name stringlengths 7 15 | statement stringlengths 75 1.66k | id stringlengths 16 16 |
|---|---|---|
putnam_1978_a6 | theorem putnam_1978_a6
(S : Finset (EuclideanSpace ℝ (Fin 2)))
(n : ℕ)
(hn : n = S.card)
(npos : n > 0)
: ({pair : Set (EuclideanSpace ℝ (Fin 2)) | ∃ P ∈ S, ∃ Q ∈ S, pair = {P, Q} ∧ dist P Q = 1}.ncard < 2 * (n : ℝ) ^ ((3 : ℝ) / 2)) :=
sorry | ff1b9c95c26614f1 |
putnam_1968_a6 | abbrev putnam_1968_a6_solution : Set ℂ[X] := sorry
-- {X - 1, -(X - 1), X + 1, -(X + 1), X^2 + X - 1, -(X^2 + X - 1), X^2 - X - 1, -(X^2 - X - 1), X^3 + X^2 - X - 1, -(X^3 + X^2 - X - 1), X^3 - X^2 - X + 1, -(X^3 - X^2 - X + 1)}
/--
Find all polynomials of the form $\sum_{0}^{n} a_{i} x^{n-i}$ with $n \ge 1$ and $a_i =... | 413e46a7acb94237 |
putnam_1992_b4 | abbrev putnam_1992_b4_solution : ℕ := sorry
-- 3984
/--
Let $p(x)$ be a nonzero polynomial of degree less than $1992$ having no nonconstant factor in common with $x^3 - x$. Let
\[
\frac{d^{1992}}{dx^{1992}} \left( \frac{p(x)}{x^3 - x} \right) = \frac{f(x)}{g(x)}
\]
for polynomials $f(x)$ and $g(x)$. Find the smallest p... | fd8317f67388b9ff |
putnam_1982_b4 | abbrev putnam_1982_b4_solution : Prop × Prop := sorry
-- (True, True)
/--
Let $n_1, n_2, \dots, n_s$ be distinct integers such that, for every integer $k$, $n_1n_2\cdots n_s$ divides $(n_1 + k)(n_2 + k) \cdots (n_s + k)$. Prove or provide a counterexample to the following claims:
\begin{enumerate}
\item
For some $i$, $... | 6d84ab14780e2a0a |
putnam_2007_b4 | abbrev putnam_2007_b4_solution : ℕ → ℕ := sorry
-- fun n ↦ 2 ^ (n + 1)
/--
Let $n$ be a positive integer. Find the number of pairs $P, Q$ of polynomials with real coefficients such that
\[
(P(X))^2 + (Q(X))^2 = X^{2n} + 1
\]
and $\deg P > \deg Q$.
-/
theorem putnam_2007_b4
(n : ℕ) (npos : n > 0) :
({(P, Q) : (Polynomia... | 0c44fc2f2b2c0006 |
putnam_2017_b4 | abbrev putnam_2017_b4_solution : ℝ := sorry
-- (log 2) ^ 2
/--
Evaluate the sum \begin{gather*} \sum_{k=0}^\infty \left( 3 \cdot \frac{\ln(4k+2)}{4k+2} - \frac{\ln(4k+3)}{4k+3} - \frac{\ln(4k+4)}{4k+4} - \frac{\ln(4k+5)}{4k+5} \right) \ = 3 \cdot \frac{\ln 2}{2} - \frac{\ln 3}{3} - \frac{\ln 4}{4} - \frac{\ln 5}{5} + 3... | 71a8b73b2110cd18 |
putnam_1970_b1 | abbrev putnam_1970_b1_solution : ℝ := sorry
-- Real.exp (2 * Real.log 5 - 4 + 2 * Real.arctan 2)
/--
Evaluate the infinite product $\lim_{n \to \infty} \frac{1}{n^4} \prod_{i = 1}^{2n} (n^2 + i^2)^{1/n}$.
