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putnam_1978_a6 | theorem putnam_1978_a6
(S : Finset (EuclideanSpace β (Fin 2)))
(n : β)
(hn : n = S.card)
(npos : n > 0)
: ({pair : Set (EuclideanSpace β (Fin 2)) | β P β S, β Q β S, pair = {P, Q} β§ dist P Q = 1}.ncard < 2 * (n : β) ^ ((3 : β) / 2)) :=
sorry | ff1b9c95c26614f1 |
putnam_1968_a6 | abbrev putnam_1968_a6_solution : Set β[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 $a_i = \pm 1$ for all $0 \le i \le n$ whose roots are all real.
-/
theorem putnam_1968_a6
: {P : β[X] | P.natDegree β₯ 1 β§ (β k β Set.Icc 0 P.natDegree, P.coeff k = 1 β¨ P.coeff k = -1) β§
β z : β, P.eval z = 0 β β r : β, r = z} = putnam_1968_a6_solution :=
sorry | 413e46a7acb94237 |
putnam_1992_b4 | abbrev putnam_1992_b4_solution : β := 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 possible degree of $f(x)$.
-/
theorem putnam_1992_b4
(IsValid : Polynomial β β Prop)
(pair : Polynomial β β Polynomial β β Prop)
(IsValid_def : β p, IsValid p β p β 0 β§ p.degree < 1992 β§ IsCoprime p (X ^ 3 - X))
(hpair : β p f, pair p f β β g : Polynomial β, iteratedDeriv 1992 (fun x β¦ p.eval x / (x ^ 3 - x)) = fun x β¦ f.eval x / g.eval x) :
IsLeast {k : β | β p f, IsValid p β§ pair p f β§ k = f.degree} putnam_1992_b4_solution :=
sorry | fd8317f67388b9ff |
putnam_1982_b4 | abbrev putnam_1982_b4_solution : Prop Γ 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 $i$, $|n_i| = 1$.
\item
If all $n_i$ are positive, then $\{n_1, n_2, \dots, n_s\} = \{1, 2, \dots, s\}$.
\end{enumerate}
-/
theorem putnam_1982_b4
(P : Finset β€ β Prop)
(P_def : β n, P n β n.Nonempty β§ β k, β i in n, i β£ β i in n, (i + k)) :
((β n, P n β 1 β n β¨ -1 β n) β putnam_1982_b4_solution.1) β§
((β n, P n β (β i β n, 0 < i) β n = Finset.Icc (1 : β€) n.card) β putnam_1982_b4_solution.2) :=
sorry | 6d84ab14780e2a0a |
putnam_2007_b4 | abbrev putnam_2007_b4_solution : β β β := sorry
-- fun n β¦ 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 : β) (npos : n > 0) :
({(P, Q) : (Polynomial β) Γ (Polynomial β) | P ^ 2 + Q ^ 2 = Polynomial.X ^ (2 * n) + 1 β§ P.degree > Q.degree}.ncard = putnam_2007_b4_solution n) :=
sorry | 0c44fc2f2b2c0006 |
putnam_2017_b4 | abbrev putnam_2017_b4_solution : β := 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 \cdot \frac{\ln 6}{6} - \frac{\ln 7}{7} \ - \frac{\ln 8}{8} - \frac{\ln 9}{9} + 3 \cdot \frac{\ln 10}{10} - \cdots . \end{gather*} (As usual, $\ln x$ denotes the natural logarithm of $x$.)
