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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 ^ 2 ^ (n + 1)) :=
sorry | 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)
: {S : β |
β x : β β β,
(β j : β, x j > 0) β§
(β' j : β, x j) = A β§
(β' j : β, (x j) ^ 2) = S}
= putnam_2000_a1_solution A :=
sorry | 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 $\{1,2,3,6\},\{4,8\},\{5,7\}$ shows that the largest $k$ is \emph{at least} $3$.]
-/
theorem putnam_2010_a1
(n : β)
(kboxes : β β Prop)
(npos : n > 0)
(hkboxes : β k : β, kboxes k =
(β boxes : Finset.Icc 1 n β Fin k, β i j : Fin k,
β x in Finset.univ.filter (boxes Β· = i), (x : β) =
β x in Finset.univ.filter (boxes Β· = j), (x : β))) :
IsGreatest kboxes (putnam_2010_a1_solution n) :=
sorry | 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)] $A_1 \cap A_2 \cap A_3 = \emptyset$.
\end{enumerate}
Express your answer in the form $2^a 3^b 5^c 7^d$, where $a,b,c,d$ are nonnegative integers.
-/
theorem putnam_1985_a1 :
let (a, b, c, d) := putnam_1985_a1_solution;
{(A1, A2, A3) : Set β€ Γ Set β€ Γ Set β€ | A1 βͺ A2 βͺ A3 = Icc 1 10 β§ A1 β© A2 β© A3 = β
}.ncard = 2 ^ a * 3 ^ b * 5 ^ c * 7 ^ d :=
sorry | 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 $n$ is a repunit, then so is $f(n)$.
-/
theorem putnam_2007_a4
(S : Set (Polynomial β))
(repunit : β β Prop)
(hrepunit : β x, repunit x β x > 0 β§ x = floor x β§ β d β (digits 10 (floor x)), d = 1)
(hS : β f, f β S β (β n : β, repunit n β repunit (f.eval n)))
: (S = putnam_2007_a4_solution) :=
sorry | 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 derivatives $f^{(k)}(0), k = 1, 2, 3, \dots$.
-/
theorem putnam_1992_a4
(f : β β β)
(hfdiff : ContDiff β β€ f)
(hf : β n : β, n > 0 β f (1 / n) = n ^ 2 / (n ^ 2 + 1))
: (β k : β, k > 0 β iteratedDeriv k f 0 = putnam_1992_a4_solution k) :=
sorry | 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 β β, (f β f' β¨ g β g') β§ hdiffeq f' g')) :
β L : β, L > 0 β§ Function.Periodic f L β§ Function.Periodic g L :=
sorry | 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 $\{x_n,x_1,x_2\}$ forms a $3$-term arithmetic progression when arranged in increasing order.
-/
theorem putnam_2022_b4
(n : β)
(hn : n β₯ 4)
(ap3 : β β β β β β Prop)
(xprog : (β β β) β Prop)
(hap3 : β x0 x1 x2, ap3 x0 x1 x2 β β o0 o1 o2 : β, (o0 < o1 β§ o1 < o2 β§ ({o0, o1, o2} : Set β) = {x0, x1, x2}) β (o1 - o0 = o2 - o1))
(hxprog : β x, xprog x β
((β i j : Fin n, i.1 β j.1 β x i.1 β x j.1) β§ (β i : Fin n, ap3 (x i.1) (x ((i.1 + 1) % n)) (x ((i.1 + 2) % n))))) :
(β x, xprog x) β n β putnam_2022_b4_solution :=
sorry | 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_1999_b5
(n : β)
(hn : n β₯ 3)
(theta : β)
(htheta : theta = 2 * Real.pi / n)
(A : Matrix (Fin n) (Fin n) β)
(hA : A = fun j k => Real.cos ((j.1 + 1) * theta + (k.1 + 1) * theta)) :
(1 + A).det = putnam_1999_b5_solution n :=
sorry | 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 each of the following forms, using (not necessarily positive) integers $x$ and $y$: (a) $x^2 + 16y^2$, (b) $4x^2 + 4xy + 5y^2$.
