name
stringlengths 7
15
| statement
stringlengths 75
1.66k
| id
stringlengths 16
16
|
|---|---|---|
putnam_1978_a4 | theorem putnam_1978_a4
(bypass : (S : Type) β [inst : Mul S] β Prop)
(hbypass : bypass = fun S [Mul S] β¦ β a b c d : S, (a * b) * (c * d) = a * d)
: ((β (S : Type) (_ : Mul S), bypass S β β a b c : S, a * b = c β (c * c = c β§ β d : S, a * d = c * d))
β§ (β (S : Type) (_ : Mul S) (_ : Fintype S), bypass S β§ (β a : S, a * a = a) β§ (β a b : S, a * b = a β§ a β b) β§ (β a b : S, a * b β a))) :=
sorry | 334c81bdac4a7a55 |
putnam_2022_a6 | abbrev putnam_2022_a6_solution : β β β := sorry
-- (fun n : β => n)
/--
Let $n$ be a positive integer. Determine, in terms of $n$, the largest integer $m$ with the following property: There exist real numbers $x_1,\dots,x_{2n}$ with $-1<x_1<x_2<\cdots<x_{2n}<1$ such that the sum of the lengths of the $n$ intervals $[x_1^{2k-1},x_2^{2k-1}],[x_3^{2k-1},x_4^{2k-1}],\dots,[x_{2n-1}^{2k-1},x_{2n}^{2k-1}]$ is equal to $1$ for all integers $k$ with $1 \leq k \leq m$.
-/
theorem putnam_2022_a6
(n : β) (hn : 0 < n) :
IsGreatest
{m : β | β x : β β β,
StrictMono x β§ -1 < x 1 β§ x (2 * n) < 1 β§
β k β Icc 1 m, β i in Icc 1 n, ((x (2 * i) : β) ^ (2 * k - 1) - (x (2 * i - 1)) ^ (2 * k - 1)) = 1}
(putnam_2022_a6_solution n) :=
sorry | 23a94c220ea3b0a5 |
putnam_2010_b2 | abbrev putnam_2010_b2_solution : β := sorry
-- 3
/--
Given that $A$, $B$, and $C$ are noncollinear points in the plane with integer coordinates such that the distances $AB$, $AC$, and $BC$ are integers, what is the smallest possible value of $AB$?
-/
theorem putnam_2010_b2
(ABCintcoords ABCintdists ABCall: EuclideanSpace β (Fin 2) β EuclideanSpace β (Fin 2) β EuclideanSpace β (Fin 2) β Prop)
(hABCintcoords : β A B C, ABCintcoords A B C β (β i : Fin 2, A i = round (A i) β§ B i = round (B i) β§ C i = round (C i)))
(hABCintdists : β A B C, ABCintdists A B C β (dist A B = round (dist A B) β§ dist A C = round (dist A C) β§ dist B C = round (dist B C)))
(hABCall : β A B C, ABCall A B C β (Β¬Collinear β {A, B, C} β§ ABCintcoords A B C β§ ABCintdists A B C)) :
IsLeast {y | β A B C, ABCall A B C β§ y = dist A B} putnam_2010_b2_solution :=
sorry | 557b3e7ddcd06891 |
putnam_2000_b2 | theorem putnam_2000_b2
: (β m n : β, m β₯ 1 β n β₯ m β n β£ Nat.gcd m n * Nat.choose n m) :=
sorry | ed011644bfce06ca |
putnam_2019_a4 | abbrev putnam_2019_a4_solution : Prop := sorry
-- False
/--
Let $f$ be a continuous real-valued function on $\mathbb{R}^3$. Suppose that for every sphere $S$ of radius $1$, the integral of $f(x,y,z)$ over the surface of $S$ equals $0$. Must $f(x,y,z)$ be identically 0?
-/
theorem putnam_2019_a4
(P : (EuclideanSpace β (Fin 3) β β) β Prop)
(P_def : β f, P f β β C, β« x in sphere C 1, f x βΞΌH[2] = 0) :
(β f, Continuous f β P f β f = 0) β putnam_2019_a4_solution :=
sorry | 47c50ab4e42f3f14 |
putnam_2009_a4 | abbrev putnam_2009_a4_solution : Prop := sorry
-- False
/--
Let $S$ be a set of rational numbers such that
\begin{enumerate}
\item[(a)] $0 \in S$;
\item[(b)] If $x \in S$ then $x+1\in S$ and $x-1\in S$; and
\item[(c)] If $x\in S$ and $x\not\in\{0,1\}$, then $\frac{1}{x(x-1)}\in S$.
\end{enumerate}
Must $S$ contain all rational numbers?
-/
theorem putnam_2009_a4
: ((β S : Set β, 0 β S β (β x β S, x + 1 β S β§ x - 1 β S) β (β x β S, x β ({0, 1} : Set β) β 1 / (x * (x - 1)) β S) β β r : β, r β S) β putnam_2009_a4_solution) :=
sorry | 5d5e527fd9614911 |
putnam_2024_a3 | abbrev putnam_2024_a3_solution : Prop := sorry
--True
/--
Let $S$ be the set of bijections $$T : \{1, 2, 3\} \times \{1, 2, ..., 2024\} \to \{1, 2, ..., 6072\}$$
such that $T(1, j) < T(2, j) < T(3, j)$ for all $j \in \{1, 2, ..., 2024\}$ and
$T(i, j) < T(i, j + 1)$ for all $i \in \{1, 2, 3\}$ and $j \in \{1, 2, ..., 2023\}$.
Do there exist $a, c$ in $\{1, 2, 3\}$ and $b$ and $d$ in $\{1, 2, ..., 2024\}$ such that
the fraction of elements $T$ in $S$ for which $T(a, b) < T(c, d)$ is at least $1/3$ and at most $2/3$?
-/
theorem putnam_2024_a3
(S : Set (β Γ β β β))
(hS : S = {T | Set.BijOn T (Finset.Icc 1 3 ΓΛ’ Finset.Icc 1 2024) (Finset.Icc 1 6072) β§
(β j β Finset.Icc 1 2024, StrictMonoOn (fun i => T (i, j)) (Set.Icc 1 3)) β§
(β i β Finset.Icc 1 3, StrictMonoOn (fun j => T (i, j)) (Set.Icc 1 2024)) β§
(β x, x β Finset.Icc 1 3 ΓΛ’ Finset.Icc 1 2024 β T x = 0)}) :
(β a β Finset.Icc 1 3, β b β Finset.Icc 1 2024, β c β Finset.Icc 1 3, β d β Finset.Icc 1 2024,
({T | T β S β§ T (a, b) < T (c, d)}.ncard / S.ncard : β) β Set.Icc (1/3) (2/3))
β putnam_2024_a3_solution :=
sorry | ba30a2d351b304b4 |
putnam_1971_b2 | abbrev putnam_1971_b2_solution : Set (β β β) := sorry
-- {fun x : β => (x^3 - x^2 - 1)/(2 * x * (x - 1))}
/--
Find all functions $F : \mathbb{R} \setminus \{0, 1\} \to \mathbb{R}$ that satisfy $F(x) + F\left(\frac{x - 1}{x}\right) = 1 + x$ for all $x \in \mathbb{R} \setminus \{0, 1\}$.
-/
theorem putnam_1971_b2
(S : Set β)
(hS : S = univ \ {0, 1})
(P : (β β β) β Prop)
(hP : P = fun (F : β β β) => β x β S, F x + F ((x - 1)/x) = 1 + x)
: (β F β putnam_1971_b2_solution, P F) β§ β f : β β β, P f β β F β putnam_1971_b2_solution, (β x β S, f x = F x) :=
sorry | 9bcd4567c23d131c |
putnam_2001_b3 | abbrev putnam_2001_b3_solution : β := sorry
-- 3
/--
For any positive integer $n$, let $\langle n \rangle$ denote the closest integer to $\sqrt{n}$. Evaluate $\sum_{n=1}^\infty \frac{2^{\langle n \rangle}+2^{-\langle n \rangle}}{2^n}$.
-/
theorem putnam_2001_b3
: β' n : Set.Ici 1, ((2 : β) ^ (round (Real.sqrt n)) + (2 : β) ^ (-round (Real.sqrt n))) / 2 ^ (n : β) = putnam_2001_b3_solution :=
sorry | ee626e2c3fb370a2 |
putnam_2011_b3 | abbrev putnam_2011_b3_solution : Prop := sorry
-- True
/--
Let $f$ and $g$ be (real-valued) functions defined on an open interval containing $0$, with $g$ nonzero and continuous at $0$. If $fg$ and $f/g$ are differentiable at $0$, must $f$ be differentiable at $0$?