-/
theorem putnam_1970_b1
: Tendsto (fun n => 1/(n^4) * ∏ i in Finset.Icc (1 : ℤ) (2*n), ((n^2 + i^2) : ℝ)^((1 : ℝ)... | dc6a98cc79df4cf9 |
putnam_2014_b2 | abbrev putnam_2014_b2_solution : ℝ := sorry
-- Real.log (4 / 3)
/--
Suppose that \( f \) is a function on the interval \([1,3]\) such that \(-1 \leq f(x) \leq 1\) for all \( x \) and \( \int_{1}^{3} f(x) \, dx = 0 \). How large can \(\int_{1}^{3} \frac{f(x)}{x} \, dx \) be?
-/
theorem putnam_2014_b2 :
IsGreatest {t |... | fb03c4d0b8832a77 |
putnam_2004_b2 | theorem putnam_2004_b2
(m n : ℕ)
(mnpos : m > 0 ∧ n > 0)
: ((m + n)! / ((m + n) ^ (m + n) : ℚ)) < (((m)! / (m ^ m : ℚ)) * ((n)! / (n ^ n : ℚ))) :=
sorry | 2d7f932132cc4d74 |
putnam_1981_b2 | abbrev putnam_1981_b2_solution : ℝ := sorry
-- 12 - 8 * Real.sqrt 2
/--
Determine the minimum value attained by $$(r - 1)^2 + (\frac{s}{r} - 1)^2 + (\frac{t}{s} - 1)^2 + (\frac{4}{t} - 1)^2$$ across all choices of real $r$, $s$, and $t$ that satisfy $1 \le r \le s \le t \le 4$.
-/
theorem putnam_1981_b2
(P : ℝ × ℝ ... | 0d4485b6f60f63fe |
putnam_1991_b2 | theorem putnam_1991_b2
(f g : ℝ → ℝ)
(fgnconst : ¬∃ c, f = Function.const ℝ c ∨ g = Function.const ℝ c)
(fgdiff : Differentiable ℝ f ∧ Differentiable ℝ g)
(fadd : ∀ x y, f (x + y) = f x * f y - g x * g y)
(gadd : ∀ x y, g (x + y) = f x * g y + g x * f y) :
(deriv f 0 = 0) → (∀ x, (f x) ^ 2 + (g x) ^ 2 = 1) ... | 2e924c47b98b8cc8 |
putnam_1986_b6 | theorem putnam_1986_b6
(n : ℕ)
(npos : n > 0)
(F : Type*) [Field F]
(A B C D : Matrix (Fin n) (Fin n) F)
(hsymm : IsSymm (A * Bᵀ) ∧ IsSymm (C * Dᵀ))
(hid : A * Dᵀ - B * Cᵀ = 1)
: (Aᵀ * D - Cᵀ * B = 1) :=
sorry | 6baaf2b59e0c65dd |
putnam_2003_b6 | theorem putnam_2003_b6
(f : ℝ → ℝ)
(hf : Continuous f)
: (∫ x in (0 : ℝ)..1, (∫ y in (0 : ℝ)..1, |f x + f y|)) ≥ (∫ x in (0 : ℝ)..1, |f x|) :=
sorry | d9614a6ce849c09d |
putnam_1974_b3 | theorem putnam_1974_b3
(α : ℝ)
(ha : Real.cos (Real.pi * α) = (1 : ℝ)/3)
: Irrational α :=
sorry | 812353adb16a64d9 |
putnam_1999_a5 | theorem putnam_1999_a5
: ∃ C : ℝ, ∀ p : Polynomial ℝ, p.degree = 1999 → ‖p.eval 0‖ ≤ C * ∫ x in (-1)..1, ‖p.eval x‖ :=
sorry | d6ec809a198deb98 |
putnam_1964_b3 | theorem putnam_1964_b3
(f : ℝ → ℝ)
(hf : Continuous f ∧ ∀ α > 0, Tendsto (fun n : ℕ ↦ f (n * α)) atTop (𝓝 0))
: (Tendsto f atTop (𝓝 0)) :=
sorry | 2f180c29c45a0c68 |
putnam_2021_a2 | abbrev putnam_2021_a2_solution : ℝ := sorry
-- Real.exp 1
/--
For every positive real number $x$, let $g(x)=\lim_{r \to 0}((x+1)^{r+1}-x^{r+1})^\frac{1}{r}$. Find $\lim_{x \to \infty}\frac{g(x)}{x}$.