-/
theorem putnam_1974_a3
(assmption : β p : β, p.Prime β§ p > 2 β ((β m n : β€, p = m^2 + n^2) β p β‘ 1 [MOD 4]))
: β p : β, ((p.Prime β§ p > 2 β§ (β x y : β€, p = x^2 + 16*y^2)) β p β putnam_1974_a3_solution.1) β§ ((p.Prime β§ p > 2 β§ (β x y : β€, p = 4*x^2 + 4*x*y + 5*y^2)) β p β putnam_1974_a3_solution.2) :=
sorry | 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 : β+ β β, (β' k, a k = 1) β§ (β k, 0 β€ a k) β§
S = β' n : β+, n / 2 ^ (n : β) * (β k in Finset.Icc 1 n, a k) ^ (1 / n : β)}
putnam_2021_b2_solution :=
sorry | 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) :=
sorry | 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 0 = 2 β§ w 0 2 = 2)
(hw4 : w (-1) (-1) = 4 β§ w 1 (-1) = 4 β§ w (-1) 1 = 4 β§ w 1 1 = 4)
(hwn4 : w 0 (-1) = -4 β§ w (-1) 0 = -4 β§ w 1 0 = -4 β§ w 0 1 = -4)
(hw12 : w 0 0 = 12)
(hw0 : β a b : β€, (|a| > 2 β¨ |b| > 2) β w a b = 0)
(hA : β S, A S = β s in S, β s' in S, w (s - s').1 (s - s').2) :
β S : Finset (β€ Γ β€), Nonempty S β A S > 0 :=
sorry | 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 that $r_A(n)=r_B(n)$ for all $n$?
-/
theorem putnam_2003_a6
(r : Set β β β β β)
(hr : β S n, r S n = β' s1 : S, β' s2 : S, if (s1 β s2 β§ s1 + s2 = n) then 1 else 0)
: (β A B : Set β, A βͺ B = β β§ A β© B = β
β§ (β n : β, r A n = r B n)) β putnam_2003_a6_solution :=
sorry | 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
\[
(1-x)^n f(x) = 1 + \sum_{i=1}^n a_i x^{b_i}.
\]
Find a simple expression (not involving any sums) for $f(1)$ in terms of $b_1, b_2, \dots, b_n$ and $n$ (but independent of $a_1, a_2, \dots, a_n$).
-/
theorem putnam_1986_a6
(n : β)
(npos : n > 0)
(a : β β β)
(b : β β β)
(bpos : β i β Finset.Icc 1 n, b i > 0)
(binj : β i β Finset.Icc 1 n, β j β Finset.Icc 1 n, b i = b j β i = j)
(f : Polynomial β)
(hf : β x : β, (1 - x) ^ n * f.eval x = 1 + β i : Finset.Icc 1 n, (a i) * x ^ (b i))
: (f.eval 1 = putnam_1986_a6_solution b n) :=
sorry | 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 proof, of the set of all continuous functions $f:\mathbb{R} \to \mathbb{R}$ such that $f(x)=f(x^2+c)$ for all $x \in \mathbb{R}$.