-/
theorem putnam_2011_b3 :
((β f g : β β β,
g 0 β 0 β ContinuousAt g 0 β
DifferentiableAt β (fun x β¦ f x * g x) 0 β
DifferentiableAt β (fun x β¦ f x / g x) 0 β
(DifferentiableAt β f 0))
β putnam_2011_b3_solution) :=
sorry | aaa710ef0e9af4cf |
putnam_1966_b6 | theorem putnam_1966_b6
(y : β β β)
(hy : Differentiable β y β§ Differentiable β (deriv y))
(diffeq : deriv (deriv y) + Real.exp * y = 0)
: β r s N : β, β x > N, r β€ y x β§ y x β€ s :=
sorry | 374a451254b51fec |
putnam_1976_b6 | theorem putnam_1976_b6
(Ο : β β β€)
(hΟ : Ο = fun N : β => β d in Nat.divisors N, (d : β€))
(quasiperfect : β β Prop)
(quasiperfect_def : β N, quasiperfect N β Ο N = 2*N + 1)
: β N : β, quasiperfect N β β m : β€, Odd m β§ m^2 = N :=
sorry | 996ef5b82e155a2a |
putnam_1979_a5 | theorem putnam_1979_a5
(S : β β β β β€)
(hS : S = fun x : β => fun n : β => Int.floor (n*x))
(P : β β Prop)
(hP : β x, P x β x^3 - 10*x^2 + 29*x - 25 = 0)
: β Ξ± Ξ² : β, Ξ± β Ξ² β§ P Ξ± β§ P Ξ² β§ β n : β, β m : β€, m > n β§ β c d : β, S Ξ± c = m β§ S Ξ² d = m :=
sorry | e119f47714efe05f |
putnam_1994_b3 | abbrev putnam_1994_b3_solution : Set β := sorry
-- Set.Iio 1
/--
Find the set of all real numbers $k$ with the following property: For any positive, differentiable function $f$ that satisfies $f'(x)>f(x)$ for all $x$, there is some number $N$ such that $f(x)>e^{kx}$ for all $x>N$.
-/
theorem putnam_1994_b3 :
{k | β f (hf : (β x, 0 < f x β§ f x < deriv f x) β§ Differentiable β f),
β N, β x > N, Real.exp (k * x) < f x} = putnam_1994_b3_solution :=
sorry | 96841a25380621c6 |
putnam_1969_a5 | theorem putnam_1969_a5
(x0 y0 t : β)
(ht : 0 < t) :
x0 = y0 β β x y u : β β β,
Differentiable β x β§
Differentiable β y β§
Continuous u β§
deriv x = - 2 β’ y + u β§
deriv y = - 2 β’ x + u β§
x 0 = x0 β§
y 0 = y0 β§
x t = 0 β§
y t = 0 :=
sorry | 25332e9b27d1d921 |
putnam_1984_b3 | abbrev putnam_1984_b3_solution : Prop := sorry
-- True
/--
Prove or disprove the following statement: If $F$ is a finite set with two or more elements, then there exists a binary operation $*$ on F such that for all $x,y,z$ in $F$,
\begin{enumerate}
\item[(i)] $x*z=y*z$ implies $x=y$ (right cancellation holds), and
\item[(ii)] $x*(y*z) \neq (x*y)*z$ (\emph{no} case of associativity holds).
\end{enumerate}
-/
theorem putnam_1984_b3
: (β (F : Type*) (_ : Fintype F), Fintype.card F β₯ 2 β (β mul : F β F β F, β x y z : F, (mul x z = mul y z β x = y) β§ (mul x (mul y z) β mul (mul x y) z))) β putnam_1984_b3_solution :=
sorry | a493ab2d09dfdae3 |
putnam_1987_b5 | theorem putnam_1987_b5
(n : β)
(npos : n > 0)
(M : Matrix (Fin (2 * n)) (Fin n) β)
(hM : β z : Matrix (Fin 1) (Fin (2 * n)) β, z * M = 0 β (Β¬β i : Fin (2 * n), z 0 i = 0) β β i : Fin (2 * n), (z 0 i).im β 0)
: (β r : Matrix (Fin (2 * n)) (Fin 1) β, β w : Matrix (Fin n) (Fin 1) β, β i : (Fin (2 * n)), ((M * w) i 0).re = r i 0) :=
sorry | 1e761ad9a668c583 |
putnam_1997_b5 | theorem putnam_1997_b5
(n : β)
(hn : n β₯ 2)
: tetration 2 n β‘ tetration 2 (n-1) [MOD n] :=
sorry | 32ebbc8716c03f4c |
putnam_2012_b5 | theorem putnam_2012_b5
(g1 g2 : β β β)
(hgim : β x : β, g1 x β₯ 1 β§ g2 x β₯ 1)
(hgbd : β B1 B2 : β, β x : β, g1 x β€ B1 β§ g2 x β€ B2)
: β h1 h2 : β β β, β x : β, sSup {((g1 s)^x * (g2 s)) | s : β} = sSup {(x * (h1 t) + h2 t) | t : β} :=
sorry | 3e1b7c0315d3d306 |
putnam_2002_b5 | theorem putnam_2002_b5
: β n : β, {b : β | b β₯ 1 β§ (Nat.digits b n).length = 3 β§ List.Palindrome (Nat.digits b n)}.ncard β₯ 2002 :=
sorry | b1ea4f2c84839df6 |
putnam_1998_a6 | theorem putnam_1998_a6
(A B C : EuclideanSpace β (Fin 2))
(hint : β i : Fin 2, β a b c : β€, A i = a β§ B i = b β§ C i = c)
(htriangle : A β B β§ A β C β§ B β C)
(harea : (dist A B + dist B C) ^ 2 < 8 * (MeasureTheory.volume (convexHull β {A, B, C})).toReal + 1)
(threesquare : (EuclideanSpace β (Fin 2)) β (EuclideanSpace β (Fin 2)) β (EuclideanSpace β (Fin 2)) β Prop)
(threesquare_def : threesquare = fun P Q R β¦ dist Q P = dist Q R β§ βͺP - Q, R - Qβ«_β = 0)
: (threesquare A B C β¨ threesquare B C A β¨ threesquare C A B) :=
sorry | 901f127e8be3074f |
putnam_2020_a1 | abbrev putnam_2020_a1_solution : β := sorry
-- 508536
/--
Find the number of positive integers $N$ satisfying: (i) $N$ is divisible by $2020$, (ii) $N$ has at most $2020$ decimal digits, (iii) The decimal digits of $N$ are a string of consecutive ones followed by a string of consecutive zeros.
-/
theorem putnam_2020_a1
: Set.ncard {x : β | (2020 β£ x) β§ (Nat.log 10 x) + 1 β€ 2020 β§ (β k l, k β₯ l β§ x = β i in Finset.range (k-l+1), 10 ^ (i+l))} = putnam_2020_a1_solution :=
sorry | 8b30c97573ce448f |
putnam_1988_a6 | abbrev putnam_1988_a6_solution : Prop := sorry
-- True
/--
If a linear transformation $A$ on an $n$-dimensional vector space has $n+1$ eigenvectors such that any $n$ of them are linearly independent, does it follow that $A$ is a scalar multiple of the identity? Prove your answer.
-/
theorem putnam_1988_a6
: (β (F V : Type*) (_ : Field F) (_ : AddCommGroup V) (_ : Module F V) (_ : FiniteDimensional F V) (n : β) (A : Module.End F V) (evecs : Set V), (n = Module.finrank F V β§ evecs β {v : V | β f : F, A.HasEigenvector f v} β§ evecs.encard = n + 1 β§ (β sevecs : Fin n β V, (Set.range sevecs β evecs β§ (Set.range sevecs).encard = n) β LinearIndependent F sevecs)) β (β c : F, A = c β’ LinearMap.id)) β putnam_1988_a6_solution :=
sorry | fcbcffc8e57f2d07 |
putnam_2015_b1 | theorem putnam_2015_b1
(f : β β β)
(nzeros : (β β β) β β β Prop)
(fdiff : ContDiff β 2 f β§ Differentiable β (iteratedDeriv 2 f))
(hnzeros : β f' : β β β, β n : β, nzeros f' n = ({x : β | f' x = 0}.encard β₯ n))
(fzeros : nzeros f 5)
: nzeros (f + 6 * deriv f + 12 * iteratedDeriv 2 f + 8 * iteratedDeriv 3 f) 2 :=
sorry | 9af92b96aa448cca |
putnam_2005_b1 | abbrev putnam_2005_b1_solution : MvPolynomial (Fin 2) β := sorry
-- (MvPolynomial.X 1 - 2 * MvPolynomial.X 0) * (MvPolynomial.X 1 - 2 * MvPolynomial.X 0 - 1)
/--
Find a nonzero polynomial $P(x,y)$ such that $P(\lfloor a \rfloor,\lfloor 2a \rfloor)=0$ for all real numbers $a$. (Note: $\lfloor \nu \rfloor$ is the greatest integer less than or equal to $\nu$.)