-/
theorem putnam_2021_a2
(g : ℝ → ℝ)
(hg : ∀ x > 0, Tendsto (fun r : ℝ => ((x + 1) ^ (r + 1) - x ^ (r + 1)) ^ (1 / r)) ... | ec3fd514b569ad4c |
putnam_2017_a2 | theorem putnam_2017_a2
(Q : ℕ → RatFunc ℚ)
(hQbase : Q 0 = 1 ∧ Q 1 = (X : ℚ[X]))
(hQn : ∀ n, Q (n + 2) = (Q (n + 1) ^ 2 - 1) / Q n)
(n : ℕ) (hn : 0 < n) :
∃ P : ℤ[X], Q n = P.map (Int.castRingHom ℚ) :=
sorry | 9e07b70eb44e1453 |
putnam_2007_a2 | abbrev putnam_2007_a2_solution : ENNReal := sorry
-- 4
/--
Find the least possible area of a convex set in the plane that intersects both branches of the hyperbola $xy=1$ and both branches of the hyperbola $xy=-1$. (A set $S$ in the plane is called \emph{convex} if for any two points in $S$ the line segment connecting ... | 60f016e566ac51cd |
putnam_1982_a2 | abbrev putnam_1982_a2_solution : Prop := sorry
-- True
/--
Let $B_n(x) = 1^x + 2^x + \dots + n^x$ and let $f(n) = \frac{B_n(\log_n 2)}{(n \log_2 n)^2}$. Does $f(2) + f(3) + f(4) + \dots$ converge?
-/
theorem putnam_1982_a2
(B : ℕ → ℝ → ℝ)
(hB : B = fun (n : ℕ) (x : ℝ) ↦ ∑ k in Finset.Icc 1 n, (k : ℝ) ^ x)
(f : ℕ → ℝ)
(... | c7dab1b8de51aa04 |
putnam_1992_a2 | abbrev putnam_1992_a2_solution : ℝ := sorry
-- 1992
/--
Define $C(\alpha)$ to be the coefficient of $x^{1992}$ in the power series about $x=0$ of $(1 + x)^\alpha$. Evaluate
\[
\int_0^1 \left( C(-y-1) \sum_{k=1}^{1992} \frac{1}{y+k} \right)\,dy.
\]
-/
theorem putnam_1992_a2
(C : ℝ → ℝ)
(hC : C = fun α ↦ taylorCoeffWithi... | 24f79b1bba75a222 |
putnam_2008_b1 | abbrev putnam_2008_b1_solution : ℕ := sorry
-- 2
def is_rational_point (p : Fin 2 → ℝ) : Prop := ∃ (a b : ℚ), a = p 0 ∧ b = p 1
def real_circle (c : EuclideanSpace ℝ (Fin 2)) (r : ℝ) : Set (EuclideanSpace ℝ (Fin 2)) := {p | dist p c = r}
/--
What is the maximum number of rational points that can lie on a circle in $\ma... | 5f802f232611746d |
putnam_2018_b1 | abbrev putnam_2018_b1_solution : Set (Fin 2 → ℤ) := sorry
-- {v : Fin 2 → ℤ | ∃ b : ℤ, 0 ≤ b ∧ b ≤ 100 ∧ Even b ∧ v = ![1, b]}
/--
Let $\mathcal{P}$ be the set of vectors defined by $\mathcal{P}=\left\{\left.\begin{pmatrix} a \\ b \end{pmatrix}\right| 0 \leq a \leq 2, 0 \leq b \leq 100,\text{ and }a,b \in \mathbb{Z}\r... | 564890e75c8a6a06 |
putnam_1985_a6 | abbrev putnam_1985_a6_solution : Polynomial ℝ := sorry
-- 6 * X ^ 2 + 5 * X + 1
/--
If $p(x)= a_0 + a_1 x + \cdots + a_m x^m$ is a polynomial with real coefficients $a_i$, then set
\[
\Gamma(p(x)) = a_0^2 + a_1^2 + \cdots + a_m^2.
