ChristianZ97/PutnamBench-lean4 huggingface dataset:
Problem: putnam_1968_b2
Tags: abstract_algebra

Let $G$ be a finite group (with a multiplicative operation), and $A$ be a subset of $G$ that contains more than half of $G$'s elements. Prove that every element of $G$ can be expressed as the product of two elements of $A$.

Solution: None.

Lean 4 formal statement:
theorem putnam_1968_b2
{G : Type*}
[Group G]
(hG : Finite G)
(A : Set G)
(hA : A.ncard > (Nat.card G : ℚ)/2)
: ∀ g : G, ∃ x ∈ A, ∃ y ∈ A, g = x * y :=
sorry
Problem: putnam_1968_b4
Tags: analysis

Suppose that $f : \mathbb{R} \to \mathbb{R}$ is continuous on $(-\infty, \infty)$ and that $\int_{-\infty}^{\infty} f(x) dx$ exists. Prove that $$\int_{-\infty}^{\infty} f\left(x - \frac{1}{x}\right) dx = \int_{-\infty}^{\infty} f(x) dx.$$

Solution: None.

Lean 4 formal statement:
theorem putnam_1968_b4
(f : ℝ → ℝ)
(hf : Continuous f ∧ ∃ r : ℝ, Tendsto (fun y => ∫ x in ball 0 y, f x) atTop (𝓝 r))
: ∃ r : ℝ, Tendsto (fun y => ∫ x in (ball 0 y \ ball 0 (1 / y)), f (x - 1/x)) atTop (𝓝 r) ∧ Tendsto (fun y => ∫ x in ball 0 y, f x) atTop (𝓝 r) :=
sorry
Problem: putnam_1968_b5
Tags: linear_algebra, number_theory, combinatorics

Let $p$ be a prime number. Find the number of distinct $2 \times 2$ matrices $$\begin{pmatrix} a & b \\ c & d \end{pmatrix}$$ such that $a, b, c, d \in \{0, 1, ..., p - 1\}$, $a + d \equiv 1 \pmod p$, and $ad - bc \equiv 0 \pmod p$.

Solution: There are $p^2 + p$ such matrices.

Lean 4 formal statement:
abbrev putnam_1968_b5_solution : ℕ → ℕ := sorry
theorem putnam_1968_b5
(p : ℕ)
(hp : Prime p)
: {M : Matrix (Fin 2) (Fin 2) (ZMod p) | M 0 0 + M 1 1 = 1 ∧ M 0 0 * M 1 1 - M 0 1 * M 1 0 = 0}.ncard = putnam_1968_b5_solution p :=
sorry
Problem: putnam_1968_b6
Tags: analysis

Prove that no sequence $\{K_n\}_{n=0}^{\infty}$ of compact (closed and bounded) sets of rational numbers has the property that every compact set of rational numbers is contained by at least one $K_n$.

Solution: None.

Lean 4 formal statement:
theorem putnam_1968_b6
: ¬∃ K : ℕ → Set ℚ, (∀ n : ℕ, IsCompact (K n)) ∧ (∀ S : Set ℚ, IsCompact S → ∃ n : ℕ, S ⊆ K n) :=
sorry
Problem: putnam_1969_a1
Tags: algebra, set_theory

What are the possible ranges (across all real inputs $x$ and $y$) of a polynomial $f(x, y)$ with real coefficients?

Solution: Show that the possibles ranges are a single point, any half-open or half-closed semi-infinite interval, or all real numbers.

Lean 4 formal statement:
abbrev putnam_1969_a1_solution : Set (Set ℝ) := sorry
theorem putnam_1969_a1
: {{z : ℝ | ∃ x : Fin 2 → ℝ, MvPolynomial.eval x f = z} | f : MvPolynomial (Fin 2) ℝ} = putnam_1969_a1_solution :=
sorry
Problem: putnam_1969_a2
Tags: linear_algebra

Let $D_n$ be the determinant of the $n$ by $n$ matrix whose value in the $i$th row and $j$th column is $|i-j|$. Show that $D_n = (-1)^{n-1} * (n-1) * (2^{n-2}).$

Solution: None.

