name stringlengths 7 15 | statement stringlengths 75 1.66k | id stringlengths 16 16 |
|---|---|---|
putnam_1977_a4 | abbrev putnam_1977_a4_solution : RatFunc β := sorry
-- RatFunc.X / (1 - RatFunc.X)
/--
Find $\sum_{n=0}^{\infty} \frac{x^{2^n}}{1 - x^{2^{n+1}}}$ as a rational function of $x$ for $x \in (0, 1)$.
-/
theorem putnam_1977_a4 :
β x β Ioo 0 1,
putnam_1977_a4_solution.eval (id β) x = β' n : β, x ^ 2 ^ n / (1 - x ^ ... | bdfc699256ecc967 |
putnam_2000_a1 | abbrev putnam_2000_a1_solution : β β Set β := sorry
-- (fun A : β => Set.Ioo 0 (A ^ 2))
/--
Let $A$ be a positive real number. What are the possible values of $\sum_{j=0}^\infty x_j^2$, given that $x_0,x_1,\ldots$ are positive numbers for which $\sum_{j=0}^\infty x_j=A$?
-/
theorem putnam_2000_a1
(A : β)
(Apos : A > 0)... | 7ea2b76275b22c17 |
putnam_2010_a1 | abbrev putnam_2010_a1_solution : β β β := sorry
-- (fun n : β => Nat.ceil ((n : β) / 2))
/--
Given a positive integer $n$, what is the largest $k$ such that the numbers $1,2,\dots,n$ can be put into $k$ boxes so that the sum of the numbers in each box is the same? [When $n=8$, the example $\{1,2,3,6\},\{4,8\},\{5,7\}$ ... | 487e03787f69ef74 |
putnam_1995_a1 | theorem putnam_1995_a1
(S : Set β)
(hS : β a β S, β b β S, a * b β S)
(T U : Set β)
(hsub : T β S β§ U β S)
(hunion : T βͺ U = S)
(hdisj : T β© U = β
)
(hT3 : β a β T, β b β T, β c β T, a * b * c β T)
(hU3 : β a β U, β b β U, β c β U, a * b * c β U)
: (β a β T, β b β T, a * b β T) β¨ (β a β U, β b β U, a * b β U) :=
sorry | 9815d3d1b44e9620 |
putnam_1985_a1 | abbrev putnam_1985_a1_solution : β Γ β Γ β Γ β := sorry
-- (10, 10, 0, 0)
/--
Determine, with proof, the number of ordered triples $(A_1, A_2, A_3)$ of sets which have the property that
\begin{enumerate}
\item[(i)] $A_1 \cup A_2 \cup A_3 = \{1,2,3,4,5,6,7,8,9,10\}$, and
\item[(ii)] $A_1 \cap A_2 \cap A_3 = \emptyset$.
... | eb93f1dfaf2655d6 |
putnam_1992_a5 | theorem putnam_1992_a5
(a : β β β)
(ha : a = fun n β¦ ite (Even {i | (digits 2 n).get i = 1}.ncard) 0 1) :
Β¬β k > 0, β m > 0, β j β€ m - 1,
a (k + j) = a (k + m + j) β§ a (k + m + j) = a (k + 2 * m + j) :=
sorry | 2fb48dc873839c8d |
putnam_1982_a5 | theorem putnam_1982_a5
(a b c d : β€)
(hpos : a > 0 β§ b > 0 β§ c > 0 β§ d > 0)
(hac : a + c β€ 1982)
(hfrac : (a : β) / b + (c : β) / d < 1)
: (1 - (a : β) / b - (c : β) / d > 1 / 1983 ^ 3) :=
sorry | cb8f331474bf0bfa |
putnam_2018_b6 | theorem putnam_2018_b6
(S : Finset (Fin 2018 β β€))
(hS : S = {s : Fin 2018 β β€ | (β i : Fin 2018, s i β ({1, 2, 3, 4, 5, 6, 10} : Set β€)) β§ (β i : Fin 2018, s i) = 3860}) :
S.card β€ 2 ^ 3860 * ((2018 : β) / 2048) ^ 2018 :=
sorry | 76242f65e0cf821a |
putnam_2008_b6 | theorem putnam_2008_b6
(n k : β)
(hnk : n > 0 β§ k > 0)
: Odd (Set.ncard {s : Equiv.Perm (Fin n) | klimited k n s}) β (n β‘ 0 [MOD 2*k+1] β¨ n β‘ 1 [MOD 2*k+1]) :=
sorry | 6faf0f60f5485143 |
putnam_2007_a5 | theorem putnam_2007_a5
(G : Type*) [Group G] [Fintype G]
(p n : β)
(hp : Nat.Prime p)
(hn : n = {g : G | orderOf g = p}.encard)
: n = 0 β¨ p β£ (n + 1) :=
sorry | b1852b36feead356 |
putnam_1970_a1 | theorem putnam_1970_a1
(a b : β)
(ha : a > 0)
(hb : b > 0)
(f : β β β)
(f_def : f = fun x : β => Real.exp (a*x) * Real.cos (b*x))
(p : β β β)
(hp : β c : β, c > 0 β§ β x β ball 0 c, β' n : β, (p n)*x^n = f x)
(S : Set β)
(S_def : S = {n : β | p n = 0})
: S = β
β¨ Β¬Finite S :=
sorry | 4860dce88963b435 |