-/
theorem putnam_1996_a6
(c : β)
(f : β β β)
(cgt0 : c > 0)
: (Continuous f β§ β x : β, f x = f (x ^ 2 + c)) β f β putnam_1996_a6_solution c :=
sorry | 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 : 1 β€ n) :
putnam_1991_a2_solution β (β A B : Matrix (Fin n) (Fin n) β,
A β B β§ A ^ 3 = B ^ 3 β§
A ^ 2 * B = B ^ 2 * A β§
Nonempty (Invertible (A ^ 2 + B ^ 2))) :=
sorry | 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)
(npos : n > 0)
(hA : β i j : Fin n, A i j = 1 / min (i.1 + 1 : β) (j.1 + 1))
: A.det = putnam_2014_a2_solution n :=
sorry | 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^2 + (C (b : β))*X + (C (c : β)) : Polynomial β).eval z = 0 β βzβ < 1) β putnam_1975_a2_solution (b, c) :=
sorry | 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 β β€)
(hquantity : quantity = fun n m => β i in Finset.range (m * n), (-1)^(i/m + i/n))
(n m : β)
(hnm : n > 0 β§ m > 0) :
quantity n m = 0 β β¨n, mβ© β putnam_1998_b4_solution :=
sorry | 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 * (a.1 + 1) * (b.1 + 1) / 2015))) / Complex.log 2 = putnam_2015_a3_solution :=
sorry | 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 exist a set of points \( S \subset \mathbb{Z}^n \) satisfying the following two conditions? \begin{enumerate} \item If \( p \) is in \( S \), then none of the neighbors of \( p \) is in \( S \). \item If \( p \in \mathbb{Z}^n \) is not in \( S \), then exactly one of the neighbors of \( p \) is in \( S \). \end{enumerate}
-/
theorem putnam_2019_b6
(n : β)
(neighbors : (Fin n β β€) β (Fin n β β€) β Prop)
(neighbors_def : β p q, neighbors p q β (β i : Fin n, abs (p i - q i) = 1 β§ β j β i, p j = q j)) :
(1 β€ n β§ β S : Set (Fin n β β€),
(β p β S, β q, neighbors p q β q β S) β§ (β p β S, {q β S | neighbors p q}.encard = 1))
β n β putnam_2019_b6_solution :=
sorry | 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 > 0 β Even (βΞ± ^ nβ - n)) β putnam_1983_a5_solution :=
sorry | 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§ β i : Fin mn.length, mn.get i β S β§ x = List.prod mn) :=
sorry | 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 an integer, and determine its value.
-/
theorem putnam_2006_a5
(n : β)
(theta : β)
(a : Set.Icc 1 n β β)
(nodd : Odd n)
(thetairr : Irrational (theta / Real.pi))
(ha : β k : Set.Icc 1 n, a k = Real.tan (theta + (k * Real.pi) / n))
: (β k : Set.Icc 1 n, a k) / (β k : Set.Icc 1 n, a k) = putnam_2006_a5_solution n :=
sorry | 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 or column, such that the numbers
contained in the selected squares are exactly $1, ..., n$?
-/
theorem putnam_2024_b1
(grid : (n : β) β β β Fin n β Fin n β β€)
(grid_def : β n k i j, grid n k i j = i.succ + j.succ - k)
(valid : β β β β Prop)
(valid_def : β n k, valid n k β 0 < n β§ 0 < k β§
β (f : Fin n β Fin n), f.Injective β§
Set.range (fun i => grid n k i (f i)) = Set.Icc (1 : β€) n) :
{ (n, k) | valid n k } = putnam_2024_b1_solution :=
sorry | 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 directions east (for $P_0P_1$), north, west, south, east, etc.;
\item the lengths of these line segments are positive and strictly increasing.
\end{itemize}
How many of the points $(x,y)$ with integer coordinates $0 \leq x \leq 2011,0 \leq y \leq 2011$ \emph{cannot} be the last point, $P_n$ of any growing spiral?