\]
Let $F(x) = 3x^2+7x+2$. Find, with proof, a polynomial $g(x)$ with real coefficients s... | 920087e2a1d120c5 |
putnam_1995_a6 | 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}... | ebeee300772c5c80 |
putnam_1977_a3 | abbrev putnam_1977_a3_solution : (ℝ → ℝ) → (ℝ → ℝ) → (ℝ → ℝ) := sorry
-- fun f g x ↦ g x - f (x - 3) + f (x - 1) + f (x + 1) - f (x + 3)
/--
Let $f, g, h$ be functions $\mathbb{R} \to \mathbb{R}$. Find an expression for $h(x)$ in terms of $f$ and $g$ such that $f(x) = \frac{h(x + 1) + h(x - 1)}{2}$ and $g(x) = \frac{h(... | 29708f4d539fb450 |
putnam_1967_a3 | abbrev putnam_1967_a3_solution : ℕ := sorry
-- 5
/--
Consider polynomial forms $ax^2-bx+c$ with integer coefficients which have two distinct zeros in the open interval $0<x<1$. Exhibit with a proof the least positive integer value of $a$ for which such a polynomial exists.
-/
theorem putnam_1967_a3 :
IsLeast
... | a1393afbd533eef4 |
putnam_2022_b2 | abbrev putnam_2022_b2_solution : Set ℕ := sorry
-- {1, 7}
/--
Let $\times$ represent the cross product in $\mathbb{R}^3$. For what positive integers $n$ does there exist a set $S \subset \mathbb{R}^3$ with exactly $n$ elements such that $S=\{v \times w:v,w \in S\}$?
-/
theorem putnam_2022_b2
(n : ℕ)
(P : Finset (Fi... | 22a37e28d74ba943 |
putnam_2010_a6 | theorem putnam_2010_a6
(f : ℝ → ℝ)
(hf : (StrictAntiOn f (Ici 0) ∧ ContinuousOn f (Ici 0) ∧ Tendsto f atTop (𝓝 0)))
: ¬∃ y : ℝ, Tendsto (fun z : ℝ => ∫ x in Ico 0 z, ((f x) - f (x + 1))/(f x)) atTop (𝓝 y) :=
sorry | 9f8b7b28c47dbfae |
putnam_2000_a6 | theorem putnam_2000_a6
(f : Polynomial ℤ)
(a : ℕ → ℤ)
(ha0 : a 0 = 0)
(ha : ∀ n : ℕ, a (n + 1) = f.eval (a n))
: ((∃ m > 0, a m = 0) → (a 1 = 0 ∨ a 2 = 0)) :=
sorry | 120b2bcf4011fc6a |
putnam_1989_b3 | abbrev putnam_1989_b3_solution : ℕ → ℝ → ℝ := sorry
-- fun n c ↦ c * n ! / (3 ^ n * ∏ m in Finset.Icc (1 : ℤ) n, (1 - 2 ^ (-m)))
/--
Let $f$ be a function on $[0,\infty)$, differentiable and satisfying
\[
f'(x)=-3f(x)+6f(2x)
\]
for $x>0$. Assume that $|f(x)|\le e^{-\sqrt{x}}$ for $x\ge 0$ (so that $f(x)$ tends rapidly ... | 2642743efb382ade |
putnam_2021_b4 | theorem putnam_2021_b4
(F : ℕ → ℕ)
(hF : ∀ n, F (n + 2) = F (n + 1) + F n)
(F01 : F 0 = 0 ∧ F 1 = 1)
: ∀ m, m > 2 → (∃ p, (∏ k : Set.Icc 1 (F m - 1), (k.1 ^ k.1)) % F m = F p) :=
sorry | 0d0051f2d86cbe59 |
putnam_1964_a5 | theorem putnam_1964_a5
(pa : (ℕ → ℝ) → Prop)
(hpa : ∀ a, pa a ↔ (∀ n : ℕ, a n > 0) ∧ ∃ L : ℝ, Tendsto (fun N ↦ ∑ n in Finset.range N, 1 / a n) atTop (𝓝 L)) :