putnam_2007_a4 | abbrev putnam_2007_a4_solution : Set (Polynomial β) := sorry
-- {f : Polynomial β | β d : β, β c β₯ (1 : β€) - d, β n : β, f.eval n = (1 / 9) * ((10 ^ c) * (9 * n + 1) ^ d - 1)}
/--
A \emph{repunit} is a positive integer whose digits in base 10 are all ones. Find all polynomials $f$ with real coefficients such that if $n... | b6a7b679b7fb5613 |
putnam_2017_a4 | theorem putnam_2017_a4
(N : β)
(score : Fin (2 * N) β Fin 11)
(hsurj : β k : Fin 11, β i : Fin (2 * N), score i = k)
(havg : (β i : Fin (2 * N), (score i : β)) / (2 * N) = 7.4)
: (β s : Finset (Fin (2 * N)), s.card = N β§ (β i in s, (score i : β)) / N = 7.4 β§ (β i in sαΆ, (score i : β)) / N = 7.4) :=
sorry | dbffa30c1b994a53 |
putnam_1992_a4 | abbrev putnam_1992_a4_solution : β β β := sorry
-- fun k β¦ ite (Even k) ((-1) ^ (k / 2) * factorial k) 0
/--
Let $f$ be an infinitely differentiable real-valued function defined on the real numbers. If
\[
f\left( \frac{1}{n} \right) = \frac{n^2}{n^2 + 1}, \qquad n = 1, 2, 3, \dots,
\]
compute the values of the derivati... | 3729b5784bfbe17a |
putnam_1982_a4 | theorem putnam_1982_a4
(hdiffeq : (β β β) β (β β β) β Prop)
(hdiffeq_def : β y z,
hdiffeq y z β
y 0 = 1 β§ z 0 = 0 β§
ContDiff β 1 y β§ ContDiff β 1 z β§
(β x : β, deriv y x = -1 * (z x)^3 β§ deriv z x = (y x)^3))
(f g : β β β)
(hfgsat : hdiffeq f g)
(hfgonly : Β¬(β f' g' : β... | 66deacfe34ae105d |
putnam_1978_b6 | theorem putnam_1978_b6
(a : β β β β β)
(ha : β i j, a i j β Icc 0 1)
(m n : β)
(mnpos : m > 0 β§ n > 0)
: ((β i in Finset.Icc 1 n, β j in Finset.Icc 1 (m * i), a i j / i) ^ 2 β€ 2 * m * β i in Finset.Icc 1 n, β j in Finset.Icc 1 (m * i), a i j) :=
sorry | c872e9c2972df93f |
putnam_1968_b6 | theorem putnam_1968_b6
: Β¬β K : β β Set β, (β n : β, IsCompact (K n)) β§ (β S : Set β, IsCompact S β β n : β, S β K n) :=
sorry | e17610b10277d891 |
putnam_1967_a5 | theorem putnam_1967_a5
(R : Set (EuclideanSpace β (Fin 2)))
(hR : Convex β R β§ (MeasureTheory.volume R).toReal > Real.pi / 4)
: β P β R, β Q β R, dist P Q = 1 :=
sorry | 7ae2860077304c40 |
putnam_2022_b4 | abbrev putnam_2022_b4_solution : Set β := sorry
-- {n : β | 3 β£ n β§ n β₯ 9}
/--
Find all integers $n$ with $n \geq 4$ for which there exists a sequence of distinct real numbers $x_1,\dots,x_n$ such that each of the sets $\{x_1,x_2,x_3\},\{x_2,x_3,x_4\},\dots,\{x_{n-2},x_{n-1},x_n\},\{x_{n-1},x_n,x_1\}$, and $\{x_n,x_1,x... | de91bdf662d58ef8 |
putnam_1977_a5 | theorem putnam_1977_a5
(p m n : β)
(hp : Nat.Prime p)
(hmgen : m β₯ n)
: (choose (p * m) (p * n) β‘ choose m n [MOD p]) :=
sorry | 978cf504d604cc5a |
putnam_1999_b5 | abbrev putnam_1999_b5_solution : β β β := sorry
-- fun n => 1 - n^2/4
/--
For an integer $n\geq 3$, let $\theta=2\pi/n$. Evaluate the determinant of the $n\times n$ matrix $I+A$, where $I$ is the $n\times n$ identity matrix and $A=(a_{jk})$ has entries $a_{jk}=\cos(j\theta+k\theta)$ for all $j,k$.
-/
theorem putnam_19... | 04a793fbf6f644e9 |
asserts | abbrev putnam_1974_a3_solution : (Set β) Γ (Set β) := sorry
-- ({p : β | p.Prime β§ p β‘ 1 [MOD 8]}, {p : β | p.Prime β§ p β‘ 5 [MOD 8]})
/--
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 eac... | db78fe08287f5cff |
putnam_2021_b2 | abbrev putnam_2021_b2_solution : β := sorry
-- 2 / 3
/--
Determine the maximum value of the sum $S = \sum_{n=1}^\infty \frac{n}{2^n}(a_1a_2 \dots a_n)^{1/n}$ over all sequences $a_1,a_2,a_3,\dots$ of nonnegative real numbers satisfying $\sum_{k=1}^\infty a_k=1$.