-/
theorem putnam_2011_a1
(IsSpiral : List (Fin 2 β β€) β Prop)
(IsSpiral_def : β P, IsSpiral P β P.length β₯ 3 β§ P[0]! = 0 β§
(β l : Fin (P.length - 1) β β, l > 0 β§ StrictMono l β§ (β i : Fin (P.length - 1),
(i.1 % 4 = 0 β (P[i] 0 + l i = P[i.1 + 1]! 0 β§ P[i] 1 = P[i.1 + 1]! 1)) β§
(i.1 % 4 = 1 β (P[i] 0 = P[i.1 + 1]! 0 β§ P[i] 1 + l i = P[i.1 + 1]! 1)) β§
(i.1 % 4 = 2 β (P[i] 0 - l i = P[i.1 + 1]! 0 β§ P[i] 1 = P[i.1 + 1]! 1)) β§
(i.1 % 4 = 3 β (P[i] 0 = P[i.1 + 1]! 0 β§ P[i] 1 - l i = P[i.1 + 1]! 1))))) :
{p | 0 β€ p 0 β§ p 0 β€ 2011 β§ 0 β€ p 1 β§ p 1 β€ 2011 β§ Β¬β spiral, IsSpiral spiral β§ spiral.getLast! = p}.encard = putnam_2011_a1_solution :=
sorry | 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 $d$ are also integers. Express another root $s$ of $P$ as a function of $r$ that does not depend on the values of $a$, $b$, $c$, or $d$.
-/
theorem putnam_1976_a4
(a b c d : β€)
(r : β)
(P Q: Polynomial β)
(hP : P = X^3 + (C (a : β))*X^2 + (C (b : β))*X - (C 1) β§ aeval r P = 0 β§ Irreducible P)
(hQ : Q = X^3 + (C (c : β))*X^2 + (C (d : β))*X + (C 1) β§ aeval (r + 1) Q = 0)
: β s : β, aeval s P = 0 β§ (s = putnam_1976_a4_solution.1 r β¨ s = putnam_1976_a4_solution.2 r) :=
sorry | 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 $\pi(\pi(k)) \equiv mk \pmod{n}$ for all $k \in \{1,2,\dots,n\}$.
-/
theorem putnam_2023_b5
(n : β) :
n β putnam_2023_b5_solution β
0 < n β§
(β m : β€, IsRelPrime m n β β p : Equiv.Perm (Fin n),
β k : Fin n, (p (p k)).1 + 1 β‘ m * (k.1 + 1) [ZMOD n]) :=
sorry | 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 that $$f(1) - 2f\left(\frac{2}{3}\right) + f\left(\frac{1}{3}\right) \ge t\left(f\left(\frac{2}{3}\right) - 2f\left(\frac{1}{3}\right) + f(0)\right).$$
For which real numbers $t \ge 0$ is $S_t$ closed under multiplication?
-/
theorem putnam_1980_b5
(T : Set β)
(hT : T = Icc 0 1)
(P : β β (β β β) β Prop)
(IsConvex : (β β β) β Prop)
(S : β β Set (β β β))
(P_def : β t f, P t f β f 1 - 2*f (2/3) + f (1/3) β₯ t*(f (2/3) - 2*f (1/3) + f 0))
(IsConvex_def : β f, IsConvex f β β u β T, β v β T, β s β T, f (s*u + (1 - s)*v) β€ s*(f u) + (1 - s)*(f v))
(hS : S = fun t : β => {f : β β β | (β x β T, f x β₯ 0) β§ StrictMonoOn f T β§ IsConvex f β§ ContinuousOn f T β§ P t f})
(t : β) (ht : t β₯ 0) :
putnam_1980_b5_solution t β (β f β S t, β g β S t, f * g β S t) :=
sorry | 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§
(β n β₯ 1, (β i in Finset.Iic n, a i β’ X ^ i : Polynomial β).roots.toFinset.card = n)) β
putnam_1990_b5_solution :=
sorry | 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 $n$, and find its value.