-/
theorem putnam_2017_b4 :
(β' k : β, (3 * log (4 * k + 2) / (4 * k + 2) - log (4 * k + 3) / (4 * k + 3) - log (4 * k + 4) / (4 * k + 4) - log (4 * k + 5) / (4 * k + 5)) = putnam_2017_b4_solution) :=
sorry | 71a8b73b2110cd18 |
putnam_1970_b1 | abbrev putnam_1970_b1_solution : β := 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) * β i in Finset.Icc (1 : β€) (2*n), ((n^2 + i^2) : β)^((1 : β)/n)) atTop (π putnam_1970_b1_solution) :=
sorry | dc6a98cc79df4cf9 |
putnam_2014_b2 | abbrev putnam_2014_b2_solution : β := 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 | β f : β β β,
(β x : Set.Icc (1 : β) 3, -1 β€ f x β§ f x β€ 1) β§
(β« x in Set.Ioo 1 3, f x = 0) β§
(β« x in Set.Ioo 1 3, (f x) / x) = t}
putnam_2014_b2_solution :=
sorry | fb03c4d0b8832a77 |
putnam_2004_b2 | theorem putnam_2004_b2
(m n : β)
(mnpos : m > 0 β§ n > 0)
: ((m + n)! / ((m + n) ^ (m + n) : β)) < (((m)! / (m ^ m : β)) * ((n)! / (n ^ n : β))) :=
sorry | 2d7f932132cc4d74 |
putnam_1981_b2 | abbrev putnam_1981_b2_solution : β := 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 : β Γ β Γ β β Prop)
(hP : P = fun (r, s, t) => 1 β€ r β§ r β€ s β§ s β€ t β§ t β€ 4)
(f : β Γ β Γ β β β)
(hf : f = fun (r, s, t) => (r - 1)^2 + (s/r - 1)^2 + (t/s - 1)^2 + (4/t - 1)^2) :
IsLeast {y | β r s t, P (r, s, t) β§ f (r, s, t) = y} putnam_1981_b2_solution :=
sorry | 0d4485b6f60f63fe |
putnam_1991_b2 | theorem putnam_1991_b2
(f g : β β β)
(fgnconst : Β¬β c, f = Function.const β c β¨ g = Function.const β c)
(fgdiff : Differentiable β f β§ Differentiable β g)
(fadd : β x y, f (x + y) = f x * f y - g x * g y)
(gadd : β x y, g (x + y) = f x * g y + g x * f y) :
(deriv f 0 = 0) β (β x, (f x) ^ 2 + (g x) ^ 2 = 1) :=
sorry | 2e924c47b98b8cc8 |
putnam_1986_b6 | theorem putnam_1986_b6
(n : β)
(npos : n > 0)
(F : Type*) [Field F]
(A B C D : Matrix (Fin n) (Fin n) F)
(hsymm : IsSymm (A * Bα΅) β§ IsSymm (C * Dα΅))
(hid : A * Dα΅ - B * Cα΅ = 1)
: (Aα΅ * D - Cα΅ * B = 1) :=
sorry | 6baaf2b59e0c65dd |
putnam_2003_b6 | theorem putnam_2003_b6
(f : β β β)
(hf : Continuous f)
: (β« x in (0 : β)..1, (β« y in (0 : β)..1, |f x + f y|)) β₯ (β« x in (0 : β)..1, |f x|) :=
sorry | d9614a6ce849c09d |
putnam_1974_b3 | theorem putnam_1974_b3
(Ξ± : β)
(ha : Real.cos (Real.pi * Ξ±) = (1 : β)/3)
: Irrational Ξ± :=
sorry | 812353adb16a64d9 |
putnam_1999_a5 | theorem putnam_1999_a5
: β C : β, β p : Polynomial β, p.degree = 1999 β βp.eval 0β β€ C * β« x in (-1)..1, βp.eval xβ :=
sorry | d6ec809a198deb98 |
putnam_1964_b3 | theorem putnam_1964_b3
(f : β β β)
(hf : Continuous f β§ β Ξ± > 0, Tendsto (fun n : β β¦ f (n * Ξ±)) atTop (π 0))
: (Tendsto f atTop (π 0)) :=
sorry | 2f180c29c45a0c68 |
putnam_2021_a2 | abbrev putnam_2021_a2_solution : β := 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 : β β β)
(hg : β x > 0, Tendsto (fun r : β => ((x + 1) ^ (r + 1) - x ^ (r + 1)) ^ (1 / r)) (π[>] 0) (π (g x)))
: Tendsto (fun x : β => g x / x) atTop (π putnam_2021_a2_solution) :=
sorry | ec3fd514b569ad4c |
putnam_2017_a2 | theorem putnam_2017_a2
(Q : β β RatFunc β)
(hQbase : Q 0 = 1 β§ Q 1 = (X : β[X]))
(hQn : β n, Q (n + 2) = (Q (n + 1) ^ 2 - 1) / Q n)
(n : β) (hn : 0 < n) :
β P : β€[X], Q n = P.map (Int.castRingHom β) :=
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 them is contained in $S$.)