-/
theorem putnam_2021_b2 :
IsGreatest
{S | β a... | 222bc354760bf00c |
putnam_1964_a3 | theorem putnam_1964_a3
(x a b : β β β)
(hxdense : range x β Ioo 0 1 β§ closure (range x) β Ioo 0 1)
(hxinj : Injective x)
(ha : a = fun n β¦ x n - sSup ({0} βͺ {p : β | p < x n β§ β i < n, p = x i}))
(hb : b = fun n β¦ sInf ({1} βͺ {p : β | p > x n β§ β i < n, p = x i}) - x n)
: (β' n : β, a n * b n * (a n + b n) = 1 / 3) :=
... | 59f3b015b84783b4 |
putnam_2013_a6 | theorem putnam_2013_a6
(w : β€ β β€ β β€)
(A : Finset (β€ Γ β€) β β€)
(hwn1 : w (-2) (-2) = -1 β§ w 2 (-2) = -1 β§ w (-2) 2 = -1 β§ w 2 2 = -1)
(hwn2 : w (-1) (-2) = -2 β§ w 1 (-2) = -2 β§ w (-2) (-1) = -2 β§ w 2 (-1) = -2 β§ w (-2) 1 = -2 β§ w 2 1 = -2 β§ w (-1) 2 = -2 β§ w 1 2 = -2)
(hw2 : w 0 (-2) = 2 β§ w (-2) 0 = 2 β§ w 2... | 97ee3ec31151a83b |
putnam_2003_a6 | abbrev putnam_2003_a6_solution : Prop := sorry
-- True
/--
For a set $S$ of nonnegative integers, let $r_S(n)$ denote the number of ordered pairs $(s_1,s_2)$ such that $s_1 \in S$, $s_2 \in S$, $s_1 \ne s_2$, and $s_1+s_2=n$. Is it possible to partition the nonnegative integers into two sets $A$ and $B$ in such a way t... | 1db2d9bacbe1fc95 |
putnam_1986_a6 | abbrev putnam_1986_a6_solution : (β β β) β β β β := sorry
-- fun b n β¦ (β i : Finset.Icc 1 n, b i) / Nat.factorial n
/--
Let $a_1, a_2, \dots, a_n$ be real numbers, and let $b_1, b_2, \dots, b_n$ be distinct positive integers. Suppose that there is a polynomial $f(x)$ satisfying the identity
\[
(1-x)^n f(x) = 1 + \sum_... | 6e635db1f8105862 |
putnam_1996_a6 | abbrev putnam_1996_a6_solution : β β Set (β β β) := sorry
-- (fun c : β => if c β€ 1 / 4 then {f : β β β | β d : β, β x : β, f x = d} else {f : β β β | ContinuousOn f (Set.Icc 0 c) β§ f 0 = f c β§ (β x > 0, f x = f (x ^ 2 + c)) β§ (β x < 0, f x = f (-x))})
/--
Let $c>0$ be a constant. Give a complete description, with proo... | 77309532097f1c06 |
putnam_1991_a2 | abbrev putnam_1991_a2_solution : Prop := sorry
-- False
/--
Let $\mathbf{A}$ and $\mathbf{B}$ be different $n \times n$ matrices with real entries. If $\mathbf{A}^3=\mathbf{B}^3$ and $\mathbf{A}^2\mathbf{B}=\mathbf{B}^2\mathbf{A}$, can $\mathbf{A}^2+\mathbf{B}^2$ be invertible?
-/
theorem putnam_1991_a2
(n : β) (hn :... | 1e9ff3559c5c5884 |
putnam_2014_a2 | abbrev putnam_2014_a2_solution : β β β := sorry
-- (fun n : β => (-1) ^ (n - 1) / ((n - 1)! * (n)!))
/--
Let $A$ be the $n \times n$ matrix whose entry in the $i$-th row and $j$-th column is $\frac{1}{\min(i,j)}$ for $1 \leq i,j \leq n$. Compute $\det(A)$.
-/
theorem putnam_2014_a2
(n : β)
(A : Matrix (Fin n) (Fin n) β... | 954747916dc3f701 |
putnam_1965_a2 | theorem putnam_1965_a2
: β n > 0, β r in Finset.Icc 0 ((n - 1)/2), ((n - 2*r) * Nat.choose n r / (n : β))^2 = (Nat.choose (2*n - 2) (n - 1))/(n : β) :=
sorry | f9f1f1e75a1487ef |
putnam_1988_b4 | theorem putnam_1988_b4
(a : β β β)
(IsPosConv : (β β β) β Prop)
(IsPosConv_def : β a' : β β β, IsPosConv a' β
(β n β₯ 1, a' n > 0) β§
(β s : β, Tendsto (fun N : β => β n : Set.Icc 1 N, a' n) atTop (π s))) :
(IsPosConv a) β IsPosConv (fun n : β => (a n) ^ ((n : β) / (n + 1))) :=
sorry | f3c0f493f646df6e |
putnam_1975_a2 | abbrev putnam_1975_a2_solution : (β Γ β) β Prop := sorry
-- fun (b, c) => c < 1 β§ c - b > -1 β§ c + b > -1
/--
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?
-/
theorem putnam_1975_a2
: β b c : β, (β z : β, (X... | 5511d29ccc594e36 |
putnam_1998_b4 | abbrev putnam_1998_b4_solution : Set (β Γ β) := sorry
-- {nm | let β¨n,mβ© := nm; multiplicity 2 n β multiplicity 2 m}
/--
Find necessary and sufficient conditions on positive integers $m$ and $n$ so that \[\sum_{i=0}^{mn-1} (-1)^{\lfloor i/m \rfloor +\lfloor i/n\rfloor}=0.\]
-/
theorem putnam_1998_b4
(quantity : β β β... | 4d0808f28adff9ad |
putnam_2005_a3 | theorem putnam_2005_a3
(p : Polynomial β)
(n : β)
(hn : 0 < n)
(g : β β β)
(pdeg : p.degree = n)
(pzeros : β z : β, p.eval z = 0 β Complex.abs z = 1)
(hg : β z : β, g z = (p.eval z) / z ^ ((n : β) / 2))
(z : β)
(hz : z β 0 β§ deriv g z = 0) :
Complex.abs z = 1 :=
sorry | 617b8318eedcb90d |
putnam_2015_a3 | abbrev putnam_2015_a3_solution : β := sorry
-- 13725
/--
Compute $\log_2 \left( \prod_{a=1}^{2015}\prod_{b=1}^{2015}(1+e^{2\pi iab/2015}) \right)$. Here $i$ is the imaginary unit (that is, $i^2=-1$).
-/
theorem putnam_2015_a3 :
Complex.log (β a : Fin 2015, β b : Fin 2015, (1 + Complex.exp (2 * Real.pi * Complex.I *... | 26c2ef115ab429e9 |
putnam_1962_a6 | theorem putnam_1962_a6
(S : Set β)
(hSadd : β a β S, β b β S, a + b β S)
(hSprod : β a β S, β b β S, a * b β S)
(hScond : β r : β, (r β S β¨ -r β S β¨ r = 0) β§ Β¬(r β S β§ -r β S) β§ Β¬(r β S β§ r = 0) β§ Β¬(-r β S β§ r = 0))
: S = { r : β | r > 0 } :=
sorry | 6f565cc92cef6182 |
putnam_1972_a6 | theorem putnam_1972_a6
(f : β β β)
(n : β€)
(hn : n β₯ 0)
(hfintegrable: IntegrableOn f (Icc 0 1))
(hfint : β i β Icc 0 (n-1), β« x in Icc 0 1, x^i*(f x) = 0)
(hfintlast : β« x in Icc 0 1, x^n*(f x) = 1)
: β S, S β Icc (0 : β) 1 β§ MeasurableSet S β§ volume S > 0 β§ β x β S, |f x| β₯ 2^n * (n + 1) :=
sorry | 467de9b23c916ce1 |
putnam_1980_a3 | abbrev putnam_1980_a3_solution : β := sorry
-- Real.pi / 4
/--
Evaluate $\int_0^{\pi/2}\frac{dx}{1+(\tan x)^{\sqrt{2}}}$.
