1)$ on a set of positive measure.

Solution: None.

Lean 4 formalization:
import Mathlib
open EuclideanGeometry Filter Topology Set MeasureTheory
theorem putnam_1972_a6
(f : ℝ → ℝ)
(n : ℤ)
(hn : n ≥ 0)
(hfintegrable: IntegrableOn f (Icc 0 1))
(hfint : ∀ i ∈ Icc 0 (n-1), ∫ x in Icc 0 1, x^i*(f x) = 0)
(hfintlast : ∫ x in Icc 0 1, x^n*(f x) = 1)
: ∃ S, S ⊆ Icc (0 : ℝ) 1 ∧ MeasurableSet S ∧ volume S > 0 ∧ ∀ x ∈ S, |f x| ≥ 2^n * (n + 1) := by
Problem putnam_1972_b1 (analysis)

Prove that no three consecutive coefficients of the power series of $$\sum_{n = 0}^{\infty} \frac{x^n(x - 1)^{2n}}{n!}$$ all equal $0$.

Solution: None.

Lean 4 formalization:
import Mathlib
open EuclideanGeometry Filter Topology Set MeasureTheory Metric
theorem putnam_1972_b1
(S : ℝ → ℝ)
(hS : S = fun (x : ℝ) => ∑' n : ℕ, x^n * (x - 1)^(2*n) / (Nat.factorial n))
(p : ℕ → ℝ)
(hp : ∃ a : ℝ, a > 0 ∧ ∀ x ∈ ball 0 a, ∑' n : ℕ, (p n)*x^n = S x)
: ¬∃ n : ℕ, p n = 0 ∧ p (n + 1) = 0 ∧ p (n + 2) = 0 := by
Problem putnam_1972_b2 (analysis)

Let $x : \mathbb{R} \to \mathbb{R}$ be a twice differentiable function whose second derivative is nonstrictly decreasing. If $x(t) - x(0) = s$, $x'(0) = 0$, and $x'(t) = v$ for some $t > 0$, find the maximum possible value of $t$ in terms of $s$ and $v$.

Solution: Show that the maximum possible time is $t = \frac{2s}{v}$.

Lean 4 formalization:
import Mathlib
open EuclideanGeometry Filter Topology Set MeasureTheory Metric
noncomputable abbrev putnam_1972_b2_solution : ℝ → ℝ → ℝ := sorry
theorem putnam_1972_b2
    (s v : ℝ)
    (hs : s > 0)
    (hv : v > 0)
    (valid : ℝ → (ℝ → ℝ) → Prop)
    (hvalid : ∀ t x, valid t x ↔
      DifferentiableOn ℝ x (Set.Icc 0 t) ∧ DifferentiableOn ℝ (deriv x) (Set.Icc 0 t) ∧
      AntitoneOn (deriv (deriv x)) (Set.Icc 0 t) ∧
      deriv x 0 = 0 ∧ deriv x t = v ∧ x t - x 0 = s)
    : IsGreatest {t | ∃ x : ℝ → ℝ, valid t x} (putnam_1972_b2_solution s v) := by
Problem putnam_1972_b3 (abstract_algebra)

Let $A$ and $B$ be two elements in a group such that $ABA = BA^2B$, $A^3 = 1$, and $B^{2n-1} = 1$ for some positive integer $n$. Prove that $B = 1$.

Solution: None.

Lean 4 formalization:
import Mathlib
open EuclideanGeometry Filter Topology Set MeasureTheory Metric
theorem putnam_1972_b3
(G : Type*) [Group G]
(A B : G)
(hab : A * B * A = B * A^2 * B ∧ A^3 = 1 ∧ (∃ n : ℤ, n > 0 ∧ B^(2*n - 1) = 1))
: B = 1 := by
Problem putnam_1972_b4 (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 formalization:
import Mathlib
open EuclideanGeometry Filter Topology Set MeasureTheory Metric
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 := by
Problem putnam_1972_b5 (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 formalization:
import Mathlib
open EuclideanGeometry Set Metric
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 := by
Problem putnam_1972_b6 (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 formalization:
import Mathlib
open EuclideanGeometry Filter Topology Set MeasureTheory Metric
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 := by
Problem putnam_1973_a1 (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 formalization:
import Mathlib
open Nat Set MeasureTheory Topology Filter
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}) := by
Problem putnam_1973_a2 (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 formalization:
import Mathlib
open Nat Set MeasureTheory Topology Filter
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)) := by
Problem putnam_1973_a3 (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 formalization:
import Mathlib
open Nat Set MeasureTheory Topology Filter
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)) := by
Problem putnam_1973_a4 (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 formalization:
import Mathlib
open Nat Set MeasureTheory Topology Filter
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 := by
Problem putnam_1973_a6 (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 formalization:
import Mathlib
open Nat Set MeasureTheory Topology Filter
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 := by
Problem putnam_1973_b1 (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 formalization:
import Mathlib
open Nat Set MeasureTheory Topology Filter
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 := by
Problem putnam_1973_b2 (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 formalization:
import Mathlib
open Nat Set MeasureTheory Topology Filter
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 := by
Problem putnam_1973_b3 (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 formalization:
import Mathlib
open Nat Set MeasureTheory Topology Filter
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 := by
Problem putnam_1973_b4 (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 formalization:
import Mathlib
open Nat Set MeasureTheory Topology Filter
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) := by
Problem putnam_1974_a1 (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 formalization:
import Mathlib
open Set
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 := by
Problem putnam_1974_a3 (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 formalization:
import Mathlib
open Set
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) := by
Problem putnam_1974_a4 (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 formalization:
import Mathlib
open Set Nat
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 := by
Problem putnam_1974_a6 (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 formalization:
import Mathlib
open Set Nat Polynomial
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 := by
Problem putnam_1974_b1 (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 formalization:
import Mathlib
open Set Nat Polynomial
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 := by
Problem putnam_1974_b2 (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 formalization:
import Mathlib
open Set Nat Polynomial Filter Topology
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) := by
Problem putnam_1974_b3 (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 formalization:
import Mathlib
open Set Nat Polynomial Filter Topology
theorem putnam_1974_b3
(α : ℝ)
(ha : Real.cos (Real.pi * α) = (1 : ℝ)/3)
: Irrational α := by
Problem putnam_1974_b4 (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 formalization:
import Mathlib
open Set Nat Polynomial Filter Topology
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⟩))) := by
Problem putnam_1974_b5 (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 formalization:
import Mathlib
open Set Nat Polynomial Filter Topology
theorem putnam_1974_b5
: ∀ n ≥ 0, ∑ i ∈ Finset.Icc (0 : ℕ) n, (n^i : ℝ)/(Nat.factorial i) > (Real.exp n)/2 := by
Problem putnam_1974_b6 (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 formalization:
import Mathlib
open Set Nat Polynomial Filter Topology
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 := by
Problem putnam_1975_a1 (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 formalization:
import Mathlib
open Polynomial
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) := by
Problem putnam_1975_a2 (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 formalization:
import Mathlib
open Polynomial
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) := by
Problem putnam_1975_a3 (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 formalization:
import Mathlib
open Polynomial
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))) := by
Problem putnam_1975_a4 (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 formalization:
import Mathlib
open Polynomial Real Complex
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) := by
Problem putnam_1975_a5 (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 formalization:
import Mathlib
open Polynomial Real Complex
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)) := by
Problem putnam_1975_b1 (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 formalization:
import Mathlib
open Polynomial Real Complex
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 := by
Problem putnam_1975_b2 (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 formalization:
import Mathlib
open Polynomial Real Complex Matrix Filter Topology
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) := by