-/
theorem putnam_2007_a2 :
IsLeast
{y | β S : Set (Fin 2 β β),
Convex β S β§
(β p β S, p 0 > 0 β§ p 1 > 0 β§ p 0 * p 1 = 1) β§
(β p β S, p 0 < 0 β§ p 1 < 0 β§ p 0 * p 1 = 1) β§
(β p β S, p 0 < 0 β§ p 1 > 0 β§ p 0 * p 1 = -1) β§
(β p β S, p 0 > 0 β§ p 1 < 0 β§ p 0 * p 1 = -1) β§
volume S = y} putnam_2007_a2_solution :=
sorry | 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 : β β β β β)
(hB : B = fun (n : β) (x : β) β¦ β k in Finset.Icc 1 n, (k : β) ^ x)
(f : β β β)
(hf : f = fun n β¦ B n (logb n 2) / (n * logb 2 n) ^ 2)
: (β L : β, Tendsto (fun N β¦ β j in Finset.Icc 2 N, f j) atTop (π L)) β putnam_1982_a2_solution :=
sorry | c7dab1b8de51aa04 |
putnam_1992_a2 | abbrev putnam_1992_a2_solution : β := 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 : β β β)
(hC : C = fun Ξ± β¦ taylorCoeffWithin (fun x β¦ (1 + x) ^ Ξ±) 1992 Set.univ 0)
: (β« y in (0)..1, C (-y - 1) * β k in Finset.Icc (1 : β) 1992, 1 / (y + k) = putnam_1992_a2_solution) :=
sorry | 24f79b1bba75a222 |
putnam_2008_b1 | abbrev putnam_2008_b1_solution : β := sorry
-- 2
def is_rational_point (p : Fin 2 β β) : Prop := β (a b : β), a = p 0 β§ b = p 1
def real_circle (c : EuclideanSpace β (Fin 2)) (r : β) : Set (EuclideanSpace β (Fin 2)) := {p | dist p c = r}
/--
What is the maximum number of rational points that can lie on a circle in $\mathbb{R}^2$ whose center is not a rational point? (A \emph{rational point} is a point both of whose coordinates are rational numbers.)
-/
theorem putnam_2008_b1 :
IsGreatest {n : β | β (c : EuclideanSpace β (Fin 2)) (r : β), Β¬ is_rational_point c β§ (Set.ncard {p : EuclideanSpace β (Fin 2) | p β real_circle c r β§ is_rational_point p} = n)} putnam_2008_b1_solution :=
sorry | 5f802f232611746d |
putnam_2018_b1 | abbrev putnam_2018_b1_solution : Set (Fin 2 β β€) := sorry
-- {v : Fin 2 β β€ | β b : β€, 0 β€ b β§ b β€ 100 β§ Even b β§ v = ![1, b]}
/--
Let $\mathcal{P}$ be the set of vectors defined by $\mathcal{P}=\left\{\left.\begin{pmatrix} a \\ b \end{pmatrix}\right| 0 \leq a \leq 2, 0 \leq b \leq 100,\text{ and }a,b \in \mathbb{Z}\right\}$. Find all $\mathbf{v} \in \mathcal{P}$ such that the set $\mathcal{P} \setminus \{\mathbf{v}\}$ obtained by omitting vector $\mathbf{v}$ from $\mathcal{P}$ can be partitioned into two sets of equal size and equal sum.
-/
theorem putnam_2018_b1
(P Pvdiff : Finset (Fin 2 β β€))
(v : Fin 2 β β€)
(hP : P = {v' : Fin 2 β β€ | 0 β€ v' 0 β§ v' 0 β€ 2 β§ 0 β€ v' 1 β§ v' 1 β€ 100})
(hPvdiff : Pvdiff = P \ ({v} : Finset (Fin 2 β β€)))
: (v β P β§ (β Q R : Finset (Fin 2 β β€),
(Q βͺ R = Pvdiff) β§ (Q β© R = β
) β§ (Q.card = R.card) β§
(β q in Q, q 0 = β r in R, r 0) β§ (β q in Q, q 1 = β r in R, r 1)))
β v β putnam_2018_b1_solution :=
sorry | 564890e75c8a6a06 |
putnam_1985_a6 | abbrev putnam_1985_a6_solution : Polynomial β := 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 such that
\begin{enumerate}
\item[(i)] $g(0)=1$, and
\item[(ii)] $\Gamma(f(x)^n) = \Gamma(g(x)^n)$
\end{enumerate}
for every integer $n \geq 1$.