-/
theorem putnam_1980_a3
: β« x in Set.Ioo 0 (Real.pi / 2), 1 / (1 + (Real.tan x) ^ (Real.sqrt 2)) = putnam_1980_a3_solution :=
sorry | 12b5470035d85430 |
putnam_2009_b6 | theorem putnam_2009_b6
(n : β) (npos : n > 0) :
(β a : β β β€,
a 0 = 0 β§ a 2009 = n β§
β i : Icc 1 2009,
((β j k : β, j < i β§ a i = a j + 2 ^ k) β¨
β b c : β, b < i β§ c < i β§ a b > 0 β§ a c > 0 β§ a i = (a b) % (a c))) :=
sorry | 7508f02673138a69 |
putnam_2019_b6 | abbrev putnam_2019_b6_solution : Set β := sorry
-- Set.Ici 1
/--
Let \( \mathbb{Z}^n \) be the integer lattice in \( \mathbb{R}^n \). Two points in \( \mathbb{Z}^n \) are called neighbors if they differ by exactly 1 in one coordinate and are equal in all other coordinates. For which integers \( n \geq 1 \) does there e... | 4f4d4be03d5e8621 |
putnam_1983_a5 | abbrev putnam_1983_a5_solution : Prop := sorry
-- True
/--
Prove or disprove that there exists a positive real number $\alpha$ such that $[\alpha_n] - n$ is even for all integers $n > 0$. (Here $[x]$ denotes the greatest integer less than or equal to $x$.)
-/
theorem putnam_1983_a5 :
(β Ξ± : β, Ξ± > 0 β§ β n : β, n > ... | f75e91361943c6d6 |
putnam_1993_a5 | theorem putnam_1993_a5
: Β¬Irrational ((β« x in Set.Ioo (-100) (-10), (((x ^ 2 - x) / (x ^ 3 - 3 * x + 1)) ^ 2)) + (β« x in Set.Ioo (1 / 101) (1 / 11), (((x ^ 2 - x) / (x ^ 3 - 3 * x + 1)) ^ 2)) + (β« x in Set.Ioo (101 / 100) (11 / 10), (((x ^ 2 - x) / (x ^ 3 - 3 * x + 1)) ^ 2))) :=
sorry | e8d5e93e7ca4a55b |
putnam_2016_a5 | theorem putnam_2016_a5
(G : Type*) [Group G]
(Gfin : Fintype G)
(g h : G)
(ghgen : Subgroup.closure {g, h} = β€ β§ Β¬Subgroup.closure {g} = β€ β§ Β¬Subgroup.closure {h} = β€)
(godd : Odd (orderOf g))
(S : Set G)
(hS : S = {g * h, gβ»ΒΉ * h, g * hβ»ΒΉ, gβ»ΒΉ * hβ»ΒΉ})
: (β x : G, β mn : List G, 1 β€ mn.length β§ mn.length β€ Gfin.card β§ ... | a543fa9d2ec27ed1 |
putnam_2006_a5 | abbrev putnam_2006_a5_solution : β β β€ := sorry
-- (fun n : β => if (n β‘ 1 [MOD 4]) then n else -n)
/--
Let $n$ be a positive odd integer and let $\theta$ be a real number such that $\theta/\pi$ is irrational. Set $a_k=\tan(\theta+k\pi/n)$, $k=1,2,\dots,n$. Prove that $\frac{a_1+a_2+\cdots+a_n}{a_1a_2 \cdots a_n}$ is a... | affc74b16c681580 |
putnam_2024_b1 | abbrev putnam_2024_b1_solution : Set (β Γ β) := sorry
--{(2*l+1, l+1) | (l : β)}
/--
Let $n$ and $k$ be positive integers. The square in the $i$th row and
$j$th column of an $n$-by-$n$ grid contains the number $i + j - k$.
For which $n$ and $k$ is it possible to select $n$ squares from the
grid, no two in the same row ... | 47b75d05243e5072 |
putnam_2011_a1 | abbrev putnam_2011_a1_solution : β := sorry
-- 10053
/--
Define a \emph{growing spiral} in the plane to be a sequence of points with integer coordinates $P_0=(0,0),P_1,\dots,P_n$ such that $n \geq 2$ and:
\begin{itemize}
\item the directed line segments $P_0P_1,P_1P_2,\dots,P_{n-1}P_n$ are in the successive coordinate ... | e072c73f8942ea7f |
putnam_2001_a1 | theorem putnam_2001_a1
(S : Type*)
[Mul S]
(hS : β a b : S, (a * b) * a = b)
: β a b : S, a * (b * a) = b :=
sorry | 6958719999ea03ee |
putnam_1976_a4 | abbrev putnam_1976_a4_solution : (β β β) Γ (β β β) := sorry
-- (fun r : β => -1/(r + 1), fun r : β => -(r + 1)/r)
/--
Let $r$ be a real root of $P(x) = x^3 + ax^2 + bx - 1$, where $a$ and $b$ are integers and $P$ is irreducible over the rationals. Suppose that $r + 1$ is a root of $x^3 + cx^2 + dx + 1$, where $c$ and $... | e3ab2d8e284193d9 |
putnam_2023_b5 | abbrev putnam_2023_b5_solution : Set β := sorry
-- {n : β | n = 1 β¨ n β‘ 2 [MOD 4]}
/--
Determine which positive integers $n$ have the following property: For all integers $m$ that are relatively prime to $n$, there exists a permutation $\pi:\{1,2,\dots,n\} \to \{1,2,\dots,n\}$ such that $\pi(\pi(k)) \equiv mk \pmod{n}$... | 8b68c4555ecedc62 |
putnam_1966_a4 | theorem putnam_1966_a4
(a : β β β€)
(ha1 : a 1 = 2)
(hai : β n β₯ 1, a (n + 1) = (if β m : β€, m^2 = a n + 1 = True then a n + 2 else a n + 1))
: β n β₯ 1, a n = n + round (Real.sqrt n) :=
sorry | f2774e3723a1b6ce |
putnam_1994_a1 | theorem putnam_1994_a1
(a : β β β)
(ha : β n β₯ 1, 0 < a n β§ a n β€ a (2 * n) + a (2 * n + 1)) :
Tendsto (fun N : β => β n : Set.Icc 1 N, a n) atTop atTop :=
sorry | 1289f74158047214 |
putnam_1980_b5 | abbrev putnam_1980_b5_solution : β β Prop := sorry
-- fun t : β => 1 β₯ t
/--
A function $f$ is convex on $[0, 1]$ if and only if $$f(su + (1-s)v) \le sf(u) + (1 - s)f(v)$$ for all $s \in [0, 1]$.