-/
theorem putnam_2015_b5
(P : β β β)
(hP : P = fun n β¦ {pi : Finset.Icc 1 n β Finset.Icc 1 n | Bijective pi β§ β i j : Finset.Icc 1 n, Nat.dist i j = 1 β Nat.dist (pi i) (pi j) β€ 2}.ncard)
: (β n : β, n β₯ 2 β
(P (n + 5) : β€) - (P (n + 4) : β€) - (P (n + 3) : β€) + (P n : β€) = putnam_2015_b5_solution) :=
sorry | 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 (EuclideanSpace β (Fin 2)) β Set (EuclideanSpace β (Fin 2)))
(harc : arc = fun P Q β¦ {R : EuclideanSpace β (Fin 2) | quadrant R β§ P 0 > R 0 β§ R 0 > Q 0})
(A B : (EuclideanSpace β (Fin 2)) β (EuclideanSpace β (Fin 2)) β β)
(hA : A = fun P Q β¦ (MeasureTheory.volume {S : EuclideanSpace β (Fin 2) | β R β arc P Q, R 0 = S 0 β§ R 1 > S 1 β§ S 1 > 0}).toReal)
(hB : B = fun P Q β¦ (MeasureTheory.volume {S : EuclideanSpace β (Fin 2) | β R β arc P Q, R 1 = S 1 β§ R 0 > S 0 β§ S 0 > 0}).toReal)
: (β f : β β β, β P Q : EuclideanSpace β (Fin 2), isarc P Q β A P Q + B P Q = f (InnerProductGeometry.angle P Q)) :=
sorry | 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_b4
(P : β Γ β β Prop)
(hP : P = fun (x, y) => x^2 + y^2 = 1)
: (β B β setOf P, IsClosed B β§ β x y : β, P (x, y) β Xor' ((x, y) β B) ((-x, -y) β B)) β putnam_1975_b4_solution :=
sorry | 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 wrong product rule is true for $x$ in $(a,b)$.
-/
theorem putnam_1988_a2
(f : β β β)
(hf : f = fun x β¦ Real.exp (x ^ 2)) :
putnam_1988_a2_solution β
(β a b : β,
a < b β§
β g : β β β,
(β x β Ioo a b, g x β 0) β§
DifferentiableOn β g (Ioo a b) β§
β x β Ioo a b, deriv (fun y β¦ f y * g y) x = (deriv f x) * (deriv g x)) :=
sorry | 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 integer $n$ such that $a_n = 2020$.
-/
theorem putnam_2020_a5
(a : β€ β β)
(ha : a = fun n : β€ => {S : Finset β | (β k β S, k > 0) β§ β k : S, Nat.fib k = n}.ncard) :
IsGreatest {n | a n = 2020} putnam_2020_a5_solution :=
sorry | 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 $$f(x, n) = \frac{{n \choose 0} + {n \choose 2}x + {n \choose 4}x^2 + \cdots}{{n \choose 1} + {n \choose 3}x + {n \choose 5}x^2 + \cdots}$$ for all real numbers $x$ and positive integers $n$. Express $f(x, n+1)$ as a rational function involving $f(x, n)$ and $x$, and find $\lim_{n \to \infty} f(x, n)$ for all $x$ for which this limit converges.
-/
theorem putnam_1965_b4
(f u v : β β β β β)
(hu : β n > 0, β x, u n x = β i in Finset.Icc 0 (n / 2), (n.choose (2 * i)) * x ^ i)
(hv : β n > 0, β x, v n x = β i in Finset.Icc 0 ((n - 1) / 2), (n.choose (2 * i + 1)) * x ^ i)
(hf : β n > 0, β x, f n x = u n x / v n x)
(n : β)
(hn : 0 < n) :
let β¨β¨p, qβ©, β¨s, gβ©β© := putnam_1965_b4_solution
(β x, v n x β 0 β v (n + 1) x β 0 β q (f n) x β 0 β f (n + 1) x = p (f n) x / q (f n) x) β§
s = {x | β l, Tendsto (fun n β¦ f n x) atTop (π l)} β§
β x β s, Tendsto (fun n β¦ f n x) atTop (π (g x)) :=
sorry | 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)
(hsum : β f β S, β g β S, (fun x β¦ (f x) + (g x)) β S)
(hcomp : β f β S, β g β S, β gnneg : Set.Ici (0 : β) β Set.Ici (0 : β), ((β x : Set.Ici (0 : β), g x = gnneg x) β (fun x β¦ f (gnneg x)) β S))
(hdiff : β f β S, β g β S, (β x : Set.Ici (0 : β), f x β₯ g x) β (fun x β¦ (f x) - (g x)) β S)
: (β f β S, β g β S, (fun x β¦ (f x) * (g x)) β S) :=
sorry | 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 \[F_n=\sum_{m=1}^{6n-1} \min(\{\frac{m}{6n}\},\{\frac{m}{3n}\}).\] (Here $\min(a,b)$ denotes the minimum of $a$ and $b$.)