Lean 4 formal statement:
theorem putnam_1969_a2
(D : (n : ℕ) → Matrix (Fin n) (Fin n) ℝ)
(hD : D = fun (n : ℕ) => λ (i : Fin n) (j : Fin n) => |(i : ℝ) - (j : ℝ)| )
: ∀ n, n ≥ 2 → (D n).det = (-1)^((n : ℤ)-1) * ((n : ℤ)-1) * 2^((n : ℤ)-2) :=
sorry
Problem: putnam_1969_a4
Tags: analysis

Show that $\int_0^1 x^x dx = \sum_{n=1}^{\infty} (-1)^{n+1}n^{-n}$.

Solution: None.

Lean 4 formal statement:
theorem putnam_1969_a4
: Tendsto (fun n => ∑ i ∈ Finset.Icc (1 : ℤ) n, (-1)^(i+1)*(i : ℝ)^(-i)) atTop (𝓝 (∫ x in Ioo (0 : ℝ) 1, x^x)) :=
sorry
Problem: putnam_1969_a5
Tags: analysis

Consider the system of differential equations $$\frac{dx}{dt} = -2y + u(t), \frac{dy}{dt} = -2x + u(t)$$ for some continuous function $u(t)$. Prove that, if $x(0) \ne y(0)$, the solution will never pass through $(0, 0)$ regardless of the choice of $u(t)$, and if $x(0) = y(0)$, a suitable $u(t)$ can be chosen for any $T > 0$ so that $(x(T), y(T)) = (0, 0)$.

Solution: None.

Lean 4 formal statement:
theorem putnam_1969_a5
    (x0 y0 t : ℝ)
    (ht : 0 < t) :
    x0 = y0 ↔ ∃ x y u : ℝ → ℝ,
      Differentiable ℝ x ∧
      Differentiable ℝ y ∧
      Continuous u ∧
      deriv x = - 2 • y + u ∧
      deriv y = - 2 • x + u ∧
      x 0 = x0 ∧
      y 0 = y0 ∧
      x t = 0 ∧
      y t = 0 :=
  sorry
Problem: putnam_1969_a6
Tags: analysis

Let $(x_n)$ be a sequence, and let $y_n = x_{n-1} + 2*x_n$ for $n \geq 2$. Suppose that $(y_n)$ converges, then prove that $(x_n)$ converges.

Solution: None.

Lean 4 formal statement:
theorem putnam_1969_a6
(x : ℕ → ℝ)
(y : ℕ → ℝ)
(hy1 : ∀ n ≥ 2, y n = x (n-1) + 2 * (x n))
(hy2 : ∃ c : ℝ, Tendsto y atTop (𝓝 c))
: ∃ C : ℝ, Tendsto x atTop (𝓝 C) :=
sorry
Problem: putnam_1969_b1
Tags: number_theory

Let $n$ be a positive integer such that $n+1$ is divisible by $24$. Prove that the sum of all the divisors of $n$ is divisible by $24$.

Solution: None.

Lean 4 formal statement:
theorem putnam_1969_b1
(n : ℕ)
(hnpos : n > 0)
(hn : 24 ∣ n + 1)
: 24 ∣ ∑ d ∈ divisors n, d :=
sorry
Problem: putnam_1969_b2
Tags: abstract_algebra

Show that a finite group can not be the union of two of its proper subgroups. Does the statement remain true if 'two' is replaced by 'three'?

Solution: Show that the statement is no longer true if 'two' is replaced by 'three'.

Lean 4 formal statement:
abbrev putnam_1969_b2_solution : Prop := sorry
theorem putnam_1969_b2
    (P : ℕ → Prop)
    (P_def : ∀ n, P n ↔ ∀ (G : Type) [Group G] [Finite G],
      ∀ H : Fin n → Subgroup G, (∀ i, H i < ⊤) → ⋃ i, (H i : Set G) < ⊤) :
    P 2 ∧ (P 3 ↔ putnam_1969_b2_solution) :=
  sorry
Problem: putnam_1969_b3
Tags: analysis

Suppose $T$ is a sequence which satisfies $T_n * T_{n+1} = n$ whenever $n \geq 1$, and also $\lim_{n \to \infty} \frac{T_n}{T_{n+1}} = 1. Show that $\pi * T_1^2 = 2$.