Let $S_t$ denote the set of all nonnegative increasing convex continuous functions $f : [0, 1] \rightarrow \mathbb{R}$ such ... | 9f672afb597c9673 |
putnam_1990_b5 | abbrev putnam_1990_b5_solution : Prop := sorry
-- True
/--
Is there an infinite sequence $a_0,a_1,a_2,\dots$ of nonzero real numbers such that for $n=1,2,3,\dots$ the polynomial $p_n(x)=a_0+a_1x+a_2x^2+\cdots+a_nx^n$ has exactly $n$ distinct real roots?
-/
theorem putnam_1990_b5 :
(β a : β β β, (β i, a i β 0) β§
... | 3521530baf9fb098 |
putnam_2015_b5 | abbrev putnam_2015_b5_solution : β := sorry
-- 4
/--
Let $P_n$ be the number of permutations $\pi$ of $\{1,2,\dots,n\}$ such that
\[
|i-j| = 1 \mbox{ implies } |\pi(i) -\pi(j)| \leq 2
\]
for all $i,j$ in $\{1,2,\dots,n\}$. Show that for $n \geq 2$, the quantity
\[
P_{n+5} - P_{n+4} - P_{n+3} + P_n
\]
does not depend on... | 058163993a1dea2e |
putnam_2005_b5 | theorem putnam_2005_b5
(n : β)
(npos : n > 0)
(P : MvPolynomial (Fin n) β)
(hderiv : β i : Fin n, (MvPolynomial.pderiv i)^[2] P = 0)
(hsumsq : β i : Fin n, (MvPolynomial.X i) ^ 2 β£ P)
: (P = 0) :=
sorry | 7cf42712c896ab48 |
putnam_1998_a2 | theorem putnam_1998_a2
(quadrant : (EuclideanSpace β (Fin 2)) β Prop)
(isarc : (EuclideanSpace β (Fin 2)) β (EuclideanSpace β (Fin 2)) β Prop)
(hquadrant : β P, quadrant P β P 0 > 0 β§ P 1 > 0 β§ dist 0 P = 1)
(hisarc : β P Q, isarc P Q β quadrant P β§ quadrant Q β§ P 0 > Q 0)
(arc : (EuclideanSpace β (Fin 2)) β (Euclidean... | edbc2f4a8d7d9f95 |
putnam_1975_b4 | abbrev putnam_1975_b4_solution : Prop := sorry
-- False
/--
Let $C = \{(x, y) \in \mathbb{R}^2 : x^2 + y^2 = 1\}$ denote the unit circle. Does there exist $B \subseteq C$ for which $B$ is topologically closed and contains exactly one point from each pair of diametrically opposite points in $C$?
-/
theorem putnam_1975_b... | 8883ccf8312e64f1 |
putnam_1988_a2 | abbrev putnam_1988_a2_solution : Prop := sorry
-- True
/--
A not uncommon calculus mistake is to believe that the product rule for derivatives says that $(fg)' = f'g'$. If $f(x)=e^{x^2}$, determine, with proof, whether there exists an open interval $(a,b)$ and a nonzero function $g$ defined on $(a,b)$ such that this wr... | d0852f2fd5c40e57 |
putnam_2020_a5 | abbrev putnam_2020_a5_solution : β€ := sorry
-- (Nat.fib 4040) - 1
/--
Let $a_n$ be the number of sets $S$ of positive integers for which
\[
\sum_{k \in S} F_k = n,
\]
where the Fibonacci sequence $(F_k)_{k \geq 1}$ satisfies $F_{k+2} = F_{k+1} + F_k$ and begins $F_1 = 1, F_2 = 1, F_3 = 2, F_4 = 3$. Find the largest int... | cf3425b1897164f4 |
putnam_1965_b4 | abbrev putnam_1965_b4_solution : ((((β β β) β (β β β)) Γ ((β β β) β (β β β))) Γ ((Set β) Γ (β β β))) := sorry
-- ((fun h : β β β => h + (fun x : β => x), fun h : β β β => h + (fun _ : β => 1)), ({x : β | x β₯ 0}, Real.sqrt))
/--
Let $$f(x, n) = \frac{{n \choose 0} + {n \choose 2}x + {n \choose 4}x^2 + \cdots}{{n \choose... | aa54494f6a069281 |
putnam_2012_b1 | theorem putnam_2012_b1
(S : Set (Set.Ici (0 : β) β β))
(rngS : β f β S, β x : Set.Ici (0 : β), f x β Set.Ici (0 : β))
(f1 : Set.Ici (0 : β) β β)
(hf1 : f1 = fun (x : Set.Ici (0 : β)) β¦ exp x - 1)
(f2 : Set.Ici (0 : β) β β)
(hf2 : f2 = fun (x : Set.Ici (0 : β)) β¦ Real.log (x + 1))
(hf1mem : f1 β S)
(hf2mem : f2 β S)
(hs... | 108f0abe029c5fde |
putnam_1987_b1 | abbrev putnam_1987_b1_solution : β := sorry
-- 1
/--
Evaluate
\[
\int_2^4 \frac{\sqrt{\ln(9-x)}\,dx}{\sqrt{\ln(9-x)}+\sqrt{\ln(x+3)}}.