∃ k : ℝ, ∀ a : ℕ → ℝ, pa a →
∑' n : ℕ, (n + 1) / (∑ i in Finset.range (n + 1), a i) ≤ k * ∑' n : ℕ, 1 / a n :=
sorry | bae539d8d621345d |
putnam_1999_b3 | abbrev putnam_1999_b3_solution : ℝ := sorry
-- 3
/--
Let $A=\{(x,y):0\leq x,y<1\}$. For $(x,y)\in A$, let \[S(x,y) = \sum_{\frac{1}{2}\leq \frac{m}{n}\leq 2} x^m y^n,\] where the sum ranges over all pairs $(m,n)$ of positive integers satisfying the indicated inequalities. Evaluate \[\lim_{(x,y)\rightarrow (1,1), (x,y... | 2ef1bf53e3ce1625 |
putnam_1991_a4 | abbrev putnam_1991_a4_solution : Prop := sorry
-- True
/--
Does there exist an infinite sequence of closed discs $D_1,D_2,D_3,\dots$ in the plane, with centers $c_1,c_2,c_3,\dots$, respectively, such that
\begin{enumerate}
\item the $c_i$ have no limit point in the finite plane,
\item the sum of the areas of the $D_i$ ... | 7c6dd778d033673e |
putnam_1973_a1 | 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})) ... | 510975ecc30f9bb9 |
putnam_2004_a4 | theorem putnam_2004_a4
(n : ℕ) (npos : n > 0)
(x : Fin n → ℝ)
(avals : ℕ → (ℕ → Fin n → ℝ) → Prop)
(havals : ∀ N a, avals N a ↔ ∀ (i : Fin N) (j : Fin n), (a i j = -1 ∨ a i j = 0 ∨ a i j = 1)) :
∃ (N : ℕ) (c : Fin N → ℚ) (a : ℕ → Fin n → ℝ),
avals N a ∧
(∏ i : Fin n, x i) = ∑ i : Fin N, c i * (∑ j : F... | 7cb0a37e7d1b26fc |
putnam_2014_a4 | abbrev putnam_2014_a4_solution : ℝ := sorry
-- 1 / 3
/--
Suppose $X$ is a random variable that takes on only nonnegative integer values, with $E\left[X\right]=1$, $E\left[X^2\right]=2$, and $E\left[X^3\right]=5$. (Here $E\left[Y\right]$ denotes the expectation of the random variable $Y$.) Determine the smallest possibl... | 97876766472c6fe4 |
putnam_1987_a1 | theorem putnam_1987_a1
(A B C D : Set (ℝ × ℝ))
(hA : A = {(x, y) : ℝ × ℝ | x ^ 2 + y ^ 2 ≠ 0 ∧ x ^ 2 - y ^ 2 = x / (x ^ 2 + y ^ 2)})
(hB : B = {(x, y) : ℝ × ℝ | x ^ 2 + y ^ 2 ≠ 0 ∧ 2 * x * y + y / (x ^ 2 + y ^ 2) = 3})
(hC : C = {(x, y) : ℝ × ℝ | x ^ 3 - 3 * x * y ^ 2 + 3 * y = 1})
(hD : D = {(x, y) : ℝ × ℝ | 3 * x ^ 2... | 23254f3dd993dacf |
putnam_2012_a1 | theorem putnam_2012_a1
(d : Fin 12 → ℝ)
(hd : ∀ i, d i ∈ Set.Ioo 1 12)
: ∃ i j k, i ≠ j ∧ i ≠ k ∧ j ≠ k ∧ d k ≥ d i ∧ d k ≥ d j ∧ (d i) ^ 2 + (d j) ^ 2 > (d k) ^ 2 :=
sorry | c05b63c7a82a9beb |
putnam_2002_a1 | abbrev putnam_2002_a1_solution : ℕ → ℕ → ℝ := sorry
-- (fun k n : ℕ => (-k) ^ n * (n)!)
/--
Let $k$ be a fixed positive integer. The $n$-th derivative of $\frac{1}{x^k-1}$ has the form $\frac{P_n(x)}{(x^k-1)^{n+1}}$ where $P_n(x)$ is a polynomial. Find $P_n(1)$.