Solution: None.

Lean 4 formal statement:
theorem putnam_1969_b3
(T : ℕ → ℝ)
(hT1 : ∀ n : ℕ, n ≥ 1 → (T n) * (T (n + 1)) = n)
(hT2 : Tendsto (fun n => (T n)/(T (n + 1))) atTop (𝓝 1))
: Real.pi * (T 1)^2 = 2 :=
sorry
Problem: putnam_1969_b4
Tags: geometry

$Γ$ is a plane curve of length 1. Show that we can find a closed rectangle of area 1/4 which covers $Γ$.

Solution: None.

Lean 4 formal statement:
theorem putnam_1969_b4
    (Γ : ℝ → EuclideanSpace ℝ (Fin 2))
    --Note: the problem doesn't say what regularity conditions we should impose on `Γ` - hopefully continuity is enough.
    (Γ_cts : ContinuousOn Γ (Set.Icc 0 1))
    (hΓ : eVariationOn Γ (Set.Icc 0 1) = 1) :
    letI : Module.Oriented ℝ (EuclideanSpace ℝ (Fin 2)) (Fin 2) := ⟨Module.Basis.orientation <| PiLp.basisFun 2 ℝ (Fin 2)⟩
    letI : Fact (Module.finrank ℝ (EuclideanSpace ℝ (Fin 2)) = 2) := ⟨finrank_euclideanSpace_fin⟩
    ∃ (a b c d : EuclideanSpace ℝ (Fin 2)), Γ '' (Set.Icc 0 1) ⊆
      convexHull ℝ {a, b, c, d} ∧ dist a b * dist b c = 1/4 ∧
      --`abcd` is a rectangle
      letI π_div_2 := Real.Angle.coe <| π / 2
      (∡ a b c = π_div_2 ∧ ∡ b c d = π_div_2 ∧ ∡ c d a = π_div_2 ∧ ∡ d a b = π_div_2) :=
  sorry
Problem: putnam_1969_b5
Tags: analysis

Let $a_1 < a_2 < a_3 < \dots$ be an increasing sequence of positive integers. Assume that the sequences $\sum_{i = 1}^{\infty} 1/(a n)$ is convergent. For any number $x$, let $k(x)$ be the number of $a_n$'s which do not exceed $x$. Show that $\lim_{x \to \infty} k(x)/x = 0$.

Solution: None.

Lean 4 formal statement:
theorem putnam_1969_b5
(a : ℕ → ℝ)
(ha : StrictMono a ∧ (∀ x : ℕ, a x > 0))
(hinvasum : ∃ C : ℝ, Tendsto (fun n => ∑ i : Fin n, 1/(a i)) atTop (𝓝 C))
(k : ℝ → ℕ)
(hk : k = fun x => {n | a n ≤ x}.ncard)
: Tendsto (fun t => (k t)/t) atTop (𝓝 0) :=
sorry
Problem: putnam_1969_b6
Tags: linear_algebra

Let $A$ be a $3 \times 2$ matrix and $B$ be a $2 \times 3$ matrix such that $$AB =
\begin{pmatrix}
8 & 2 & -2 \\
2 & 5 & 4 \\
-2 & 4 & 5
\end{pmatrix}.
$$ Prove that $$BA =
\begin{pmatrix}
9 & 0 \\
0 & 9
\end{pmatrix}.$$

Solution: None.

Lean 4 formal statement:
theorem putnam_1969_b6
(A : Matrix (Fin 3) (Fin 2) ℝ)
(B : Matrix (Fin 2) (Fin 3) ℝ)
(p : Fin 3 → Fin 3 → ℝ)
(hp : p 0 0 = 8 ∧ p 0 1 = 2 ∧ p 0 2 = -2 ∧
p 1 0 = 2 ∧ p 1 1 = 5 ∧ p 1 2 = 4 ∧
p 2 0 = -2 ∧ p 2 1 = 4 ∧ p 2 2 = 5)
(hAB : A * B = Matrix.of p)
: B * A = 9 * (1 : Matrix (Fin 2) (Fin 2) ℝ) :=
sorry
Problem: putnam_1970_a1
Tags: analysis

Prove that, for all $a > 0$ and $b > 0$, the power series of $e^{ax} \cos (bx)$ with respect to $x$ has either zero or infinitely many zero coefficients.