-/
theorem putnam_1997_b1
(F : β β β)
(hF : F = fun (n : β) => β m in Finset.Icc 1 (6 * n - 1), min (dist_to_int (m/(6*n)) ) (dist_to_int (m/(3*n))))
: β n, n > 0 β F n = putnam_1997_b1_solution n :=
sorry | 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 Multiset.card a > 0 β§ (β i β a, i > 0) β§ a.sum = 1979) :
P putnam_1979_a1_solution β§ β a : Multiset β, P a β putnam_1979_a1_solution.prod β₯ a.prod :=
sorry | 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
: {{z : β | β x : Fin 2 β β, MvPolynomial.eval x f = z} | f : MvPolynomial (Fin 2) β} = putnam_1969_a1_solution :=
sorry | 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)] $|f'(x)| \leq |g(x)|$ for all $x$,
\item[(d)] $|g'(x)| \leq |f(x)|$ for all $x$, and
\item[(e)] $f(r) = 0$. \end{enumerate}
-/
theorem putnam_2023_a3
: sInf {r > 0 | β f g : β β β,
Differentiable β f β§ Differentiable β g β§
(β x : β, |deriv f x| β€ |g x| β§ |deriv g x| β€ |f x|) β§
f 0 > 0 β§ g 0 = 0 β§ f r = 0} = putnam_2023_a3_solution :=
sorry | 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 $A^{n_1}B^{n_2} \cdots A^{n_{k-1}}B^{n_k}$, where the $n_i$ can assume any integer values and $A^0 = B^0 = 1$. Further assume that $A^4 = B^7 = ABA^{-1}B = 1$, but $A^2 \ne 1$ and $B \ne 1$. Find the number of elements of $G$ than can be written as $C^2$ for some $C \in G$ and express each such square as a word in $A$ and $B$.
-/
theorem putnam_1976_b2
(G : Type*) [Group G]
(A B : G)
(word : List (β€ Γ β€) β G)
(hword : word = fun w : List (β€ Γ β€) => (List.map (fun t : β€ Γ β€ => A^(t.1)*B^(t.2)) w).prod)
(hG : β g : G, β w : List (β€ Γ β€), g = word w)
(hA : A^4 = 1 β§ A^2 β 1)
(hB : B^7 = 1 β§ B β 1)
(h1 : A*B*A^(-(1 : β€))*B = 1)
(S : Set G)
(hS : S = {g : G | β C : G, C^2 = g})
: S.ncard = putnam_1976_b2_solution.1 β§ S = {word w | w β putnam_1976_b2_solution.2} :=
sorry | 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 disjoint from $S$ ($A$ or $B$ may be empty). Let $L_S$ be the number of linear partitions of $S$. For each positive integer $n$, find the maximum of $L_S$ over all sets $S$ of $n$ points.