-/
theorem putnam_1985_a6
(Ξ : Polynomial β β β)
(f : Polynomial β)
(hΞ : Ξ = fun p β¦ β k in Finset.range (p.natDegree + 1), coeff p k ^ 2)
(hf : f = 3 * X ^ 2 + 7 * X + 2) :
let g := putnam_1985_a6_solution;
g.eval 0 = 1 β§ β n : β, n β₯ 1 β Ξ (f ^ n) = Ξ (g ^ n) :=
sorry | 920087e2a1d120c5 |
putnam_1995_a6 | theorem putnam_1995_a6
(SM : (n : β) β Set (Matrix (Fin 3) (Fin n) β€))
(Srowsums : (n : β) β Matrix (Fin 3) (Fin n) β€ β Set β€)
(sumsprog : (Set β€) β Prop)
(hSM : β n, SM n = {M : Matrix (Fin 3) (Fin n) β€ | β j : Fin n, {M i j | i : Fin 3} = {1, 2, 3}})
(hSrowsums : β n M, Srowsums n M = {β j : Fin n, M i j | i : Fin 3})
(hsumsprog : β sums, sumsprog sums β sums.encard = 3 β§ (β a b c : β€, {a, b, c} = sums β§ b = a + 1 β§ c = a + 2))
: β n β₯ 1995, {M : (SM n) | sumsprog (Srowsums n M)}.encard β₯ 4 * {M : (SM n) | (Srowsums n M).encard = 1}.encard :=
sorry | ebeee300772c5c80 |
putnam_1977_a3 | abbrev putnam_1977_a3_solution : (β β β) β (β β β) β (β β β) := sorry
-- fun f g x β¦ 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 $g(x) = \frac{h(x + 4) + h(x - 4)}{2}$.
-/
theorem putnam_1977_a3
(f g h : β β β)
(hf : β x, f x = (h (x + 1) + h (x - 1)) / 2)
(hg : β x, g x = (h (x + 4) + h (x - 4)) / 2) :
h = putnam_1977_a3_solution f g :=
sorry | 29708f4d539fb450 |
putnam_1967_a3 | abbrev putnam_1967_a3_solution : β := 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
{a | β P : Polynomial β€,
P.degree = 2 β§
(β z1 z2 : Set.Ioo (0 : β) 1, z1 β z2 β§ aeval (z1 : β) P = 0 β§ aeval (z2 : β) P = 0) β§
P.coeff 2 = a β§ a > 0}
putnam_1967_a3_solution :=
sorry | a1393afbd533eef4 |
putnam_2022_b2 | abbrev putnam_2022_b2_solution : Set β := 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 : β)
(P : Finset (Fin 3 β β) β Prop)
(P_def : β S : Finset (Fin 3 β β), P S β (S = {u : Fin 3 β β | β v w : S, u = crossProduct v w})) :
(0 < n β§ β S : Finset (Fin 3 β β), S.card = n β§ P S) β n β putnam_2022_b2_solution :=
sorry | 22a37e28d74ba943 |
putnam_2010_a6 | theorem putnam_2010_a6
(f : β β β)
(hf : (StrictAntiOn f (Ici 0) β§ ContinuousOn f (Ici 0) β§ Tendsto f atTop (π 0)))
: Β¬β y : β, Tendsto (fun z : β => β« x in Ico 0 z, ((f x) - f (x + 1))/(f x)) atTop (π y) :=
sorry | 9f8b7b28c47dbfae |
putnam_2000_a6 | theorem putnam_2000_a6
(f : Polynomial β€)
(a : β β β€)
(ha0 : a 0 = 0)
(ha : β n : β, a (n + 1) = f.eval (a n))
: ((β m > 0, a m = 0) β (a 1 = 0 β¨ a 2 = 0)) :=
sorry | 120b2bcf4011fc6a |
putnam_1989_b3 | abbrev putnam_1989_b3_solution : β β β β β := sorry
-- fun n c β¦ c * n ! / (3 ^ n * β m in Finset.Icc (1 : β€) 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 to $0$ as $x$ increases). For $n$ a non-negative integer, define
\[
\mu_n=\int_0^\infty x^n f(x)\,dx
\]
(sometimes called the $n$th moment of $f$).
\begin{enumerate}
\item[a)] Express $\mu_n$ in terms of $\mu_0$.