-/
theorem putnam_2002_a1
(k : ℕ)
(P : ℕ → Polynomial ℝ)
... | 690aeb9eb356d8d6 |
putnam_1975_a4 | abbrev putnam_1975_a4_solution : ℕ → Polynomial ℤ := sorry
-- fun m => ∑ j in Finset.range ((m - 1) / 2), Polynomial.X ^ (2 * j + 1)
/--
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}... | 42ef33d7d1a63b0f |
putnam_1998_b2 | abbrev putnam_1998_b2_solution : ℝ → ℝ → ℝ := sorry
-- fun a b => if a > b then Real.sqrt (2*a^2 + 2*b^2) else 0
/--
Given a point $(a,b)$ with $0<b<a$, determine the minimum perimeter of a triangle with one vertex at $(a,b)$, one on the $x$-axis, and one on the line $y=x$. You may assume that a triangle of minimum pe... | 76a5b677756fc36f |
putnam_1965_a4 | theorem putnam_1965_a4
{G B : Type*}
[Fintype G] [Nonempty G]
[Fintype B] [Nonempty B]
(dances : G → B → Prop)
(h : (¬∃ b : B, ∀ g : G, dances g b) ∧ ∀ g : G, ∃ b : B, dances g b)
: ∃ g h : G, ∃ b c : B, dances g b ∧ dances h c ∧ ¬dances h b ∧ ¬dances g c :=
sorry | c3b2cfd8c834a4ea |
putnam_1988_b2 | abbrev putnam_1988_b2_solution : Prop := sorry
-- True
/--
Prove or disprove: If $x$ and $y$ are real numbers with $y \geq 0$ and $y(y+1) \leq (x+1)^2$, then $y(y-1) \leq x^2$.
-/
theorem putnam_1988_b2
: (∀ x y : ℝ, (y ≥ 0 ∧ y * (y + 1) ≤ (x + 1) ^ 2) → (y * (y - 1) ≤ x ^ 2)) ↔ putnam_1988_b2_solution :=
sorry | 700ca10252fd3ed5 |
putnam_2020_b5 | theorem putnam_2020_b5
(z : Fin 4 → ℂ)
(hzle1 : ∀ n, ‖z n‖ = 1)
(hzne1 : ∀ n, z n ≠ 1)
: 3 - z 0 - z 1 - z 2 - z 3 + (z 0) * (z 1) * (z 2) * (z 3) ≠ 0:=
sorry | a284f1a6ea86518a |
putnam_2015_a5 | theorem putnam_2015_a5
(q : ℕ)
(Nq : ℕ)
(qodd : Odd q)
(qpos : q > 0)
(hNq : Nq = {a : ℕ | 0 < a ∧ a < (q : ℝ) / 4 ∧ Nat.gcd a q = 1}.encard)
: Odd Nq ↔ ∃ p k : ℕ, q = p ^ k ∧ k > 0 ∧ p.Prime ∧ (p % 8 = 5 ∨ p % 8 = 7) :=
sorry | 267a3f6aa3488410 |
putnam_2005_a5 | abbrev putnam_2005_a5_solution : ℝ := sorry
-- Real.pi * (Real.log 2) / 8
/--
Evaluate $\int_0^1 \frac{\ln(x+1)}{x^2+1}\,dx$.
-/
theorem putnam_2005_a5 :
∫ x in (0:ℝ)..1, (Real.log (x+1))/(x^2 + 1) = putnam_2005_a5_solution :=
sorry | 3cb3373ce77445e8 |
putnam_1980_a5 | theorem putnam_1980_a5
(P : Polynomial ℝ)
(Pnonconst : P.degree > 0) :
Set.Finite {x : ℝ |
0 = (∫ t in (0)..x, P.eval t * Real.sin t) ∧
0 = (∫ t in (0)..x, P.eval t * Real.cos t)} :=
sorry | 4e4d5149a5135c1d |
putnam_1990_a5 | abbrev putnam_1990_a5_solution : Prop := sorry
-- False
/--
If $\mathbf{A}$ and $\mathbf{B}$ are square matrices of the same size such that $\mathbf{ABAB}=\mathbf{0}$, does it follow that $\mathbf{BABA}=\mathbf{0}$?