Solution: None.

Lean 4 formal statement:
theorem putnam_1970_a1
(a b : ℝ)
(ha : a > 0)
(hb : b > 0)
(f : ℝ → ℝ)
(f_def : f = fun x : ℝ => Real.exp (a*x) * Real.cos (b*x))
(p : ℕ → ℝ)
(hp : ∃ c : ℝ, c > 0 ∧ ∀ x ∈ ball 0 c, ∑' n : ℕ, (p n)*x^n = f x)
(S : Set ℕ)
(S_def : S = {n : ℕ | p n = 0})
: S = ∅ ∨ ¬Finite S :=
sorry
Problem: putnam_1970_a2
Tags: analysis, algebra

Let $A$, $B$, $C$, $D$, $E$, $F$, and $G$ be real numbers satisfying $B^2 - 4AC < 0$. Prove that there exists some $\delta > 0$ such that no points $(x, y)$ in the punctured disk $0 < x^2 + y^2 < \delta$ satisfy $$Ax^2 + Bxy + Cy^2 + Dx^3 + Ex^2y + Fxy^2 + Gy^3 = 0.$$

Solution: None.

Lean 4 formal statement:
theorem putnam_1970_a2
(A B C D E F G : ℝ)
(hle : B^2 - 4*A*C < 0)
: ∃ δ > 0, ¬∃ x y : ℝ, x^2 + y^2 ∈ Set.Ioo 0 (δ^2) ∧
A*x^2 + B*x*y + C*y^2 + D*x^3 + E*x^2*y + F*x*y^2 + G*y^3 = 0 :=
sorry
Problem: putnam_1970_a3
Tags: number_theory

Find the length of the longest possible sequence of equal nonzero digits (in base 10) in which a perfect square can terminate. Also, find the smallest square that attains this length.

Solution: The maximum attainable length is $3$; the smallest such square is $38^2 = 1444$.

Lean 4 formal statement:
abbrev putnam_1970_a3_solution : ℕ × ℕ := sorry
theorem putnam_1970_a3
(L : ℕ → ℕ)
(hL : ∀ n : ℕ, L n ≤ (Nat.digits 10 n).length ∧
(∀ k : ℕ, k < L n → (Nat.digits 10 n)[k]! = (Nat.digits 10 n)[0]!) ∧
(L n ≠ (Nat.digits 10 n).length → (Nat.digits 10 n)[L n]! ≠ (Nat.digits 10 n)[0]!))
: (∃ n : ℕ, (Nat.digits 10 (n^2))[0]! ≠ 0 ∧ L (n^2) = putnam_1970_a3_solution.1) ∧
(∀ n : ℕ, (Nat.digits 10 (n^2))[0]! ≠ 0 → L (n^2) ≤ putnam_1970_a3_solution.1) ∧
(∃ m : ℕ, m^2 = putnam_1970_a3_solution.2) ∧
L (putnam_1970_a3_solution.2) = putnam_1970_a3_solution.1 ∧
(Nat.digits 10 putnam_1970_a3_solution.2)[0]! ≠ 0 ∧
∀ n : ℕ, (Nat.digits 10 (n^2))[0]! ≠ 0 ∧ L (n^2) = putnam_1970_a3_solution.1 → n^2 ≥ putnam_1970_a3_solution.2 :=
sorry
Problem: putnam_1970_a4
Tags: analysis

Suppose $(x_n)$ is a sequence such that $\lim_{n \to \infty} (x_n - x_{n-2} = 0$. Prove that $\lim_{n \to \infty} \frac{x_n - x_{n-1}}{n} = 0$.

Solution: None.

Lean 4 formal statement:
theorem putnam_1970_a4
(x : ℕ → ℝ)
(hxlim : Tendsto (fun n => x (n+2) - x n) atTop (𝓝 0))
: Tendsto (fun n => (x (n+1) - x (n))/(n+1)) atTop (𝓝 0) :=
sorry
Problem: putnam_1970_b1
Tags: analysis

Evaluate the infinite product $\lim_{n \to \infty} \frac{1}{n^4} \prod_{i = 1}^{2n} (n^2 + i^2)^{1/n}$.