-/
theorem putnam_2005_b1
: putnam_2005_b1_solution β 0 β§ β a : β, MvPolynomial.eval (fun n : Fin 2 => if (n = 0) then (Int.floor a : β) else (Int.floor (2 * a))) putnam_2005_b1_solution = 0 :=
sorry | 5b82fc7c3e7bc40b |
putnam_1972_b4 | theorem putnam_1972_b4
(n : β)
(hn : n > 1)
(vars : β€ β β€ β β€ β (Fin 3 β β€))
(hvars : vars = fun a b c β¦ fun i β¦ ite (i = 0) a (ite (i = 1) b c))
: β P : MvPolynomial (Fin 3) β€, β x : β€, x = MvPolynomial.eval (vars (x^n) (x^(n+1)) (x + x^(n+2))) P :=
sorry | fb9052ce93417245 |
putnam_1980_b1 | abbrev putnam_1980_b1_solution : Set β := sorry
-- {c : β | c β₯ 1 / 2}
/--
For which real numbers $c$ is $(e^x+e^{-x})/2 \leq e^{cx^2}$ for all real $x$?
-/
theorem putnam_1980_b1
(c : β)
: (β x : β, (exp x + exp (-x)) / 2 β€ exp (c * x ^ 2)) β c β putnam_1980_b1_solution :=
sorry | d74af7f38f92cd8f |
putnam_1990_b1 | abbrev putnam_1990_b1_solution : Set (β β β) := sorry
-- {fun x : β => (Real.sqrt 1990) * Real.exp x, fun x : β => -(Real.sqrt 1990) * Real.exp x}
/--
Find all real-valued continuously differentiable functions $f$ on the real line such that for all $x$, $(f(x))^2=\int_0^x [(f(t))^2+(f'(t))^2]\,dt+1990$.
-/
theorem putnam_1990_b1
(P : (β β β) β Prop)
(P_def : β f, P f β β x,
(f x) ^ 2 = (β« t in (0 : β)..x, (f t) ^ 2 + (deriv f t) ^ 2) + 1990)
(f : β β β) :
(ContDiff β 1 f β§ P f) β f β putnam_1990_b1_solution :=
sorry | 2661f3261c98e78b |
putnam_1984_a5 | abbrev putnam_1984_a5_solution : β Γ β Γ β Γ β Γ β := sorry
-- (1, 9, 8, 4, 25)
/--
Let $R$ be the region consisting of all triples $(x,y,z)$ of nonnegative real numbers satisfying $x+y+z \leq 1$. Let $w=1-x-y-z$. Express the value of the triple integral $\iiint_R x^1y^9z^8w^4\,dx\,dy\,dz$ in the form $a!b!c!d!/n!$, where $a$, $b$, $c$, $d$, and $n$ are positive integers.
-/
theorem putnam_1984_a5
(R : Set (Fin 3 β β))
(w : (Fin 3 β β) β β)
(hR : R = {p | (β i : Fin 3, p i β₯ 0) β§ p 0 + p 1 + p 2 β€ 1})
(hw : β p, w p = 1 - p 0 - p 1 - p 2) :
let (a, b, c, d, n) := putnam_1984_a5_solution;
a > 0 β§ b > 0 β§ c > 0 β§ d > 0 β§ n > 0 β§
(β« p in R, (p 0) ^ 1 * (p 1) ^ 9 * (p 2) ^ 8 * (w p) ^ 4 = ((a)! * (b)! * (c)! * (d)! : β) / (n)!) :=
sorry | fe6312600dff5dd7 |
putnam_1969_b3 | theorem putnam_1969_b3
(T : β β β)
(hT1 : β n : β, n β₯ 1 β (T n) * (T (n + 1)) = n)
(hT2 : Tendsto (fun n => (T n)/(T (n + 1))) atTop (π 1))
: Real.pi * (T 1)^2 = 2 :=
sorry | 16372ab48fd3908e |
putnam_1994_a5 | theorem putnam_1994_a5
(r : β β β)
(S : Set β)
(rpos : β n, r n > 0)
(rlim : Tendsto r atTop (π 0))
(hS : S = {x | β i : Fin 1994 β β, (β j k : Fin 1994, j < k β i j < i k) β§ (x = β j : Fin 1994, r (i j))}) :
β a b : β, a < b β (β c d : β, a β€ c β§ c < d β§ d β€ b β§ (Set.Ioo c d) β© S = β
) :=
sorry | 5dbd1f9747ed51a5 |
putnam_1979_b3 | abbrev putnam_1979_b3_solution : β β β€ := sorry
-- fun n β¦ (n - 1) / 2
/--
Let $F$ be a finite field with $n$ elements, and assume $n$ is odd. Suppose $x^2 + bx + c$ is an irreducible polynomial over $F$. For how many elements $d \in F$ is $x^2 + bx + c + d$ irreducible?
-/
theorem putnam_1979_b3
(F : Type*) [Field F] [Fintype F]
(n : β)
(hn : n = Fintype.card F)
(nodd : Odd n)
(b c : F)
(p : Polynomial F)
(hp : p = X ^ 2 + (C b) * X + (C c) β§ Irreducible p)
: ({d : F | Irreducible (p + (C d))}.ncard = putnam_1979_b3_solution n) :=
sorry | a7d55170758e337c |
putnam_2023_b1 | abbrev putnam_2023_b1_solution : β β β β β := sorry
-- (fun m n : β => Nat.choose (m + n - 2) (m - 1))
/--
Consider an $m$-by-$n$ grid of unit squares, indexed by $(i,j)$ with $1 \leq i \leq m$ and $1 \leq j \leq n$. There are $(m-1)(n-1)$ coins, which are initially placed in the squares $(i,j)$ with $1 \leq i \leq m-1$ and $1 \leq j \leq n-1$. If a coin occupies the square $(i,j)$ with $i \leq m-1$ and $j \leq n-1$ and the squares $(i+1,j)$, $(i,j+1)$, and $(i+1,j+1)$ are unoccupied, then a legal move is to slide the coin from $(i,j)$ to $(i+1,j+1)$. How many distinct configurations of coins can be reached starting from the initial configuration by a (possibly empty) sequence of legal moves?
-/
theorem putnam_2023_b1
(m n : β)
(initcoins : β β β β Bool)
(hinitcoins : initcoins = (fun i j : β => (i β€ m - 2 β§ j β€ n - 2 : Bool)))
(IsLegalMove : (β β β β Bool) β (β β β β Bool) β Prop)
(IsLegalMove_def : β coins1 coins2,
IsLegalMove coins1 coins2 β
β i j, i < m - 1 β§ j < n - 1 β§
coins1 i j β§ !coins1 (i + 1) j β§ !coins1 i (j + 1) β§ !coins1 (i + 1) (j + 1) β§
!coins2 i j β§ !coins2 (i + 1) j β§ !coins2 i (j + 1) β§ coins2 (i + 1) (j + 1) β§
(β i' j', ((i', j') β (i, j) β§ (i', j') β (i + 1, j) β§ (i', j') β (i, j + 1) β§ (i', j') β (i + 1, j + 1))
β coins1 i' j' = coins2 i' j'))
(IsLegalSeq : List (β β β β Bool) β Prop)
(IsLegalSeq_def : β seq, IsLegalSeq seq β seq.length β₯ 1 β§ seq[0]! = initcoins β§ (β i < seq.length - 1, IsLegalMove seq[i]! seq[i + 1]!))