\item[b)] Prove that the sequence $\{\mu_n \frac{3^n}{n!}\}$ always converges, and that the limit is $0$ only if $\mu_0=0$.
\end{enumerate}
-/
theorem putnam_1989_b3
(f : β β β)
(hfdiff : Differentiable β f)
(hfderiv : β x > 0, deriv f x = -3 * f x + 6 * f (2 * x))
(hdecay : β x β₯ 0, |f x| β€ Real.exp (- βx))
(ΞΌ : β β β)
(ΞΌ_def : β n, ΞΌ n = β« x in Set.Ioi 0, x ^ n * f x) :
(β n, ΞΌ n = putnam_1989_b3_solution n (ΞΌ 0)) β§
(β L, Tendsto (fun n β¦ (ΞΌ n) * 3 ^ n / n !) atTop (π L)) β§
(Tendsto (fun n β¦ (ΞΌ n) * 3 ^ n / n !) atTop (π 0) β ΞΌ 0 = 0) :=
sorry | 2642743efb382ade |
putnam_2021_b4 | theorem putnam_2021_b4
(F : β β β)
(hF : β n, F (n + 2) = F (n + 1) + F n)
(F01 : F 0 = 0 β§ F 1 = 1)
: β m, m > 2 β (β p, (β 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 : (β β β) β Prop)
(hpa : β a, pa a β (β n : β, a n > 0) β§ β L : β, Tendsto (fun N β¦ β n in Finset.range N, 1 / a n) atTop (π L)) :
β k : β, β a : β β β, pa a β
β' n : β, (n + 1) / (β i in Finset.range (n + 1), a i) β€ k * β' n : β, 1 / a n :=
sorry | bae539d8d621345d |
putnam_1999_b3 | abbrev putnam_1999_b3_solution : β := 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 \[\lim_{(x,y)\rightarrow (1,1), (x,y)\in A} (1-xy^2)(1-x^2y)S(x,y).\]
-/
theorem putnam_1999_b3
(A : Set (β Γ β))
(hA : A = {xy | 0 β€ xy.1 β§ xy.1 < 1 β§ 0 β€ xy.2 β§ xy.2 < 1})
(S : β β β β β)
(hS : S = fun x y => β' m : β, β' n : β, if (m > 0 β§ n > 0 β§ 1/2 β€ m/n β§ m/n β€ 2) then x^m * y^n else 0)
: Tendsto (fun xy : (β Γ β) => (1 - xy.1 * xy.2^2) * (1 - xy.1^2 * xy.2) * (S xy.1 xy.2)) (π[A] β¨1,1β©) (π putnam_1999_b3_solution) :=
sorry | 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 $D_i$ is finite, and
\item every line in the plane intersects at least one of the $D_i$?
\end{enumerate}
-/
theorem putnam_1991_a4 :
(β (c : β β EuclideanSpace β (Fin 2)) (r : β β β),
(Β¬ β p, MapClusterPt p atTop c) β§
(Summable <| fun i β¦ r i ^ 2) β§
(β L : AffineSubspace β (EuclideanSpace β (Fin 2)),
Module.finrank β L.direction = 1 β β i, (βL β© closedBall (c i) (r i)).Nonempty)) β
putnam_1991_a4_solution :=
sorry | 7c6dd778d033673e |
putnam_1973_a1 | theorem putnam_1973_a1
(A B C X Y Z : EuclideanSpace β (Fin 2))
(hnoncol : Β¬Collinear β {A, B, C})
(hX : X β segment β B C)
(hY : Y β segment β C A)
(hZ : Z β segment β A B)
: ((dist B X β€ dist X C β§ dist C Y β€ dist Y A β§ dist A Z β€ dist Z B) β
volume (convexHull β {X, Y, Z}) β₯ (1/4) * volume (convexHull β {A, B, C})) β§
sInf {volume (convexHull β {A, Z, Y}), volume (convexHull β {B, X, Z}), volume (convexHull β {C, Y, X})} β€ volume (convexHull β {X, Y, Z}) :=
sorry | 510975ecc30f9bb9 |
putnam_2004_a4 | theorem putnam_2004_a4
(n : β) (npos : n > 0)
(x : Fin n β β)
(avals : β β (β β Fin n β β) β Prop)
(havals : β N a, avals N a β β (i : Fin N) (j : Fin n), (a i j = -1 β¨ a i j = 0 β¨ a i j = 1)) :
β (N : β) (c : Fin N β β) (a : β β Fin n β β),
avals N a β§
(β i : Fin n, x i) = β i : Fin N, c i * (β j : Fin n, a i j * x j) ^ n :=
sorry | 7cb0a37e7d1b26fc |
putnam_2014_a4 | abbrev putnam_2014_a4_solution : β := 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 possible value of the probability of the event $X=0$.