Solution: Show that the solution is $e^{2 \log(5) - 4 + 2 arctan(2)}$.

Lean 4 formal statement:
noncomputable abbrev putnam_1970_b1_solution : ℝ := sorry
theorem putnam_1970_b1
: Tendsto (fun n => 1/(n^4) * ∏ i ∈ Finset.Icc (1 : ℤ) (2*n), ((n^2 + i^2) : ℝ)^((1 : ℝ)/n)) atTop (𝓝 putnam_1970_b1_solution) :=
sorry
Problem: putnam_1970_b2
Tags: analysis, algebra

Let $H$ be a polynomial of degree at most $3$ and $T$ be a positive real number. Show that the average value of $H(t)$ over the interval $[-T, T]$ equals the average of $H\left(-\frac{T}{\sqrt{3}}\right)$ and $H\left(\frac{T}{\sqrt{3}}\right)$.

Solution: None.

Lean 4 formal statement:
theorem putnam_1970_b2
(T : ℝ)
(H : Polynomial ℝ)
(hT : T > 0)
(hH : H.degree ≤ 3)
: (H.eval (-T / Real.sqrt 3) + H.eval (T / Real.sqrt 3))/2 = ⨍ t in Set.Icc (-T) T, H.eval t :=
sorry
Problem: putnam_1970_b3
Tags: analysis

A closed subset $S$ of $\mathbb{R}^2$ lies in $a < x < b$. Show that its projection on the y-axis is closed.

Solution: None.

Lean 4 formal statement:
theorem putnam_1970_b3
(S : Set (ℝ × ℝ))
(a b : ℝ)
(hab : a < b)
(hS : ∀ s ∈ S, s.1 ∈ Ioo a b)
(hSclosed : IsClosed S)
: IsClosed {y | ∃ x : ℝ, ⟨x,y⟩ ∈ S} :=
sorry
Problem: putnam_1970_b4
Tags: analysis

Let $x : \mathbb{R} \to \mathbb{R}$ be a twice differentiable function satisfying $x(1) - x(0) = 1$, $x'(0) = x'(1) = 0$, and $|x'(t)| \le \frac{3}{2}$ for all $t \in (0, 1)$. Prove that there exists some $t \in [0, 1]$ such that $|x''(t)| \ge \frac{9}{2}$.

Solution: None.

Lean 4 formal statement:
theorem putnam_1970_b4
(x : ℝ → ℝ)
(hdiff : DifferentiableOn ℝ x (Set.Icc 0 1) ∧ DifferentiableOn ℝ (deriv x) (Set.Icc 0 1))
(hx : x 1 - x 0 = 1)
(hv : deriv x 0 = 0 ∧ deriv x 1 = 0)
(hs : ∀ t ∈ Set.Ioo 0 1, |deriv x t| ≤ 3/2)
: ∃ t ∈ Set.Icc 0 1, |(deriv (deriv x)) t| ≥ 9/2 :=
sorry
Problem: putnam_1970_b5
Tags: analysis

Let $u_n$ denote the function $u_n(x) = -n$ if $x \leq -n$, $x$ if $-n < x \leq n$, and $n$ otherwise. Let $F$ be a function on the reals. Show that $F$ is continuous if and only if $u_n \circ F$ is continuous for all natural numbers $n$.

Solution: None.

Lean 4 formal statement:
theorem putnam_1970_b5
(ramp : ℤ → (ℝ → ℝ))
(ramp_def : ramp = fun (n : ℤ) => (fun (x : ℝ) => if x ≤ -n then (-n : ℝ) else (if -n < x ∧ x ≤ n then x else (n : ℝ))))
(F : ℝ → ℝ)
: Continuous F ↔ (∀ n : ℕ, Continuous ((ramp n) ∘ F)) :=
sorry
Problem: putnam_1970_b6
Tags: geometry

Prove that if a quadrilateral with side lengths $a$, $b$, $c$, and $d$ and area $\sqrt{abcd}$ can be circumscribed to a circle (i.e., a circle can be inscribed in it), then it must be cyclic (i.e., it can be inscribed in a circle).

Solution: None.