(mnpos : 1 < m β§ 1 < n)
: {config : β β β β Bool | β seq : List (β β β β Bool), IsLegalSeq seq β§ config = seq.getLast!}.encard = putnam_2023_b1_solution m n :=
sorry | 7cf9991a66d49206 |
putnam_2011_a5 | theorem putnam_2011_a5
(F : (Fin 2 β β) β β)
(g : β β β)
(vec : β β β β (Fin 2 β β))
(Fgrad : (Fin 2 β β) β (Fin 2 β β))
(parallel : (Fin 2 β β) β (Fin 2 β β) β Prop)
(hparallel : parallel = (fun u v : Fin 2 β β => β c : β, u = c β’ v))
(Fgdiff : ContDiff β 2 F β§ ContDiff β 2 g)
(prop1 : β uu : Fin 2 β β, uu 0 = uu 1 β F uu = 0)
(prop2 : β x : β, g x > 0 β§ x ^ 2 * g x β€ 1)
(hvec : β x y : β, (vec x y) 0 = x β§ (vec x y) 1 = y)
(hFgrad : β uv : Fin 2 β β, Fgrad uv 0 = deriv (fun x : β => F (vec x (uv 1))) (uv 0) β§ Fgrad uv 1 = deriv (fun y : β => F (vec (uv 0) y)) (uv 1))
(prop3 : β uv : Fin 2 β β, Fgrad uv = 0 β¨ parallel (Fgrad uv) (vec (g (uv 0)) (-g (uv 1))))
: β C : β, β n β₯ 2, β x : Fin (n + 1) β β, sInf {Fxx : β | β i j : Fin (n + 1), i β j β§ Fxx = |F (vec (x i) (x j))|} β€ C / n :=
sorry | a0914c24d8c29742 |
putnam_2001_a5 | theorem putnam_2001_a5
: β! an : β€ Γ β, let (a, n) := an; a > 0 β§ n > 0 β§ a^(n+1) - (a+1)^n = 2001 :=
sorry | 2dd9ca5be69002fc |
putnam_1971_a4 | theorem putnam_1971_a4
(Ξ΅ : β)
(hΞ΅ : 0 < Ξ΅ β§ Ξ΅ < 1)
(P : β β β β MvPolynomial (Fin 2) β)
(hP : P = fun (n : β) (Ξ΄ : β) => (MvPolynomial.X 0 + MvPolynomial.X 1)^n * ((MvPolynomial.X 0)^2 - (MvPolynomial.C (2 - Ξ΄))*(MvPolynomial.X 0)*(MvPolynomial.X 1) + (MvPolynomial.X 1)^2))
: β N : β, β n β₯ N, β i : Fin 2 ββ β, MvPolynomial.coeff i (P n Ξ΅) β₯ 0 :=
sorry | 52cdd9017602529a |
putnam_2024_b5 | theorem putnam_2024_b5
(IsQualifyingSeq : {q : β} β (Fin q β β) β β β Prop)
(IsQualifyingSeq_def : β q w z, β [NeZero q],
IsQualifyingSeq w z β Monotone w β§ 1 β€ w 0 β§ w (-1 : Fin q) β€ z)
(k m : β) [NeZero k] [NeZero m]
(f : β β β)
(f_def : β n,
f n = {(x, y, z) : (Fin k β β) Γ (Fin m β β) Γ β |
IsQualifyingSeq x z β§ IsQualifyingSeq y z β§ z β€ n}.ncard) :
β P : β[X],
(β n > 0, f n = P.eval (n : β)) β§
(β i, 0 β€ P.coeff i) :=
sorry | 534c4adee8639ec0 |
putnam_2016_a1 | abbrev putnam_2016_a1_solution : β := sorry
-- 8
/--
Find the smallest positive integer $j$ such that for every polynomial $p(x)$ with integer coefficients and for every integer $k$, the integer \[ p^{(j)}(k) = \left. \frac{d^j}{dx^j} p(x) \right|_{x=k} \] (the $j$-th derivative of $p(x)$ at $k$) is divisible by 2016.
-/
theorem putnam_2016_a1 :
IsLeast {j : β | 0 < j β§ β P : β€[X], β k : β€, 2016 β£ (derivative^[j] P).eval k} putnam_2016_a1_solution :=
sorry | 03c43aebbf02eb17 |
putnam_2006_a1 | abbrev putnam_2006_a1_solution : β := sorry
-- 6 * Real.pi ^ 2
/--
Find the volume of the region of points $(x,y,z)$ such that
\[
(x^2 + y^2 + z^2 + 8)^2 \leq 36(x^2 + y^2).
\]
-/
theorem putnam_2006_a1
: ((MeasureTheory.volume {(x, y, z) : β Γ β Γ β | (x ^ 2 + y ^ 2 + z ^ 2 + 8) ^ 2 β€ 36 * (x ^ 2 + y ^ 2)}).toReal = putnam_2006_a1_solution) :=
sorry | 36e42ea769659c68 |
putnam_1983_a1 | abbrev putnam_1983_a1_solution : β := sorry
-- 2301
/--
How many positive integers $n$ are there such that $n$ is an exact divisor of at least one of the numbers $10^{40},20^{30}$?
-/
theorem putnam_1983_a1
: {n : β€ | n > 0 β§ (n β£ 10 ^ 40 β¨ n β£ 20 ^ 30)}.encard = putnam_1983_a1_solution :=
sorry | 494eae0724242169 |
putnam_1993_a1 | abbrev putnam_1993_a1_solution : β := sorry
-- 4 / 9
/--
The horizontal line $y=c$ intersects the curve $y=2x-3x^3$ in the first quadrant as in the figure. Find $c$ so that the areas of the two shaded regions are equal. [Figure not included. The first region is bounded by the $y$-axis, the line $y=c$ and the curve; the other lies under the curve and above the line $y=c$ between their two points of intersection.]
-/
theorem putnam_1993_a1
: 0 < putnam_1993_a1_solution β§ putnam_1993_a1_solution < (4 * Real.sqrt 2) / 9 β§ (β« x in Set.Ioo 0 ((Real.sqrt 2) / 3), max (putnam_1993_a1_solution - (2 * x - 3 * x ^ 3)) 0) = (β« x in Set.Ioo 0 ((Real.sqrt 6) / 3), max ((2 * x - 3 * x ^ 3) - putnam_1993_a1_solution) 0) :=
sorry | 581e929d21e2c8ee |
putnam_2009_b2 | abbrev putnam_2009_b2_solution : Set β := sorry
-- Ioc (1 / 3) 1
/--
A game involves jumping to the right on the real number line. If $a$ and $b$ are real numbers and $b > a$, the cost of jumping from $a$ to $b$ is $b^3-ab^2$. For what real numbers $c$ can one travel from $0$ to $1$ in a finite number of jumps with total cost exactly $c$?
-/
theorem putnam_2009_b2
: ({c : β | β s : β β β, s 0 = 0 β§ StrictMono s β§ (β n : β, s n = 1 β§ ((β i in Finset.range n, ((s (i + 1)) ^ 3 - (s i) * (s (i + 1)) ^ 2)) = c))} = putnam_2009_b2_solution) :=
sorry | fac39177bfae5cf0 |
putnam_2019_b2 | abbrev putnam_2019_b2_solution : β := sorry
-- 8/Real.pi^3
/--
For all $n \geq 1$, let
\[
a_n = \sum_{k=1}^{n-1} \frac{\sin \left( \frac{(2k-1)\pi}{2n} \right)}{\cos^2 \left( \frac{(k-1)\pi}{2n} \right) \cos^2 \left( \frac{k\pi}{2n} \right)}.
\]
Determine
\[
\lim_{n \to \infty} \frac{a_n}{n^3}.
\]
-/
theorem putnam_2019_b2
(a : β β β)
(ha : a = fun n : β => β k : Icc (1 : β€) (n - 1),
Real.sin ((2*k - 1)*Real.pi/(2*n))/((Real.cos ((k - 1)*Real.pi/(2*n))^2)*(Real.cos (k*Real.pi/(2*n))^2)))
: Tendsto (fun n : β => (a n)/n^3) atTop (π putnam_2019_b2_solution) :=
sorry | e0048563f684a2c4 |
putnam_1962_a2 | abbrev putnam_1962_a2_solution : Set (β β β) := sorry
-- {f : β β β | β a c : β, a β₯ 0 β§ f = fun x β¦ a / (1 - c * x) ^ 2}
/--
Find every real-valued function $f$ whose domain is an interval $I$ (finite or infinite) having 0 as a left-hand endpoint, such that for every positive member $x$ of $I$ the average of $f$ over the closed interval $[0, x]$ is equal to the geometric mean of the numbers $f(0)$ and $f(x)$.
-/
theorem putnam_1962_a2
(P : Set β β (β β β) β Prop)
(P_def : β s f, P s f β 0 β€ f β§ β x β s, β¨ t in Ico 0 x, f t = β(f 0 * f x)) :
(β f,
(P (Ioi 0) f β β g β putnam_1962_a2_solution, EqOn f g (Ici 0)) β§
(β e > 0, P (Ioo 0 e) f β β g β putnam_1962_a2_solution, EqOn f g (Ico 0 e))) β§
β f β putnam_1962_a2_solution, P (Ioi 0) f β¨ (β e > 0, P (Ioo 0 e) f) :=
sorry | 31a647955d81d399 |
putnam_1972_a2 | theorem putnam_1972_a2
: (β (S : Type*) (_ : Mul S), (β x y : S, x * (x * y) = y β§ ((y * x) * x) = y) β (β x y : S, x * y = y * x)) β§ β (S : Type*) (_ : Mul S), (β x y : S, x * (x * y) = y β§ ((y * x) * x) = y) β§ Β¬(β x y z : S, x * (y * z) = (x * y) * z) :=
sorry | 031f9358a653e9dc |
putnam_1965_a6 | theorem putnam_1965_a6
(u v m : β)
(hu : 0 < u)
(hv : 0 < v)
(hm : 1 < m) :
(βα΅ (x > 0) (y > 0),
u * x + v * y = 1 β§
x ^ m + y ^ m = 1 β§
u = x ^ (m - 1) β§
v = y ^ (m - 1)) β
β n, u ^ n + v ^ n = 1 β§ mβ»ΒΉ + nβ»ΒΉ = 1 :=
sorry | 85e0f90cb4308080 |
putnam_2002_a3 | theorem putnam_2002_a3
(n Tn : β€)
(hn : n β₯ 2)
(hTn : Tn = Set.ncard {S : Set β€ | S β Set.Icc 1 n β§ Nonempty S β§ β k : β€, k = ((1 : β) / S.ncard) * (β' s : S, s.1)})
: Even (Tn - n) :=
sorry | 292e7385d3d83d07 |
putnam_2012_a3 | abbrev putnam_2012_a3_solution : β β β := sorry
-- fun x : β => Real.sqrt (1 - x^2)
/--
Let $f: [-1, 1] \to \mathbb{R}$ be a continuous function such that
\begin{itemize}
\item[(i)]
$f(x) = \frac{2-x^2}{2} f \left( \frac{x^2}{2-x^2} \right)$ for every $x$ in $[-1, 1]$,
\item[(ii)]
$f(0) = 1$, and
\item[(iii)]
$\lim_{x \to 1^-} \frac{f(x)}{\sqrt{1-x}}$ exists and is finite.