-/
theorem putnam_2014_a4
(Xrandvar : (β β β) β Prop)
(hXrandvar : Xrandvar = (fun X : β β β => (β n : β, X n β Set.Icc 0 1) β§ β' n : β, X n = 1))
(E : (β β β) β (β β β) β β)
(hE : E = (fun (X : β β β) (f : β β β) => β' n : β, f n * X n))
: sInf {X0 : β | β X : β β β, Xrandvar X β§ E X (fun x : β => x) = 1 β§ E X (fun x : β => x ^ 2) = 2 β§ E X (fun x : β => x ^ 3) = 5 β§ X0 = X 0} = putnam_2014_a4_solution :=
sorry | 97876766472c6fe4 |
putnam_1987_a1 | theorem putnam_1987_a1
(A B C D : Set (β Γ β))
(hA : A = {(x, y) : β Γ β | x ^ 2 + y ^ 2 β 0 β§ x ^ 2 - y ^ 2 = x / (x ^ 2 + y ^ 2)})
(hB : B = {(x, y) : β Γ β | x ^ 2 + y ^ 2 β 0 β§ 2 * x * y + y / (x ^ 2 + y ^ 2) = 3})
(hC : C = {(x, y) : β Γ β | x ^ 3 - 3 * x * y ^ 2 + 3 * y = 1})
(hD : D = {(x, y) : β Γ β | 3 * x ^ 2 * y - 3 * x - y ^ 3 = 0})
: A β© B = C β© D := sorry | 23254f3dd993dacf |
putnam_2012_a1 | theorem putnam_2012_a1
(d : Fin 12 β β)
(hd : β i, d i β Set.Ioo 1 12)
: β i j k, i β j β§ i β k β§ j β k β§ d k β₯ d i β§ d k β₯ d j β§ (d i) ^ 2 + (d j) ^ 2 > (d k) ^ 2 :=
sorry | c05b63c7a82a9beb |
putnam_2002_a1 | abbrev putnam_2002_a1_solution : β β β β β := sorry
-- (fun k n : β => (-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 : β)
(P : β β Polynomial β)
(kpos : k > 0)
(Pderiv : β n x, iteratedDeriv n (fun x' : β => 1 / (x' ^ k - 1)) x = ((P n).eval x) / ((x ^ k - 1) ^ (n + 1)))
: β n, (P n).eval 1 = putnam_2002_a1_solution k n :=
sorry | 690aeb9eb356d8d6 |
putnam_1975_a4 | abbrev putnam_1975_a4_solution : β β Polynomial β€ := sorry
-- fun m => β 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 $\theta$ \[ a_k \theta^k + a_{k-1}\theta^{k-1} + \dots + a_1\theta + a_0\], with integer coefficients $a_i$.
-/
theorem putnam_1975_a4
(m : β)
(hm : Odd m β§ m > 1)
(ΞΈ : β)
(hΞΈ : ΞΈ = cexp (2 * Real.pi * I / (2 * m)))
: 1/(1 - ΞΈ) = Polynomial.aeval ΞΈ (putnam_1975_a4_solution m) :=
sorry | 42ef33d7d1a63b0f |
putnam_1998_b2 | abbrev putnam_1998_b2_solution : β β β β β := 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 perimeter exists.