\]
-/
theorem putnam_1987_b1
: (β« x in (2)..4, sqrt (log (9 - x)) / (sqrt (log (9 - x)) + sqrt (log (x + 3))) = putnam_1987_b1_solution) :=
sorry | a347b5725a26d120 |
putnam_1997_b1 | abbrev putnam_1997_b1_solution : β β β := sorry
-- fun n => n
noncomputable def dist_to_int : β β β := fun r => |r - round r|
/--
Let $\{x\}$ denote the distance between the real number $x$ and the nearest integer. For each positive integer $n$, evaluate \[F_n=\sum_{m=1}^{6n-1} \min(\{\frac{m}{6n}\},\{\frac{m}{3n}\}).... | 5bd3e81075dd3306 |
putnam_1979_a1 | abbrev putnam_1979_a1_solution : Multiset β := sorry
-- Multiset.replicate 659 3 + {2}
/--
For which positive integers $n$ and $a_1, a_2, \dots, a_n$ with $\sum_{i = 1}^{n} a_i = 1979$ does $\prod_{i = 1}^{n} a_i$ attain the greatest value?
-/
theorem putnam_1979_a1
(P : Multiset β β Prop)
(hP : β a, P a β Mult... | e8978e84d7f5ee7e |
putnam_1969_a1 | abbrev putnam_1969_a1_solution : Set (Set β) := sorry
-- {{x} | x : β} βͺ {Set.Ici x | x : β} βͺ {Set.Iic x | x : β} βͺ {Set.Iio x | x : β} βͺ {Set.Ioi x | x : β} βͺ {Set.univ}
/--
What are the possible ranges (across all real inputs $x$ and $y$) of a polynomial $f(x, y)$ with real coefficients?
-/
theorem putnam_1969_a1
: ... | ff753e018045e9be |
putnam_1966_b2 | theorem putnam_1966_b2
(S : β€ β Set β€)
(hS : S = fun n : β€ => {n, n + 1, n + 2, n + 3, n + 4, n + 5, n + 6, n + 7, n + 8, n + 9})
: β n : β€, n > 0 β (β k β S n, β m β S n, k β m β IsCoprime m k) :=
sorry | 35bb601cf2f0caf3 |
putnam_2023_a3 | abbrev putnam_2023_a3_solution : β := sorry
-- Real.pi / 2
/--
Determine the smallest positive real number $r$ such that there exist differentiable functions $f\colon \mathbb{R} \to \mathbb{R}$ and $g\colon \mathbb{R} \to \mathbb{R}$ satisfying
\begin{enumerate}
\item[(a)] $f(0) > 0$,
\item[(b)] $g(0) = 0$,
\item[(c... | 4724b10205cfca70 |
putnam_1976_b2 | abbrev putnam_1976_b2_solution : β Γ Set (List (β€ Γ β€)) := sorry
-- (8, {[(0, 0)], [(2, 0)], [(0, 1)], [(0, 2)], [(0, 3)], [(0, 4)], [(0, 5)], [(0, 6)]})
/--
Let $G$ be a group generated by two elements $A$ and $B$; i.e., every element of $G$ can be expressed as a finite word $A^{n_1}B^{n_2} \cdots A^{n_{k-1}}B^{n_k}$,... | de2aed7f95f0feb7 |
putnam_1971_b6 | theorem putnam_1971_b6
(Ξ΄ : β€ β β€)
(hΞ΄ : Ξ΄ = fun n => sSup {t | Odd t β§ t β£ n})
: β x : β€, x > 0 β |β i in Finset.Icc 1 x, (Ξ΄ i)/(i : β) - 2*x/3| < 1 :=
sorry | 8d08b04315529ab6 |
putnam_2006_b3 | abbrev putnam_2006_b3_solution : β β β := sorry
-- (fun n : β => (Nat.choose n 2) + 1)
/--
Let $S$ be a finite set of points in the plane. A linear partition of $S$ is an unordered pair $\{A,B\}$ of subsets of $S$ such that $A \cup B=S$, $A \cap B=\emptyset$, and $A$ and $B$ lie on opposite sides of some straight line ... | db27465358ae78eb |
putnam_2016_b3 | theorem putnam_2016_b3
(S : Finset (Fin 2 β β))
(hS : β A β S, β B β S, β C β S, MeasureTheory.volume (convexHull β {A, B, C}) β€ 1)
: β A' B' C' : Fin 2 β β, MeasureTheory.volume (convexHull β {A', B', C'}) = 4 β§ convexHull β {A', B', C'} β S :=
sorry | 034e637da2727cb6 |
putnam_1993_b3 | abbrev putnam_1993_b3_solution : β Γ β := sorry
-- (5 / 4, -1 / 4)
/--
Two real numbers $x$ and $y$ are chosen at random in the interval $(0,1)$ with respect to the uniform distribution. What is the probability that the closest integer to $x/y$ is even? Express the answer in the form $r+s\pi$, where $r$ and $s$ are rat... | d64ec33e9c311628 |
putnam_2010_b6 | theorem putnam_2010_b6
(n : β)
(npos : n β₯ 1)
(A : Matrix (Fin n) (Fin n) β)
(Apow : β β Matrix (Fin n) (Fin n) β)
(hApow : β k > 0, Apow k = (fun i j : Fin n => (A i j) ^ k))
: (β k β Set.Icc 1 (n + 1), A ^ k = Apow k) β (β k β₯ 1, A ^ k = Apow k) :=
sorry | 19bb4e7341655210 |
putnam_1977_b3 | abbrev putnam_1977_b3_solution : Prop := sorry
-- False
/--
An ordered triple $(a, b, c)$ of positive irrational numbers with $a + b + c = 1$ is considered $\textit{balanced}$ if all three elements are less than $\frac{1}{2}$. If a triple is not balanced, we can perform a ``balancing act'' $B$ defined by $B(a, b, c) = ... | e97219934df9a5c5 |
putnam_2022_a2 | abbrev putnam_2022_a2_solution : β β β := sorry
-- fun n => 2*n - 2
/--
Let $n$ be an integer with $n \geq 2$. Over all real polynomials $p(x)$ of degree $n$, what is the largest possible number of negative coefficients of $p(x)^2$?