\end{itemize}
Prove that $f$ is unique, and express $f(x)$ in closed form.
-/
theorem putnam_2012_a3
(S : Set β)
(hS : S = Set.Icc (-1 : β) 1)
(fsat : (β β β) β Prop)
(hfsat : fsat = fun f : β β β => ContinuousOn f S β§
(β x β S, f x = ((2 - x^2)/2)*f (x^2/(2 - x^2))) β§ f 0 = 1 β§
(β y : β, leftLim (fun x : β => (f x)/Real.sqrt (1 - x)) 1 = y))
: fsat putnam_2012_a3_solution β§ β f : β β β, fsat f β β x β S, f x = putnam_2012_a3_solution x :=
sorry | bfdb8265cfca8d43 |
putnam_1997_a3 | abbrev putnam_1997_a3_solution : β := sorry
-- Real.sqrt (Real.exp 1)
/--
Evaluate \begin{gather*} \int_0^\infty \left(x-\frac{x^3}{2}+\frac{x^5}{2\cdot 4}-\frac{x^7}{2\cdot 4\cdot 6}+\cdots\right) \\ \left(1+\frac{x^2}{2^2}+\frac{x^4}{2^2\cdot 4^2}+\frac{x^6}{2^2\cdot 4^2 \cdot 6^2}+\cdots\right)\,dx. \end{gather*}
-/
theorem putnam_1997_a3
(series1 series2 : β β β)
(hseries1 : series1 = fun x => β' n : β, (-1)^n * x^(2*n + 1)/(β i : Finset.range n, 2 * ((i : β) + 1)))
(hseries2 : series2 = fun x => β' n : β, x^(2*n)/(β i : Finset.range n, (2 * ((i : β) + 1))^2))
: Tendsto (fun t => β« x in Set.Icc 0 t, series1 x * series2 x) atTop (π (putnam_1997_a3_solution)) :=
sorry | 023a3b90970ffb65 |
putnam_2014_a6 | abbrev putnam_2014_a6_solution : β β β := sorry
-- (fun n : β => n ^ n)
/--
Let \( n \) be a positive integer. What is the largest \( k \) for which there exist \( n \times n \) matrices \( M_1, \ldots, M_k \) and \( N_1, \ldots, N_k \) with real entries such that for all \( i \) and \( j \), the matrix product \( M_i N_j \) has a zero entry somewhere on its diagonal if and only if \( i \neq j \)?
-/
theorem putnam_2014_a6
(n : β)
(kex : β β Prop)
(npos : n > 0)
(hkex : β k β₯ 1, kex k = β M N : Fin k β Matrix (Fin n) (Fin n) β, β i j : Fin k, ((β p : Fin n, (M i * N j) p p = 0) β i β j))
: (putnam_2014_a6_solution n β₯ 1 β§ kex (putnam_2014_a6_solution n)) β§ (β k β₯ 1, kex k β k β€ putnam_2014_a6_solution n) :=
sorry | c4537e006b5bd630 |
putnam_2004_a6 | theorem putnam_2004_a6
(f : (Set.Icc (0 : β) 1 Γ Set.Icc (0 : β) 1) β β)
(fcont : Continuous f)
: (β« y : Set.Icc (0 : β) 1, (β« x : Set.Icc (0 : β) 1, f (x, y)) ^ 2) + (β« x : Set.Icc (0 : β) 1, (β« y : Set.Icc (0 : β) 1, f (x, y)) ^ 2) β€ (β« y : Set.Icc (0 : β) 1, (β« x : Set.Icc (0 : β) 1, f (x, y))) ^ 2 + (β« y : Set.Icc (0 : β) 1, (β« x : Set.Icc (0 : β) 1, (f (x, y)) ^ 2)) :=
sorry | 8500322d6a4a454f |
putnam_1973_a3 | theorem putnam_1973_a3
(b : β€ β β)
(hb : b = fun (n : β€) => sInf {k + (n : β)/(k : β) | k > (0 : β€)})
: β n : β€, n > 0 β floor (b n) = floor (Real.sqrt (4 * n + 1)) :=
sorry | b0683d5d232f6eb4 |
putnam_1963_a3 | abbrev putnam_1963_a3_solution : (β β β) β β β β β β β β := sorry
-- fun (f : β β β) (n : β) (x : β) (t : β) β¦ (x - t) ^ (n - 1) * (f t) / ((n - 1)! * t ^ n)
/--
Find an integral formula (i.e., a function $z$ such that $y(x) = \int_{1}^{x} z(t) dt$) for the solution of the differential equation $$\delta (\delta - 1) (\delta - 2) \cdots (\delta - n + 1) y = f(x)$$ with the initial conditions $y(1) = y'(1) = \cdots = y^{(n-1)}(1) = 0$, where $n \in \mathbb{N}$, $f$ is continuous for all $x \ge 1$, and $\delta$ denotes $x\frac{d}{dx}$.
-/
theorem putnam_1963_a3
(P : β β (β β β) β (β β β))
(hP : P 0 = id β§ β i y, P (i + 1) y = P i (fun x β¦ x * deriv y x - i * y x))
(n : β)
(hn : 0 < n)
(f y : β β β)
(hf : ContinuousOn f (Ici 1))
(hy : ContDiffOn β n y (Ici 1)) :
(β i < n, deriv^[i] y 1 = 0) β§ (Ici 1).EqOn (P n y) f β
β x β₯ 1, y x = β« t in (1 : β)..x, putnam_1963_a3_solution f n x t :=
sorry | 73704981f558c00d |
putnam_1991_a6 | theorem putnam_1991_a6
(nabsum : β β β Γ (β β β) β Prop)
(agt bge bg1 bg2 : β Γ (β β β) β Prop)
(A g B: β β β)
(hnabsum : βα΅ (n β₯ 1) (ab), nabsum n ab β
(ab.1 β₯ 1 β§ (β i < ab.1, ab.2 i > 0) β§
(β i β₯ ab.1, ab.2 i = 0) β§ (β i : Fin ab.1, ab.2 i) = n))
(hA : β n β₯ 1, A n = {a | nabsum n a β§
(β i : Fin (a.1 - 2), a.2 i > a.2 (i + 1) + a.2 (i + 2)) β§ 1 < a.1 β a.2 (a.1 - 2) > a.2 (a.1 - 1)}.encard)
(hg : g 0 = 1 β§ g 1 = 2 β§ (β j β₯ 2, g j = g (j - 1) + g (j - 2) + 1))
(hB : β n β₯ 1, B n = {b | nabsum n b β§
(β i : Fin (b.1 - 1), b.2 i β₯ b.2 (i + 1)) β§
(β i : Fin b.1, β j : β, b.2 i = g j) β§
(β k : β, b.2 0 = g k β§ (β j β€ k, β i : Fin b.1, b.2 i = g j))}.encard) :
β n β₯ 1, (A n) = (B n) :=
sorry | 1815b5f2f0f92c84 |
putnam_1986_a2 | abbrev putnam_1986_a2_solution : β := sorry
-- 3
/--
What is the units (i.e., rightmost) digit of
\[
\left\lfloor \frac{10^{20000}}{10^{100}+3}\right\rfloor ?
\]
-/
theorem putnam_1986_a2
: (Nat.floor ((10 ^ 20000 : β) / (10 ^ 100 + 3)) % 10 = putnam_1986_a2_solution) :=
sorry | fb3e62a24123156a |
putnam_1996_a2 | abbrev putnam_1996_a2_solution : (EuclideanSpace β (Fin 2)) β (EuclideanSpace β (Fin 2)) β Set (EuclideanSpace β (Fin 2)) := sorry
-- (fun O1 O2 : EuclideanSpace β (Fin 2) => {p : EuclideanSpace β (Fin 2) | dist p (midpoint β O1 O2) β₯ 1 β§ dist p (midpoint β O1 O2) β€ 2})
/--
Let $C_1$ and $C_2$ be circles whose centers are $10$ units apart, and whose radii are $1$ and $3$. Find, with proof, the locus of all points $M$ for which there exist points $X$ on $C_1$ and $Y$ on $C_2$ such that $M$ is the midpoint of the line segment $XY$.