-/
theorem putnam_1998_b2
(a b : β)
(hab : 0 < b β§ b < a)
: sInf {d : β | β (c : β) (x : β), d = Real.sqrt ((a - c)^2 + (b - 0)^2) + Real.sqrt ((c - x)^2 + (0 - x)^2) + Real.sqrt ((a - x)^2 + (b - x)^2) β§
Real.sqrt ((a - c)^2 + (b - 0)^2) + Real.sqrt ((c - x)^2 + (0 - x)^2) > Real.sqrt ((a - x)^2 + (b - x)^2) β§
Real.sqrt ((a - c)^2 + (b - 0)^2) + Real.sqrt ((a - x)^2 + (b - x)^2) > Real.sqrt ((c - x)^2 + (0 - x)^2) β§
Real.sqrt ((c - x)^2 + (0 - x)^2) + Real.sqrt ((a - x)^2 + (b - x)^2) > Real.sqrt ((a - c)^2 + (b - 0)^2)}
= putnam_1998_b2_solution a b :=
sorry | 76a5b677756fc36f |
putnam_1965_a4 | theorem putnam_1965_a4
{G B : Type*}
[Fintype G] [Nonempty G]
[Fintype B] [Nonempty B]
(dances : G β B β Prop)
(h : (Β¬β b : B, β g : G, dances g b) β§ β g : G, β b : B, dances g b)
: β g h : G, β b c : B, dances g b β§ dances h c β§ Β¬dances h b β§ Β¬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
: (β x y : β, (y β₯ 0 β§ y * (y + 1) β€ (x + 1) ^ 2) β (y * (y - 1) β€ x ^ 2)) β putnam_1988_b2_solution :=
sorry | 700ca10252fd3ed5 |
putnam_2020_b5 | theorem putnam_2020_b5
(z : Fin 4 β β)
(hzle1 : β n, βz nβ = 1)
(hzne1 : β n, z n β 1)
: 3 - z 0 - z 1 - z 2 - z 3 + (z 0) * (z 1) * (z 2) * (z 3) β 0:=
sorry | a284f1a6ea86518a |
putnam_2015_a5 | theorem putnam_2015_a5
(q : β)
(Nq : β)
(qodd : Odd q)
(qpos : q > 0)
(hNq : Nq = {a : β | 0 < a β§ a < (q : β) / 4 β§ Nat.gcd a q = 1}.encard)
: Odd Nq β β p k : β, q = p ^ k β§ k > 0 β§ p.Prime β§ (p % 8 = 5 β¨ p % 8 = 7) :=
sorry | 267a3f6aa3488410 |
putnam_2005_a5 | abbrev putnam_2005_a5_solution : β := sorry
-- Real.pi * (Real.log 2) / 8
/--
Evaluate $\int_0^1 \frac{\ln(x+1)}{x^2+1}\,dx$.
-/
theorem putnam_2005_a5 :
β« x in (0:β)..1, (Real.log (x+1))/(x^2 + 1) = putnam_2005_a5_solution :=
sorry | 3cb3373ce77445e8 |
putnam_1980_a5 | theorem putnam_1980_a5
(P : Polynomial β)
(Pnonconst : P.degree > 0) :
Set.Finite {x : β |
0 = (β« t in (0)..x, P.eval t * Real.sin t) β§
0 = (β« 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 β
(β n β₯ 1, β A B : Matrix (Fin n) (Fin n) β,
A * B * A * B = 0 β B * A * B * A = 0) :=
sorry | d85e9ab4cc895b8e |
putnam_1993_a3 | theorem putnam_1993_a3
(c : β β β β β)
(hc : β n β₯ 1, β m β₯ 1, c n m = {f : Finset (Fin n) β Fin m | β A B : Finset (Fin n), f (A β© B) = min (f A) (f B)}.encard)
: β n β₯ 1, β m β₯ 1, c n m = β' j : Set.Icc 1 m, (j : β€) ^ n :=
sorry | e01f69df5f1aec80 |
putnam_1983_a3 | theorem putnam_1983_a3
(p : β)
(F : β β β)
(poddprime : Odd p β§ p.Prime)
(hF : β n : β, F n = β i in Finset.range (p - 1), (i + 1) * n ^ i)
: β a β Finset.Icc 1 p, β b β Finset.Icc 1 p, a β b β Β¬(F a β‘ F b [MOD p]) :=
sorry | 066ed7591ccb30cb |
putnam_2006_a3 | theorem putnam_2006_a3
(x : β€ β β€)
(hxlo : β k : β€, 0 β€ k β§ k β€ 2006 β x k = k)
(hxhi : β k : β€, k β₯ 2006 β x (k + 1) = x k + x (k - 2005))
: (β i : β, i > 0 β§ β j : Finset.range 2005, 2006 β£ x (i + j)) :=
sorry | 3d5bd9427d1bb632 |
putnam_2016_a3 | abbrev putnam_2016_a3_solution : β := 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
\[
\int_0^1 f(x)\,dx.