Lean 4 formal statement:
theorem putnam_1970_b6
(L : ZMod 4 → (EuclideanSpace ℝ (Fin 2)))
(S : Set (EuclideanSpace ℝ (Fin 2)))
(S_def : S = {L i | i : ZMod 4})
(hSquad : S.ncard = 4 ∧ ∀ s ⊆ S, s.ncard = 3 → ¬ Collinear ℝ s)
(hlens : dist (L 0) (L 1) > 0 ∧ dist (L 1) (L 2) > 0 ∧ dist (L 2) (L 3) > 0 ∧ dist (L 3) (L 0) > 0)
(horder : ∀ i : ZMod 4, segment ℝ (L i) (L (i + 1)) ∩ interior (convexHull ℝ S) = ∅)
(hcircum : ∃ (O: EuclideanSpace ℝ (Fin 2)) (r : ℝ), O ∈ convexHull ℝ S ∧ r > 0 ∧ ∀ i : ZMod 4,
∃! I : EuclideanSpace ℝ (Fin 2), Collinear ℝ {I, L i, L (i + 1)} ∧ dist O I = r)
(harea : (MeasureTheory.volume (convexHull ℝ S)).toReal =
Real.sqrt (dist (L 0) (L 1) * dist (L 1) (L 2) * dist (L 2) (L 3) * dist (L 3) (L 0)))
: Cospherical S :=
sorry
Problem: putnam_1971_a1
Tags: geometry, combinatorics

Let $S$ be a set of $9$ lattice points (points with integer coordinates) in $3$-dimensional Euclidean space. Prove that there exists a lattice point along the interior of some line segment that joins two distinct points in $S$.

Solution: None.

Lean 4 formal statement:
theorem putnam_1971_a1
(S : Set (ℤ × ℤ × ℤ))
(hS : S.ncard = 9)
(L : (ℤ × ℤ × ℤ) × (ℤ × ℤ × ℤ) → Set (ℝ × ℝ × ℝ))
(hL : L = fun ((a, b, c), (d, e, f)) =>
{(t*a + (1-t)*d, t*b + (1-t)*e, t*c + (1-t)*f) | t ∈ Ioo (0 : ℝ) 1})
: ∃ x y z : ℤ, ∃ P Q : ℤ × ℤ × ℤ, P ∈ S ∧ Q ∈ S ∧ P ≠ Q ∧ ((x : ℝ), (y : ℝ), (z : ℝ)) ∈ L (P, Q) :=
sorry
Problem: putnam_1971_a2
Tags: algebra

Determine all polynomials $P(x)$ such that $P(x^2 + 1) = (P(x))^2 + 1$ and $P(0) = 0$.

Solution: Show that the only such polynomial is the identity function.

Lean 4 formal statement:
abbrev putnam_1971_a2_solution : Set (Polynomial ℝ) := sorry
theorem putnam_1971_a2
    (P : Polynomial ℝ) :
    (P.eval 0 = 0 ∧ (∀ x : ℝ, P.eval (x^2 + 1) = (P.eval x)^2 + 1)) ↔ P ∈ putnam_1971_a2_solution :=
  sorry
Problem: putnam_1971_a3
Tags: geometry

The three vertices of a triangle of sides $a,b,c$ are lattice points and lie on a circle of radius $R$. Show that $abc \geq 2R$.

Solution: None.

Lean 4 formal statement:
theorem putnam_1971_a3
    (a b c : ℝ × ℝ)
    (R : ℝ)
    (habclattice : a.1 = round a.1 ∧ a.2 = round a.2 ∧ b.1 = round b.1 ∧ b.2 = round b.2 ∧ c.1 = round c.1 ∧ c.2 = round c.2)
    (habcneq : a ≠ b ∧ a ≠ c ∧ b ≠ c)
    (hR : R > 0)
    (hcircle : ∃ C : ℝ × ℝ,
      √((a.1 - C.1)^2 + (a.2 - C.2)^2) = R ∧
      √((b.1 - C.1)^2 + (b.2 - C.2)^2) = R ∧
      √((c.1 - C.1)^2 + (c.2 - C.2)^2) = R) :
    (√((a.1 - b.1)^2 + (a.2 - b.2)^2)) * (√((a.1 - c.1)^2 + (a.2 - c.2)^2)) * (√((b.1 - c.1)^2 + (b.2 - c.2)^2)) ≥ 2 * R :=
  sorry
Problem: putnam_1971_a4
Tags: analysis

Show that for $\epsilon \in (0,1)$, the expression $(x + y)^n (x^2 - 2-\epsilon)xy + y^2)$ is a polynomial with positive coefficients for $n$ sufficiently large, where $n$ is an integer.