-/
theorem putnam_1996_a2
(O1 O2 : EuclideanSpace β (Fin 2))
(C1 C2 : Set (EuclideanSpace β (Fin 2)))
(hC1 : C1 = sphere O1 1)
(hC2 : C2 = sphere O2 3)
(hO1O2 : dist O1 O2 = 10)
: {M : EuclideanSpace β (Fin 2) | β X Y, X β C1 β§ Y β C2 β§ M = midpoint β X Y} = putnam_1996_a2_solution O1 O2 :=
sorry | fc130f2776bfe01e |
putnam_2013_a2 | theorem putnam_2013_a2
(S : Set β€)
(hS : S = {n : β€ | n > 0 β§ Β¬β m : β€, m ^ 2 = n})
(P : β€ β List β€ β Prop)
(hP : β n a, P n a β
a.length > 0 β§ n < a[0]! β§
(β m : β€, m ^ 2 = n * a.prod) β§
(β i : Fin (a.length - 1), a[i] < a[i+(1:β)]))
(T : β€ β Set β€)
(hT : T = fun n : β€ => {m : β€ | β a : List β€, P n a β§ a[a.length - 1]! = m})
(f : β€ β β€)
(hf : β n β S, ((β r β T n, f n = r) β§ β r β T n, f n β€ r)) :
InjOn f S :=
sorry | 619904f5c957907e |
putnam_2003_a2 | theorem putnam_2003_a2
(n : β)
(hn : 0 < n)
(a b : Fin n β β)
(abnneg : β i, a i β₯ 0 β§ b i β₯ 0) :
(β i, a i) ^ ((1 : β) / n) +
(β i, b i) ^ ((1 : β) / n) β€
(β i, (a i + b i)) ^ ((1 : β) / n) :=
sorry | a4617d509a48d8bf |
putnam_1989_b1 | abbrev putnam_1989_b1_solution : β€ Γ β€ Γ β€ Γ β€ := sorry
-- (4, 2, -5, 3)
/--
A dart, thrown at random, hits a square target. Assuming that any two parts of the target of equal area are equally likely to be hit, find the probability that the point hit is nearer to the center than to any edge. Express your answer in the form $(a\sqrt{b}+c)/d$, where $a$, $b$, $c$, $d$ are integers and $b$, $d$ are positive.
-/
theorem putnam_1989_b1
(square Scloser perimeter: Set (EuclideanSpace β (Fin 2)))
(center : EuclideanSpace β (Fin 2))
(square_def : square = {p | β i : Fin 2, p i β Set.Icc 0 1})
(perimeter_def : perimeter = {p β square | p 0 = 0 β¨ p 0 = 1 β¨ p 1 = 0 β¨ p 1 = 1})
(center_def : center = (fun i : Fin 2 => 1 / 2))
(hScloser : Scloser = {p β square | β q β perimeter, dist p center < dist p q}) :
let (a, b, c, d) := putnam_1989_b1_solution;
b > 0 β§ d > 0 β§ (Β¬β n : β€, n^2 = b) β§
(volume Scloser).toReal / (volume square).toReal = (a * Real.sqrt b + c) / d :=
sorry | c4b8a6b60b2360cb |
putnam_2000_a4 | theorem putnam_2000_a4
: β y : β, Tendsto (fun B : β => β« x in Set.Ioo 0 B, Real.sin x * Real.sin (x ^ 2)) atTop (π y) :=
sorry | 547e3ad75b9ab1a4 |
putnam_2010_a4 | theorem putnam_2010_a4
: β n : β, n > 0 β Β¬Nat.Prime (10^10^10^n + 10^10^n + 10^n - 1) :=
sorry | 67bf2958bcb42226 |
putnam_1967_a1 | theorem putnam_1967_a1
(n : β) (hn : n > 0)
(a : Set.Icc 1 n β β)
(f : β β β)
(hf : f = (fun x : β => β i : Set.Icc 1 n, a i * Real.sin (i * x)))
(flesin : β x : β, abs (f x) β€ abs (Real.sin x))
: abs (β i : Set.Icc 1 n, i * a i) β€ 1 :=
sorry | 3ce146116b935699 |
putnam_1977_a1 | abbrev putnam_1977_a1_solution : β := sorry
-- -7 / 8
/--
Show that if four distinct points of the curve $y = 2x^4 + 7x^3 + 3x - 5$ are collinear, then their average $x$-coordinate is some constant $k$. Find $k$.
-/
theorem putnam_1977_a1
(y : β β β)
(hy : y = fun x β¦ 2 * x ^ 4 + 7 * x ^ 3 + 3 * x - 5)
(S : Finset β)
(hS : S.card = 4)
: (Collinear β {P : Fin 2 β β | P 0 β S β§ P 1 = y (P 0)} β (β x in S, x) / 4 = putnam_1977_a1_solution) :=
sorry | c520a08077a2b926 |
putnam_1978_b2 | abbrev putnam_1978_b2_solution : β := sorry
-- 7 / 4
/--
Find
\[
\sum_{i=1}^{\infty} \sum_{j=1}^{\infty} \frac{1}{i^2j + 2ij + ij^2}.
\]
-/
theorem putnam_1978_b2
: (β' i : β+, β' j : β+, (1 : β) / (i ^ 2 * j + 2 * i * j + i * j ^ 2) = putnam_1978_b2_solution) :=
sorry | 9abff58508c044d4 |
putnam_1995_a4 | theorem putnam_1995_a4
(n : β)
(hn : n > 0)
(necklace : Fin n β β€)
(hnecklacesum : β i : Fin n, necklace i = n - 1)
: β cut, β k, β i : {j : Fin n | j.1 β€ k}, necklace (cut + i) β€ k :=
sorry | 06a2770f00322a73 |
putnam_1968_b2 | theorem putnam_1968_b2
{G : Type*}
[Group G]
(hG : Finite G)
(A : Set G)
(hA : A.ncard > (Nat.card G : β)/2)
: β g : G, β x β A, β y β A, g = x * y :=
sorry | c341f40734ff7d56 |
putnam_1985_a4 | abbrev putnam_1985_a4_solution : Set (Fin 100) := sorry
-- {87}
/--
Define a sequence $\{a_i\}$ by $a_1=3$ and $a_{i+1}=3^{a_i}$ for $i \geq 1$. Which integers between $00$ and $99$ inclusive occur as the last two digits in the decimal expansion of infinitely many $a_i$?
-/
theorem putnam_1985_a4
(a : β β β)
(ha1 : a 1 = 3)
(ha : β i β₯ 1, a (i + 1) = 3 ^ a i) :
{k : Fin 100 | β N : β, β i β₯ N, a i % 100 = k} = putnam_1985_a4_solution :=
sorry | ea7f31b1aadfb47a |
putnam_2018_b3 | abbrev putnam_2018_b3_solution : Set β := sorry
-- {2^2, 2^4, 2^16, 2^256}
/--
Find all positive integers $n < 10^{100}$ for which simultaneously $n$ divides $2^n$, $n-1$ divides $2^n-1$, and $n-2$ divides $2^n - 2$.
-/
theorem putnam_2018_b3
(n : β) (hn : 0 < n) :
(n < 10^100 β§ ((n : β€) β£ (2^n : β€) β§ (n - 1 : β€) β£ (2^n - 1 : β€) β§ (n - 2 : β€) β£ (2^n - 2 : β€))) β n β putnam_2018_b3_solution :=
sorry | bf804d431255cba5 |
putnam_2008_b3 | abbrev putnam_2008_b3_solution : β := sorry
-- Real.sqrt 2 / 2
/--
What is the largest possible radius of a circle contained in a $4$-dimensional hypercube of side length $1$?
-/
theorem putnam_2008_b3
(H : Set (EuclideanSpace β (Fin 4)))
(H_def : H = {P : Fin 4 β β | β i : Fin 4, |P i| β€ 1 / 2})
(contains : β β Prop)
(contains_def : β r, contains r β
βα΅ (A : AffineSubspace β (EuclideanSpace β (Fin 4))) (C β A),
Module.finrank β A.direction = 2 β§
sphere C r β© A β H) :
IsGreatest contains putnam_2008_b3_solution :=
sorry | e79c95a0c0cf7ce7 |
putnam_1980_b6 | theorem putnam_1980_b6
(G : β€ Γ β€ β β)
(hG : β d n : β, d β€ n β (d = 1 β G (d, n) = 1/(n : β)) β§ (d > 1 β G (d, n) = (d/(n : β))*β i in Finset.Icc d n, G ((d : β€) - 1, (i : β€) - 1)))
: β d p : β, 1 < d β§ d β€ p β§ Prime p β Β¬p β£ (G (d, p)).den :=
sorry | 46eacf38ed14cb7c |
putnam_2005_b6 | theorem putnam_2005_b6
(n : β)
(v : Equiv.Perm (Fin n) β β)
(npos : n β₯ 1)
(hv : β p : Equiv.Perm (Fin n), v p = Set.encard {i : Fin n | p i = i})
: (β p : Equiv.Perm (Fin n), (Equiv.Perm.signAux p : β€) / (v p + 1 : β)) = (-1) ^ (n + 1) * (n / (n + 1 : β)) :=
sorry | 0cdb99c50d245314 |
putnam_2015_b6 | abbrev putnam_2015_b6_solution : β := sorry
-- Real.pi ^ 2 / 16
/--
For each positive integer $k$, let $A(k)$ be the number of odd divisors of $k$ in the interval $[1,\sqrt{2k})$. Evaluate $\sum_{k=1}^\infty (-1)^{k-1}\frac{A(k)}{k}$.