-/
theorem putnam_2006_b3
(IsLinearPartition : Finset (Fin 2 β β) β Finset (Finset (Fin 2 β β)) β Prop)
(IsLinearPartition_def : β S AB, IsLinearPartition S AB β
(AB.card = 2 β§ β A β AB, β B β AB,
A β B β§ (A βͺ B = S) β§ (A β© B = β
) β§
(β m b : β,
(β p β A, p 1 > m * p 0 + b) β§
(β p β B, p 1 < m * p 0 + b))))
(L : Finset (Fin 2 β β) β β)
(hL : β S, L S = {AB | IsLinearPartition S AB}.encard)
(n : β) (npos : 0 < n) :
IsGreatest {L S | (S) (hS : S.card = n)} (putnam_2006_b3_solution n) :=
sorry | 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 rational numbers.
-/
theorem putnam_1993_b3 :
let (r, s) := putnam_1993_b3_solution;
(MeasureTheory.volume
{p : Fin 2 β β | 0 < p β§ p < 1 β§ Even (round (p 0 / p 1))}
).toReal
= r + s * Real.pi :=
sorry | 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 $B(a, b, c) = (f(a), f(b), f(c))$, where $f(x) = 2x - 1$ if $x > 1/2$ and $f(x) = 2x$ otherwise. Will finitely many iterations of this balancing act always eventually produce a balanced triple?
-/
theorem putnam_1977_b3
(P balanced: β Γ β Γ β β Prop)
(B : β Γ β Γ β β β Γ β Γ β)
(hP : P = fun (a, b, c) => Irrational a β§ Irrational b β§ Irrational c β§ a > 0 β§ b > 0 β§ c > 0 β§ a + b + c = 1)
(hbalanced : balanced = fun (a, b, c) => a < 1/2 β§ b < 1/2 β§ c < 1/2)
(hB : B = fun (a, b, c) => (ite (a > 1/2) (2*a - 1) (2*a), ite (b > 1/2) (2*b - 1) (2*b), ite (c > 1/2) (2*c - 1) (2*c)))
: (β t : β Γ β Γ β, P t β β n : β, balanced (B^[n] t)) β putnam_1977_b3_solution :=
sorry | 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 P = n})
(negs : β[X] β β)
(hnegs : β P : β[X], negs P = β i in Finset.range (P.natDegree + 1), if P.coeff i < 0 then 1 else 0)
: sSup {negs (P^2) | P β S} = putnam_2022_a2_solution n :=
sorry | 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) :
((1 : β) / a + (1 : β) / b = (3 : β) / 2018) β (β¨a, bβ© β putnam_2018_a1_solution) :=
sorry | 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 $$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} A(x, y)e^{-x^2 - y^2} dx dy$$ as a polynomial in $g$.
-/
theorem putnam_1982_b2
(A : β Γ β β β)
(g I : β)
(hA : A = fun (x, y) => {(m, n) : β€ Γ β€ | m^2 + n^2 β€ x^2 + y^2}.ncard)
(hg : g = β' k : β, Real.exp (-k^2))
(hI : I = β« y : β, β« x : β, A (x, y) * Real.exp (-x^2 - y^2))
: I = putnam_1982_b2_solution.eval g :=
sorry | 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 smallest positive integer $a$ that occurs in any of these expressions?
-/
theorem putnam_2017_b2
(S : β€ β β β β€)
(p : β€ β β β Prop)
(q : β€ β Prop)
(hS : S = fun (a : β€) k β¦ β i in Finset.range k, (a + i))
(hp : β N k, p N k β β a > 0, S a k = N)
(hq : β N, q N β p N 2017 β§ β k : β, k > 1 β k β 2017 β Β¬p N k) :
IsLeast {a : β€ | q (S a 2017)} putnam_2017_b2_solution :=
sorry | 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 $P_k$. Let $R$ denote the map obtained by applying, in order, $R_1$, then $R_2, \dots$, then $R_n$. For an arbitrary point $(x,y)$, find, and simplify, the coordinates of $R(x,y)$.