-/
theorem putnam_1990_a5 :
putnam_1990_a5_solution ↔
(∀ n ≥ 1, ∀ A B : Matrix (Fin n) (Fin n) ℝ,
... | d85e9ab4cc895b8e |
putnam_1993_a3 | theorem putnam_1993_a3
(c : ℕ → ℕ → ℕ)
(hc : ∀ n ≥ 1, ∀ m ≥ 1, c n m = {f : Finset (Fin n) → Fin m | ∀ A B : Finset (Fin n), f (A ∩ B) = min (f A) (f B)}.encard)
: ∀ n ≥ 1, ∀ m ≥ 1, c n m = ∑' j : Set.Icc 1 m, (j : ℤ) ^ n :=
sorry | e01f69df5f1aec80 |
putnam_1983_a3 | theorem putnam_1983_a3
(p : ℕ)
(F : ℕ → ℕ)
(poddprime : Odd p ∧ p.Prime)
(hF : ∀ n : ℕ, F n = ∑ i in Finset.range (p - 1), (i + 1) * n ^ i)
: ∀ a ∈ Finset.Icc 1 p, ∀ b ∈ Finset.Icc 1 p, a ≠ b → ¬(F a ≡ F b [MOD p]) :=
sorry | 066ed7591ccb30cb |
putnam_2006_a3 | theorem putnam_2006_a3
(x : ℤ → ℤ)
(hxlo : ∀ k : ℤ, 0 ≤ k ∧ k ≤ 2006 → x k = k)
(hxhi : ∀ k : ℤ, k ≥ 2006 → x (k + 1) = x k + x (k - 2005))
: (∃ i : ℕ, i > 0 ∧ ∀ j : Finset.range 2005, 2006 ∣ x (i + j)) :=
sorry | 3d5bd9427d1bb632 |
putnam_2016_a3 | abbrev putnam_2016_a3_solution : ℝ := sorry
-- 3 * Real.pi / 8
/--
Suppose that $f$ is a function from $\mathbb{R}$ to $\mathbb{R}$ such that
\[
f(x) + f\left( 1 - \frac{1}{x} \right) = \arctan x
\]
for all real $x \neq 0$. (As usual, $y = \arctan x$ means $-\pi/2 < y < \pi/2$ and $\tan y = x$.) Find
\[
\int_0^1 f(x)\,... | a360124adaf6920b |
putnam_1971_a6 | theorem putnam_1971_a6
(c : ℝ)
(hc : ∀ n : ℤ, n > 0 → ∃ m : ℤ, (n : ℝ)^c = m)
: ∃ m : ℤ, m ≥ 0 ∧ c = m :=
sorry | 494c27a38e1c0165 |
putnam_2023_b3 | abbrev putnam_2023_b3_solution : ℕ → ℝ := sorry
-- (fun n : ℕ ↦ (2 * (n : ℝ) + 2) / 3)
/--
A sequence $y_1, y_2, \ldots, y_k$ of real numbers is called zigzag if $k = 1$, or if $y_2 - y_1, y_3 - y_2, \ldots, y_k - y_{k-1}$ are nonzero and alternate in sign. Let $X_1, X_2,\ldots, X_n$ be chosen independently from the un... | 53a5afe7c805acaa |
putnam_1966_a2 | theorem putnam_1966_a2
(r : ℝ)
(A B C : EuclideanSpace ℝ (Fin 2))
(hABC : ¬Collinear ℝ {A, B, C})
(a b c p : ℝ)
(ha : a = dist B C)
(hb : b = dist C A)
(hc : c = dist A B)
(hp : p = (dist B C + dist C A + dist A B)/2)
(hr : ∃ I : EuclideanSpace ℝ (Fin 2),
(∃! P : EuclideanSpace ℝ (Fin 2), dist I P = r ∧ Collinear ℝ {P,... | a32f88b0f5565572 |
putnam_1976_a2 | 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)
(... | 6ebee7bcee66f89c |
putnam_1969_b1 | theorem putnam_1969_b1
(n : ℕ)
(hnpos : n > 0)
(hn : 24 ∣ n + 1)
: 24 ∣ ∑ d in divisors n, d :=
sorry | bbdc3aeb399fc02d |
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