-/
theorem putnam_2015_b6
(A : β β β)
(hA : β k > 0, A k = {j : β | Odd j β§ j β£ k β§ j < Real.sqrt (2 * k)}.encard) :
Tendsto (fun K : β β¦ β k in Finset.Icc 1 K, (-1 : β) ^ ((k : β) - 1) * (A k / (k : β)))
atTop (π putnam_2015_b6_solution) :=
sorry | 839a9f57b374c516 |
putnam_1962_b3 | theorem putnam_1962_b3
(S : Set (EuclideanSpace β (Fin 2)))
(hS : Convex β S β§ 0 β S)
(htopo : (0 β interior S) β¨ IsClosed S)
(hray : β P : EuclideanSpace β (Fin 2), P β 0 β β Q : EuclideanSpace β (Fin 2), SameRay β P Q β§ Q β S)
: Bornology.IsBounded S :=
sorry | 6388dbd795a4f407 |
putnam_1972_b3 | theorem putnam_1972_b3
(G : Type*) [Group G]
(A B : G)
(hab : A * B * A = B * A^2 * B β§ A^3 = 1 β§ (β n : β€, n > 0 β§ B^(2*n - 1) = 1))
: B = 1 :=
sorry | b205109bad09b688 |
putnam_1988_a1 | abbrev putnam_1988_a1_solution : β := sorry
-- 6
/--
Let $R$ be the region consisting of the points $(x,y)$ of the cartesian plane satisfying both $|x|-|y| \leq 1$ and $|y| \leq 1$. Find the area of $R$.
-/
theorem putnam_1988_a1
(R : Set (Fin 2 β β))
(hR : R = {p | |p 0| - |p 1| β€ 1 β§ |p 1| β€ 1}) :
(volume R).toReal = putnam_1988_a1_solution :=
sorry | 8b20e82014063e18 |
putnam_2020_a6 | abbrev putnam_2020_a6_solution : β := sorry
-- Real.pi / 4
/--
For a positive integer $N$, let $f_N$ be the function defined by
\[
f_N(x) = \sum_{n=0}^N \frac{N+1/2-n}{(N+1)(2n+1)} \sin((2n+1)x).
\]
Determine the smallest constant $M$ such that $f_N(x) \leq M$ for all $N$ and all real $x$.
-/
theorem putnam_2020_a6
(f : β€ β (β β β))
(hf : f = fun N : β€ => fun x : β =>
β n in Finset.Icc 0 N, (N + 1/2 - n)/((N + 1)*(2*n + 1)) * Real.sin ((2*n + 1)*x))
: putnam_2020_a6_solution = sSup {y | βα΅ (N > 0) (x : β), y = f N x} :=
sorry | 41004573a19b5c80 |
putnam_1997_b2 | theorem putnam_1997_b2
(f g : β β β)
(hg : β x : β, g x β₯ 0)
(hfderiv1 : ContDiff β 1 f)
(hfderiv2 : Differentiable β (deriv f))
(hfg : β x : β, f x + iteratedDeriv 2 f x = -x * g x * deriv f x)
: IsBounded (range (fun x => |f x|)) :=
sorry | b0c4e241cf0ed0c5 |
putnam_1987_b2 | theorem putnam_1987_b2
(r s t : β)
(hsum : r + s β€ t)
: (β i : Finset.range (s + 1), (choose s i : β) / (choose t (r + i)) = ((t + 1) : β) / ((t + 1 - s) * choose (t - s) r)) :=
sorry | 7d8b12efa243a2df |
putnam_1969_a2 | theorem putnam_1969_a2
(D : (n : β) β Matrix (Fin n) (Fin n) β)
(hD : D = fun (n : β) => Ξ» (i : Fin n) (j : Fin n) => |(i : β) - (j : β)| )
: β n, n β₯ 2 β (D n).det = (-1)^((n : β€)-1) * ((n : β€)-1) * 2^((n : β€)-2) :=
sorry | cd5ce63c0d77a487 |
putnam_1979_a2 | abbrev putnam_1979_a2_solution : β β Prop := sorry
-- fun k : β => k β₯ 0
/--
For which real numbers $k$ does there exist a continuous function $f : \mathbb{R} \to \mathbb{R}$ such that $f(f(x)) = kx^9$ for all real $x$?
-/
theorem putnam_1979_a2
: β k : β, (β f : β β β, Continuous f β§ β x : β, f (f x) = k*x^9) β putnam_1979_a2_solution k :=
sorry | 707705262d9f6bc9 |
putnam_1994_b4 | theorem putnam_1994_b4
(matgcd : Matrix (Fin 2) (Fin 2) β€ β β€)
(A : Matrix (Fin 2) (Fin 2) β€)
(d : β β β€)
(hmatgcd : β M, matgcd M = Int.gcd (Int.gcd (Int.gcd (M 0 0) (M 0 1)) (M 1 0)) (M 1 1))
(hA : A 0 0 = 3 β§ A 0 1 = 2 β§ A 1 0 = 4 β§ A 1 1 = 3)
(hd : β n β₯ 1, d n = matgcd (A ^ n - 1))
: Tendsto d atTop atTop :=
sorry | edfba96dbbe691ff |
putnam_2011_b4 | theorem putnam_2011_b4
(games : Fin 2011 β Fin 2011 β Bool)
(T W : Matrix (Fin 2011) (Fin 2011) β)
(hT : T = (fun p1 p2 => ({g | games g p1 = games g p2}.ncard : β)))
(hW : W = (fun p1 p2 => ({g | games g p1 β§ !games g p2}.ncard - {g | !games g p1 β§ games g p2}.ncard : β)))
: β n : β, (T + Complex.I β’ W).det = n β§ (2 ^ 2010) β£ n :=
sorry | 2db5d4e5012aa7e5 |
putnam_2001_b4 | abbrev putnam_2001_b4_solution : Prop := sorry
-- True
/--
Let $S$ denote the set of rational numbers different from $\{-1,0,1\}$. Define $f:S\rightarrow S$ by $f(x)=x-1/x$. Prove or disprove that \[\bigcap_{n=1}^\infty f^{(n)}(S) = \emptyset,\] where $f^{(n)}$ denotes $f$ composed with itself $n$ times.
-/
theorem putnam_2001_b4
(S : Set β)
(hS : S = univ \ {-1, 0, 1})
(f : S β S)
(hf : β x : S, f x = x - 1 / (x : β))
: β n β Ici 1, f^[n] '' univ = β
β putnam_2001_b4_solution :=
sorry | 612d88fa19e431c1 |
putnam_1976_b1 | abbrev putnam_1976_b1_solution : β Γ β := sorry
-- (4, 1)
/--
Find $$\lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^{n}\left(\left\lfloor \frac{2n}{k} \right\rfloor - 2\left\lfloor \frac{n}{k} \right\rfloor\right).$$ Your answer should be in the form $\ln(a) - b$, where $a$ and $b$ are positive integers.
-/
theorem putnam_1976_b1
: Tendsto (fun n : β => ((1 : β)/n)*β k in Finset.Icc (1 : β€) n, (Int.floor ((2*n)/k) - 2*Int.floor (n/k))) atTop
(π (Real.log putnam_1976_b1_solution.1 - putnam_1976_b1_solution.2)) :=
sorry | 12148cb024c95406 |
putnam_1966_b1 | theorem putnam_1966_b1
(n : β)
(hn : n β₯ 3)
(L : ZMod n β (EuclideanSpace β (Fin 2)))
(hsq : β i : ZMod n, L i 0 β Set.Icc 0 1 β§ L i 1 β Set.Icc 0 1)
(hnoncol : β i j k : ZMod n, i β j β§ j β k β§ k β i β Β¬Collinear β {L i, L j, L k})
(hconvex : β i : ZMod n, segment β (L i) (L (i + 1)) β© interior (convexHull β {L j | j : ZMod n}) = β
)
: β i : Fin n, (dist (L i) (L (i + 1)))^2 β€ 4 :=
sorry | ea44d5a94a362efa |
putnam_2024_a4 | abbrev putnam_2024_a4_solution : Set β := sorry
--{7}
/--
Find all primes $p > 5$ for which there exists an integer $a$ and an integer $r$ satisfying $1 \le r \le p - 1$ with the following property:
the sequence $1, a, a^2, ..., a^{p-5}$ can be rearranged to form a sequence $b_0, ..., b_{p-5}$ such that $b_n - b_{n-1} - r$ is divisible by $p$ for $1 \le n \le p - 5$.