-/
theorem putnam_2022_a2
(n : β)
(hn : n β₯ 2)
(S : Set β[X])
(hS : S = {P | natDegree ... | ec96680e03bd2ff6 |
putnam_1967_b3 | theorem putnam_1967_b3
(f g : β β β)
(fgcont : Continuous f β§ Continuous g)
(fgperiod : Function.Periodic f 1 β§ Function.Periodic g 1)
: Tendsto (fun n : β€ => β« x in Set.Ioo 0 1, f x * g (n * x)) atTop (π ((β« x in Set.Ioo 0 1, f x) * (β« x in Set.Ioo 0 1, g x))) :=
sorry | 23bcf79916c4033c |
putnam_1985_b6 | theorem putnam_1985_b6
(n : β)
(npos : n > 0)
(G : Finset (Matrix (Fin n) (Fin n) β))
(groupG : (β g β G, β h β G, g * h β G) β§ 1 β G β§ (β g β G, β h β G, g * h = 1))
(hG : β M in G, Matrix.trace M = 0)
: (β M in G, M = 0) :=
sorry | c9538cd13c8765d1 |
putnam_1995_b6 | theorem putnam_1995_b6
(S : β β Set β)
(hS : S = fun (Ξ± : β) => {x : β | β n : β, n β₯ 1 β§ x = floor (n * Ξ±)}) :
Β¬ β Ξ± Ξ² Ξ³,
Ξ± > 0 β§ Ξ² > 0 β§ Ξ³ > 0 β§
(S Ξ±) β© (S Ξ²) = β
β§ (S Ξ²) β© (S Ξ³) = β
β§ (S Ξ±) β© (S Ξ³) = β
β§
Set.Ici 1 = (S Ξ±) βͺ (S Ξ²) βͺ (S Ξ³) :=
sorry | 076c5abea64bdd8a |
putnam_2008_a1 | theorem putnam_2008_a1
(f : β β β β β)
(hf : β x y z : β, f x y + f y z + f z x = 0)
: β g : β β β, β x y : β, f x y = g x - g y :=
sorry | eab40aa77c2eb040 |
putnam_2018_a1 | abbrev putnam_2018_a1_solution : Set (β€ Γ β€) := sorry
-- {β¨673, 1358114β©, β¨674, 340033β©, β¨1009, 2018β©, β¨2018, 1009β©, β¨340033, 674β©, β¨1358114, 673β©}
/--
Find all ordered pairs $(a,b)$ of positive integers for which $\frac{1}{a} + \frac{1}{b} = \frac{3}{2018}$.
-/
theorem putnam_2018_a1
(a b : β€)
(h : 0 < a β§ 0 < b) ... | 4da0c14353099928 |
putnam_1982_b2 | abbrev putnam_1982_b2_solution : Polynomial β := sorry
-- C Real.pi * (2*X - 1)^2
/--
Let $A(x, y)$ denote the number of points $(m, n)$ with integer coordinates $m$ and $n$ where $m^2 + n^2 \le x^2 + y^2$. Also, let $g = \sum_{k = 0}^{\infty} e^{-k^2}$. Express the value $$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty... | 8f7a6b32376ab494 |
putnam_1992_b2 | theorem putnam_1992_b2
(Q : β β β β β)
(hQ : Q = fun n k β¦ coeff ((1 + X + X ^ 2 + X ^ 3) ^ n) k)
(n k : β) :
Q n k = β j in Finset.Iic k, choose n j * (if 2 * j β€ k then choose n (k - 2 * j) else 0) :=
sorry | 7859590c71b5bc24 |
putnam_2017_b2 | abbrev putnam_2017_b2_solution : β := sorry
-- 16
/--
Suppose that a positive integer $N$ can be expressed as the sum of $k$ consecutive positive integers
\[
N = a + (a+1) +(a+2) + \cdots + (a+k-1)
\]
for $k=2017$ but for no other values of $k>1$. Considering all positive integers $N$ with this property, what is the sm... | 373c86091be2689b |
putnam_2007_b2 | theorem putnam_2007_b2
(f : β β β)
(hf : ContDiffOn β 1 f (Icc 0 1))
(hfint : β« x in (0)..1, f x = 0)
(max : β)
(heqmax : β x β Icc (0 : β) 1, |deriv f x| = max)
(hmaxub : β x β Icc (0 : β) 1, |deriv f x| β€ max)
: (β Ξ± β (Ioo (0 : β) 1), |β« x in (0)..Ξ±, f x| β€ (1 / 8) * max) :=
sorry | 98b78a7cff5249c2 |
putnam_2004_b4 | abbrev putnam_2004_b4_solution : β β β β β := sorry
-- fun n z β¦ z + n
/--
Let $n$ be a positive integer, $n \ge 2$, and put $\theta = 2 \pi / n$. Define points $P_k = (k,0)$ in the $xy$-plane, for $k = 1, 2, \dots, n$. Let $R_k$ be the map that rotates the plane counterclockwise by the angle $\theta$ about the point $... | 561daef8fae6dfcb |
putnam_2014_b4 | theorem putnam_2014_b4
(n : β)
(P: Polynomial β)
(npos : n > 0)
(Px : P.degree = n β§ β k β Set.Icc 0 n, P.coeff k = 2 ^ (k * (n - k)))
: β r β P.roots, r.im = 0 :=
sorry | 01fb7b3d7113347a |
putnam_1963_b1 | abbrev putnam_1963_b1_solution : β€ := sorry
-- 2
/--
For what integer $a$ does $x^2-x+a$ divide $x^{13}+x+90$?