-/
theorem putnam_2004_b4
(n : β)
(nge2 : n β₯ 2)
(R Rk : β β β β β)
(hR : R 0 = id β§ β k : β, R (k + 1) = Rk (k + 1) β R k)
(hRk : Rk = fun (k : β) (Q : β) β¦ k + Complex.exp (Complex.I * 2 * Real.pi / n) * (Q - k))
: (R n = putnam_2004_b4_solution n) :=
sorry | 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 $0$, $1$, $2$, $4$, and $5$, then it also contains a matrix of rank $3$. [The rank of a nonzero matrix $M$ is the largest $k$ such that the entries of some $k$ rows and some $k$ columns form a $k$ by $k$ matrix with a nonzero determinant.]
-/
theorem putnam_1981_b4
(VAB : Set (Matrix (Fin 5) (Fin 7) β) β Prop)
(Vrank : Set (Matrix (Fin 5) (Fin 7) β) β β β Prop)
(hVAB : β V, VAB V = (β A β V, β B β V, β r s : β, r β’ A + s β’ B β V))
(hVrank : β V k, Vrank V k = β A β V, A.rank = k) :
putnam_1981_b4_solution β
(β V,
VAB V β Vrank V 0 β Vrank V 1 β
Vrank V 2 β Vrank V 4 β Vrank V 5 β
Vrank V 3) :=
sorry | 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_a4
(S : β β Set (EuclideanSpace β (Fin 2)))
(hS : S = fun R => ball (0 : EuclideanSpace β (Fin 2)) R)
(I : β β β)
(hI : I = fun R => β« p in S R,
(1 + 2*(p 0)^2)/(1 + (p 0)^4 + 6*(p 0)^2*(p 1)^2 + (p 1)^4) - (1 + (p 1)^2)/(2 + (p 0)^4 + (p 1)^4)) :
Tendsto I atTop (π putnam_2021_a4_solution) :=
sorry | 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 i (j + 1)))
(hF : F = {g : β€ β β€ | β e : Fin 10 β Fin 2, g = (f 0)^[e 0] β (f 1)^[e 1] β (f 2)^[e 2] β (f 3)^[e 3] β (f 4)^[e 4] β (f 5)^[e 5] β (f 6)^[e 6] β (f 7)^[e 7] β (f 8)^[e 8] β (f 9)^[e 9]})
: β A : Finset β€, A.Nonempty β {g β F | g '' A = A}.card β€ 512 :=
sorry | 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 $f(5^{a_1}+5^{a_2}+\dots+5^{a_k})$ depends only on the sum $a_1+a_2+\dots+a_k$.
\item[(b)] Assuming part (a), we can define $g(s)=f(5^{a_1}+5^{a_2}+\dots+5^{a_k})$, where $s=a_1+a_2+\dots+a_k$. Find the least positive integer $p$ for which $g(s)=g(s + p)$, for all $s \geq 1$, or else show that no such $p$ exists.
\end{enumerate}
-/
theorem putnam_1984_a6
(f : β β β)
(hf : β n, some (f n) = (Nat.digits 10 (n !)).find? (fun d β¦ d β 0))
(IsPeriodicFrom : β β (β β β) β β β Prop)
(IsPeriodicFrom_def : β x f p, IsPeriodicFrom x f p β Periodic (f β (Β· + x)) p)
(P : β β (β β β) β β β Prop)
(P_def : β x g p, P x g p β if p = 0 then (β q > 0, Β¬ IsPeriodicFrom x g q) else
IsLeast {q | 0 < q β§ IsPeriodicFrom x g q} p) :
β g : β β β,
(βα΅ (k > 0) (a : Fin k β β) (ha : Injective a), f (β i, 5 ^ (a i)) = g (β i, a i)) β§
P 1 g putnam_1984_a6_solution :=
sorry | 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 : 1 β gset)
(hgsetgen : Subgroup.closure gset = β€)
: β b β Set.Ioo (0 : β) 1, β C > 0, Tendsto (fun m : β => (1 / b ^ (2 * m)) * (β x : G, (mgprob m x - 1 / n) ^ 2)) atTop (π C) :=
sorry | 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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