Solution: None.

Lean 4 formal statement:
theorem putnam_1971_a4
(ε : ℝ)
(hε : 0 < ε ∧ ε < 1)
(P : ℕ → ℝ → MvPolynomial (Fin 2) ℝ)
(hP : P = fun (n : ℕ) (δ : ℝ) => (MvPolynomial.X 0 + MvPolynomial.X 1)^n * ((MvPolynomial.X 0)^2 - (MvPolynomial.C (2 - δ))*(MvPolynomial.X 0)*(MvPolynomial.X 1) + (MvPolynomial.X 1)^2))
: ∃ N : ℕ, ∀ n ≥ N, ∀ i : Fin 2 →₀ ℕ, MvPolynomial.coeff i (P n ε) ≥ 0 :=
sorry
Problem: putnam_1971_a5
Tags: number_theory

After each play of a certain game of solitaire, the player receives either $a$ or $b$ points, where $a$ and $b$ are positive integers with $a > b$; scores accumulate from play to play. If there are $35$ unattainable scores, one of which is $58$, find $a$ and $b$.

Solution: Show that the solution is $a = 11$ and $b = 8$.

Lean 4 formal statement:
abbrev putnam_1971_a5_solution : ℤ × ℤ := sorry
theorem putnam_1971_a5
    (a b : ℤ)
    (hab : a > 0 ∧ b > 0 ∧ a > b)
    (pab : ℤ → ℤ → Prop)
    (hpab : ∀ x y, pab x y ↔
      {s : ℕ | ¬∃ m n : ℕ, m*x + n*y = s}.ncard = 35 ∧
      ¬∃ m n : ℕ, m*x + n*y = 58) :
    pab a b ↔ a = putnam_1971_a5_solution.1 ∧ b = putnam_1971_a5_solution.2 :=
  sorry
Problem: putnam_1971_a6
Tags: number_theory

Let $c$ be a real number such that $n^c$ is an integer for every positive integer $n$. Show that $c$ is a non-negative integer.

Solution: None.

Lean 4 formal statement:
theorem putnam_1971_a6
(c : ℝ)
(hc : ∀ n : ℤ, n > 0 → ∃ m : ℤ, (n : ℝ)^c = m)
: ∃ m : ℤ, m ≥ 0 ∧ c = m :=
sorry
Problem: putnam_1971_b1
Tags: abstract_algebra

Let $S$ be a set and let $\cdot$ be a binary operation on $S$ satisfying the two following laws: (1) for all $x$ in $S$, $x = x \cdot x$, (2) for all $x,y,z$ in $S$, $(x \cdot y) \cdot z) = (y \cdot z) \cdot x$. Show that $\cdot$ is associative and commutative.

Solution: None.

Lean 4 formal statement:
theorem putnam_1971_b1
(S : Type*) [Mul S]
(hself : ∀ x : S, x * x = x)
(h2 : ∀ x y z : S, (x * y) * z = (y * z) * x)
: (∀ x y z : S, (x * y) * z = x * (y * z)) ∧ (∀ x y : S, x * y = y * x) :=
sorry
Problem: putnam_1971_b2
Tags: algebra

Find all functions $F : \mathbb{R} \setminus \{0, 1\} \to \mathbb{R}$ that satisfy $F(x) + F\left(\frac{x - 1}{x}\right) = 1 + x$ for all $x \in \mathbb{R} \setminus \{0, 1\}$.

Solution: The only such function is $F(x) = \frac{x^3 - x^2 - 1}{2x(x - 1)}$.