-/
theorem putnam_1963_b1
: β a : β€, (X^2 - X + (C a)) β£ (X ^ 13 + X + (C 90)) β a = putnam_1963_b1_solution :=
sorry | c5e5393838a15bf2 |
putnam_1973_b1 | theorem putnam_1973_b1
(n : β)
(a : Finset.Icc 1 (2 * n + 1) β β€)
(h_remove : β S : Finset (Finset.Icc 1 (2 * n + 1)), S.card = 2*n β β T, T β S β§ T.card = n β§ β i in T, a i = β i in (S \ T), a i)
: β i j : Finset.Icc 1 (2 * n + 1), a i = a j :=
sorry | 76098aa58176ac9a |
putnam_1991_b4 | theorem putnam_1991_b4
(p : β)
(podd : Odd p)
(pprime : Prime p)
: (β j : Fin (p + 1), (p.choose j) * ((p + j).choose j)) β‘ (2 ^ p + 1) [MOD (p ^ 2)] :=
sorry | 07843bc3544afc32 |
putnam_1981_b4 | abbrev putnam_1981_b4_solution : Prop := sorry
-- False
/--
Let $V$ be a set of $5$ by $7$ matrices, with real entries and with the property that $rA+sB \in V$ whenever $A,B \in V$ and $r$ and $s$ are scalars (i.e., real numbers). \emph{Prove or disprove} the following assertion: If $V$ contains matrices of ranks $0$, ... | 79ba19c2861f11da |
putnam_1964_b5 | theorem putnam_1964_b5
(a b : β β β)
(ha : StrictMono a β§ β n : β, a n > 0)
(hb : b 0 = a 0 β§ β n : β, b (n + 1) = lcm (b n) (a (n + 1)))
: (β L : β, Tendsto (fun N β¦ β n in Finset.range N, (1 : β) / b n) atTop (π L)) :=
sorry | be50b373a74a6ea3 |
putnam_1989_a3 | theorem putnam_1989_a3
(z : β)
(hz : 11 * z ^ 10 + 10 * I * z ^ 9 + 10 * I * z - 11 = 0)
: (βzβ = 1) :=
sorry | 44470eaaa9d4eb43 |
putnam_2021_a4 | abbrev putnam_2021_a4_solution : β := sorry
-- ((Real.sqrt 2) / 2) * Real.pi * Real.log 2
/--
Let
\[
I(R) = \iint_{x^2+y^2 \leq R^2} \left( \frac{1+2x^2}{1+x^4+6x^2y^2+y^4} - \frac{1+y^2}{2+x^4+y^4} \right)\,dx\,dy.
\]
Find
\[
\lim_{R \to \infty} I(R),
\]
or show that this limit does not exist.
-/
theorem putnam_2021_a... | ede795b18a7e6da3 |
putnam_1974_b5 | theorem putnam_1974_b5
: β n β₯ 0, β i in Finset.Icc (0 : β) n, (n^i : β)/(Nat.factorial i) > (Real.exp n)/2 :=
sorry | 7f86dc460ad5461f |
putnam_1999_a3 | theorem putnam_1999_a3
(f : β β β)
(hf : f = fun x β¦ 1 / (1 - 2 * x - x ^ 2))
(a : β β β)
(hf' : βαΆ x in π 0, Tendsto (fun N : β β¦ β n in Finset.range N, (a n) * x ^ n) atTop (π (f x)))
(n : β) :
β m : β, (a n) ^ 2 + (a (n + 1)) ^ 2 = a m :=
sorry | 271b78e8fdb16719 |
putnam_1994_a6 | theorem putnam_1994_a6
(f : Fin 10 β Equiv.Perm β€)
(mijcomp : β β (β β Fin 10) β β β (β€ β β€))
(F : Finset (β€ β β€))
(hf: β n : β€, β m : β, β i : β β Fin 10, m β₯ 1 β§ (mijcomp m i 0) 0 = n)
(hmijcomp : β m β₯ 1, β (i : β β Fin 10) (j : Fin m), mijcomp m i j = if (j = m - 1) then (f (i j) : β€ β β€) else (f (i j) β mijcomp m ... | 743c0e8f3075473d |
putnam_1984_a6 | abbrev putnam_1984_a6_solution : β := sorry
-- 4
/--
Let $n$ be a positive integer, and let $f(n)$ denote the last nonzero digit in the decimal expansion of $n!$. For instance, $f(5)=2$.
\begin{enumerate}
\item[(a)] Show that if $a_1,a_2,\dots,a_k$ are \emph{distinct} nonnegative integers, then $f(5^{a_1}+5^{a_2}+\dots... | d93bff612e25223c |
putnam_2011_a6 | theorem putnam_2011_a6
(G : Type*) [CommGroup G] [Fintype G]
(n : β)
(gset : Set G)
(k : β)
(mgprob : β β G β β)
(hmgprob : mgprob = (fun (m : β) (x : G) => ({f : Fin m β gset | (β i : Fin m, (f i).1) = x}.ncard : β) / k ^ m))
(hn : n = (Fintype.elems : Finset G).card)
(hk : gset.encard = k)
(hgsetprop : k < n)
(hgset1... | 1f0bf8b6abf4487f |
putnam_2023_b2 | abbrev putnam_2023_b2_solution : β := sorry
-- 3
/--
For each positive integer $n$, let $k(n)$ be the number of ones in the binary representation of $2023 * n$. What is the minimum value of $k(n)$?
-/
theorem putnam_2023_b2
: sInf {(digits 2 (2023*n)).sum | n > 0} = putnam_2023_b2_solution :=
sorry | 2a5047b969278943 |
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