\]
-/
theorem putnam_2016_a3
(f : β β β)
(hf : β x : β, x β 0 β f x + f (1 - 1 / x) = arctan x)
: (β« x in (0)..1, f x = putnam_2016_a3_solution) :=
sorry | a360124adaf6920b |
putnam_1971_a6 | theorem putnam_1971_a6
(c : β)
(hc : β n : β€, n > 0 β β m : β€, (n : β)^c = m)
: β m : β€, m β₯ 0 β§ c = m :=
sorry | 494c27a38e1c0165 |
putnam_2023_b3 | abbrev putnam_2023_b3_solution : β β β := sorry
-- (fun n : β β¦ (2 * (n : β) + 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 uniform distribution on $[0,1]$. Let $a(X_1, X_2, \ldots, X_n)$ be the largest value of $k$ for which there exists an increasing sequence of integers $i_1, i_2, \ldots, i_k$ such that $X_{i_1}, X_{i_2}, \ldots, X_{i_k}$ is zigzag. Find the expected value of $a(X_1, X_2, \ldots, X_n)$ for $n \ge 2$.
-/
theorem putnam_2023_b3
(IsZigZag : {k : β} β (Fin k β β) β Prop)
(IsZigZag_def : β (k : β) [NeZero k] (y : Fin k β β),
IsZigZag y β k = 1 β¨ ((β i, i + 1 < k β y (i + 1) β y i)) β§
(β i, i + 2 < k β (y (i + 2) < y (i + 1) β y i < y (i + 1))))
(n : β)
(hn : 2 β€ n)
(a : (Fin n β Icc (0 : β) 1) β β)
(ha : β x, IsGreatest {k | β i : Fin k βͺo Fin n, IsZigZag ((β) β x β i)} (a x)) :
πΌ[(β) β a] = putnam_2023_b3_solution n :=
sorry | 53a5afe7c805acaa |
putnam_1966_a2 | theorem putnam_1966_a2
(r : β)
(A B C : EuclideanSpace β (Fin 2))
(hABC : Β¬Collinear β {A, B, C})
(a b c p : β)
(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 : β I : EuclideanSpace β (Fin 2),
(β! P : EuclideanSpace β (Fin 2), dist I P = r β§ Collinear β {P, B, C}) β§
(β! Q : EuclideanSpace β (Fin 2), dist I Q = r β§ Collinear β {Q, C, A}) β§
(β! R : EuclideanSpace β (Fin 2), dist I R = r β§ Collinear β {R, A, B}) β§
(β Z : EuclideanSpace β (Fin 2), dist I Z β€ r β Z β convexHull β {A, B, C}))
: 1/(p - a)^2 + 1/(p - b)^2 + 1/(p - c)^2 β₯ 1/r^2 :=
sorry | a32f88b0f5565572 |
putnam_1976_a2 | theorem putnam_1976_a2
(P Q : MvPolynomial (Fin 2) β€)
(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 : β β MvPolynomial (Fin 2) β€)
(hF : β n, F n = (X 0 + X 1) ^ n - X 0 ^ n - X 1 ^ n)
(hG : β n, G n = (X 0 + X 1) ^ n + X 0 ^ n + X 1 ^ n)
(n : β)
(hn : 0 < n) :
β A : MvPolynomial (Fin 2) β€,
F n = aeval ![P, Q] A β¨
G n = aeval ![P, Q] A :=
sorry | 6ebee7bcee66f89c |
putnam_1969_b1 | theorem putnam_1969_b1
(n : β)
(hnpos : n > 0)
(hn : 24 β£ n + 1)
: 24 β£ β d in divisors n, d :=
sorry | bbdc3aeb399fc02d |
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