Lean 4 formal statement:
abbrev putnam_1971_b2_solution : Set (ℝ → ℝ) := sorry
theorem putnam_1971_b2
(S : Set ℝ)
(hS : S = univ \ {0, 1})
(P : (ℝ → ℝ) → Prop)
(hP : P = fun (F : ℝ → ℝ) => ∀ x ∈ S, F x + F ((x - 1)/x) = 1 + x)
: (∀ F ∈ putnam_1971_b2_solution, P F) ∧ ∀ f : ℝ → ℝ, P f → ∃ F ∈ putnam_1971_b2_solution, (∀ x ∈ S, f x = F x) :=
sorry
Problem: putnam_1971_b3
Tags: algebra

If two cars travel around a track at constant speeds of one lap per hour, starting from the same point but at different times, prove that the total amount of time for which the first car has completed exactly twice as many laps as the second is exactly $1$ hour.

Solution: None.

Lean 4 formal statement:
theorem putnam_1971_b3
(T : ℝ)
(hT : T > 0)
: MeasureTheory.volume {t : ℝ | t ≥ T ∧ Nat.floor t = 2 * (Nat.floor (t - T))} = 1 :=
sorry
Problem: putnam_1971_b6
Tags: number_theory

Let $\delta(x) be the greatest odd divisor of the positive integer $x$. Show that $|\sum_{n = 1}^x \delta(n)/n - 2x/3| < 1$ for all positive integers $x$.

Solution: None.

Lean 4 formal statement:
theorem putnam_1971_b6
(δ : ℤ → ℤ)
(hδ : δ = fun n => sSup {t | Odd t ∧ t ∣ n})
: ∀ x : ℤ, x > 0 → |∑ i ∈ Finset.Icc 1 x, (δ i)/(i : ℚ) - 2*x/3| < 1 :=
sorry
Problem: putnam_1972_a1
Tags: algebra

Show that there are no four consecutive binomial coefficients ${n \choose r}, {n \choose (r+1)}, {n \choose (r+2)}, {n \choose (r+3)}$ where $n,r$ are positive integers and $r+3 \leq n$, which are in arithmetic progression.

Solution: None.

Lean 4 formal statement:
theorem putnam_1972_a1
    (n : ℕ) (hn : n > 0)
    (fourAP : ℤ → ℤ → ℤ → ℤ → Prop)
    (hfourAP : ∀ n1 n2 n3 n4, fourAP n1 n2 n3 n4 ↔ n4-n3 = n3-n2 ∧ n3-n2 = n2-n1) :
    ¬ ∃ r : ℕ, r > 0 ∧ r + 3 ≤ n ∧ fourAP (n.choose r) (n.choose (r+1)) (n.choose (r+2)) (n.choose (r+3)) :=
  sorry
Problem: putnam_1972_a2
Tags: abstract_algebra

Let $S$ be a set and $\cdot$ be a binary operation on $S$ satisfying: (1) for all $x,y$ in $S$, $x \cdot (x \cdot y) = y$ (2) for all $x,y$ in $S$, $(y \cdot x) \cdot x = y$. Show that $\cdot$ is commutative but not necessarily associative.

Solution: None.

Lean 4 formal statement:
theorem putnam_1972_a2
: (∀ (S : Type*) (_ : Mul S), (∀ x y : S, x * (x * y) = y ∧ ((y * x) * x) = y) → (∀ x y : S, x * y = y * x)) ∧ ∃ (S : Type*) (_ : Mul S), (∀ x y : S, x * (x * y) = y ∧ ((y * x) * x) = y) ∧ ¬(∀ x y z : S, x * (y * z) = (x * y) * z) :=
sorry
Problem: putnam_1972_a3
Tags: 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 formal statement:
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 :=
  sorry
Problem: putnam_1972_a5
Tags: number_theory

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

Solution: None.

Lean 4 formal statement:
theorem putnam_1972_a5
(n : ℕ)
(hn : n > 1)
: ¬((n : ℤ) ∣ 2^n - 1) :=
sorry
Problem: putnam_1972_a6
Tags: 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 formal statement:
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) :=
sorry
Problem: putnam_1972_b1
Tags: 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 formal statement:
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 :=
sorry
Problem: putnam_1972_b2
Tags: 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 formal statement:
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) :=
  sorry
Problem: putnam_1972_b3
Tags: 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 formal statement:
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 :=
sorry