-/
theorem putnam_2024_a4 :
{p : β | p.Prime β§ 5 < p β§ β a r : β€, 1 β€ r β§ r β€ p - 1 β§
β e : β β β, Set.BijOn e (Set.Icc 0 (p - 5 : β)) (Set.Icc 0 (p - 5 : β)) β§
β n, 1 β€ n β§ n β€ (p - 5 : β) β
(p : β€) β£ a ^ (e n : β) - a ^ (e (n - 1) : β) - r} =
putnam_2024_a4_solution :=
sorry | a7409229837cb472 |
putnam_2009_a3 | abbrev putnam_2009_a3_solution : β := sorry
-- 0
/--
Let $d_n$ be the determinant of the $n \times n$ matrix whose entries, from left to right and then from top to bottom, are $\cos 1, \cos 2, \dots, \cos n^2$. (For example,\[ d_3 = \left|\begin{matrix} \cos 1 & \cos 2 & \cos 3 \\ \cos 4 & \cos 5 & \cos 6 \\ \cos 7 & \cos 8 & \cos 9 \end{matrix} \right|. \]The argument of $\cos$ is always in radians, not degrees.) Evaluate $\lim_{n\to\infty} d_n$.
-/
theorem putnam_2009_a3
(cos_matrix : (n : β) β Matrix (Fin n) (Fin n) β)
(hM : β n : β, β i j : Fin n, (cos_matrix n) i j = Real.cos (1 + n * i + j))
: Tendsto (fun n => (cos_matrix n).det) atTop (π putnam_2009_a3_solution) :=
sorry | 16bcb3911e43c1bf |
putnam_2019_a3 | abbrev putnam_2019_a3_solution : β := sorry
-- 2019^(-(1:β)/2019)
/--
Given real numbers $b_0, b_1, \dots, b_{2019}$ with $b_{2019} \neq 0$, let $z_1,z_2,\dots,z_{2019}$ be
the roots in the complex plane of the polynomial
\[
P(z) = \sum_{k=0}^{2019} b_k z^k.
\]
Let $\mu = (|z_1| + \cdots + |z_{2019}|)/2019$ be the average of the distances from $z_1,z_2,\dots,z_{2019}$ to the origin. Determine the largest constant $M$ such that $\mu \geq M$ for all choices of $b_0,b_1,\dots, b_{2019}$ that satisfy
\[
1 \leq b_0 < b_1 < b_2 < \cdots < b_{2019} \leq 2019.
\]
-/
theorem putnam_2019_a3
(v : Polynomial β β Prop)
(hv : v = fun b => b.degree = 2019 β§ 1 β€ (b.coeff 0).re β§ (b.coeff 2019).re β€ 2019 β§
(β i : Fin 2020, (b.coeff i).im = 0) β§ (β i : Fin 2019, (b.coeff i).re < (b.coeff (i + 1)).re))
(ΞΌ : Polynomial β β β)
(hΞΌ : ΞΌ = fun b => (Multiset.map (fun Ο : β => βΟβ) (Polynomial.roots b)).sum/2019) :
IsGreatest {M : β | β b, v b β ΞΌ b β₯ M} putnam_2019_a3_solution :=
sorry | c0caf67c7210190b |
putnam_2022_a1 | abbrev putnam_2022_a1_solution : Set (β Γ β) := sorry
-- {(a, b) | (a = 0 β§ b = 0) β¨ 1 β€ |a| β¨ (0 < |a| β§ |a| < 1 β§ letI rm := (1 - β(1 - a ^ 2)) / a; letI rp := (1 + β(1 - a ^ 2)) / a; (b < Real.log (1 + rm ^ 2) - a * rm β¨ b > Real.log (1 + rp ^ 2) - a * rp))}
/--
Determine all ordered pairs of real numbers $(a,b)$ such that the line $y = ax+b$ intersects the curve $y = \ln(1+x^2)$ in exactly one point.
-/
theorem putnam_2022_a1
: {(a, b) | β! x : β, a * x + b = Real.log (1 + x^2)} = putnam_2022_a1_solution :=
sorry | e5d6a4cd681e259f |
putnam_2000_b5 | theorem putnam_2000_b5
(S : β β Set β€)
(hSfin : β n, Set.Finite (S n))
(hSpos : β n, β s β S n, s > 0)
(hSdef : β n, β a, a β S (n + 1) β Xor' (a - 1 β S n) (a β S n))
: (β n, β N β₯ n, S N = S 0 βͺ {M : β€ | M - N β S 0}) :=
sorry | 8ec0022a4579ff4a |
putnam_2010_b5 | abbrev putnam_2010_b5_solution : Prop := sorry
-- False
/--
Is there a strictly increasing function $f: \mathbb{R} \to \mathbb{R}$ such that $f'(x) = f(f(x))$ for all $x$?
-/
theorem putnam_2010_b5 :
(β f : β β β, StrictMono f β§ Differentiable β f β§ (β x : β, deriv f x = f (f x))) β putnam_2010_b5_solution :=
sorry | 10fb3d8ff1be8a64 |
putnam_1978_a3 | abbrev putnam_1978_a3_solution : β := sorry
-- 2
/--
Let $p(x) = 2(x^6 + 1) + 4(x^5 + x) + 3(x^4 + x^2) + 5x^3$. For $k$ with $0 < k < 5$, let
\[
I_k = \int_0^{\infty} \frac{x^k}{p(x)} \, dx.
\]
For which $k$ is $I_k$ smallest?
-/
theorem putnam_1978_a3
(p : Polynomial β)
(hp : p = 2 * (X ^ 6 + 1) + 4 * (X ^ 5 + X) + 3 * (X ^ 4 + X ^ 2) + 5 * X ^ 3)
(I : β β β)
(hI : I = fun k β¦ β« x in Ioi 0, x ^ k / p.eval x) :
IsLeast {y | β k β Ioo 0 5, I k = y} putnam_1978_a3_solution :=
sorry | 6dfa01bb29a42f75 |
putnam_1985_b5 | abbrev putnam_1985_b5_solution : β := sorry
-- sqrt (Real.pi / 1985) * exp (-3970)
/--
Evaluate $\int_0^\infty t^{-1/2}e^{-1985(t+t^{-1})}\,dt$. You may assume that $\int_{-\infty}^\infty e^{-x^2}\,dx = \sqrt{\pi}$.
-/
theorem putnam_1985_b5
(fact : β« x in univ, exp (- x ^ 2) = sqrt (Real.pi))
: (β« t in Set.Ioi 0, t ^ (- (1 : β) / 2) * exp (-1985 * (t + t ^ (-(1 : β)))) = putnam_1985_b5_solution) :=
sorry | a3d6513e2e95d266 |
putnam_1968_a3 | theorem putnam_1968_a3
(Ξ± : Type*) [Finite Ξ±] :
β (n : β) (s : Fin (2 ^ n) β Set Ξ±),
s 0 = β
β§
(β t, β! i, s i = t) β§
(β i, i + 1 < 2 ^ n β (s i β s (i + 1)).ncard = 1) :=
sorry | 669b05b58514fda1 |
putnam_1992_b1 | abbrev putnam_1992_b1_solution : β β β€ := sorry
-- fun n β¦ 2 * n - 3
/--
Let $S$ be a set of $n$ distinct real numbers. Let $A_S$ be the set of numbers that occur as averages of two distinct elements of $S$. For a given $n \geq 2$, what is the smallest possible number of elements in $A_S$?
-/
theorem putnam_1992_b1
(n : β) (hn : n β₯ 2)
(A : Finset β β Set β)
(hA : A = fun S β¦ {x | β a β S, β b β S, a β b β§ (a + b) / 2 = x}) :
IsLeast {k : β€ | β S : Finset β, S.card = n β§ k = (A S).ncard} (putnam_1992_b1_solution n) :=
sorry | ae38b0d5b9256206 |
putnam_2018_a2 | abbrev putnam_2018_a2_solution : β β β := sorry
-- (fun n : β => if n = 1 then 1 else -1)
/--
Let \( S_1, S_2, \ldots, S_{2^n-1} \) be the nonempty subsets of \( \{1, 2, \ldots, n\} \) in some order, and let \( M \) be the \( (2^n - 1) \times (2^n - 1) \) matrix whose \((i, j)\) entry is $m_{ij} = \begin{cases} 0 & \text{if } S_i \cap S_j = \emptyset; \\ 1 & \text{otherwise}. \end{cases} $ Calculate the determinant of \( M \).
-/
theorem putnam_2018_a2
(n : β)
(S : Fin (2 ^ n - 1) β Set β)
(M : Matrix (Fin (2 ^ n - 1)) (Fin (2 ^ n - 1)) β)
(npos : n β₯ 1)
(hS : Set.range S = (Set.Icc 1 n).powerset \ {β
})
(hM : β i j, M i j = if (S i β© S j = β
) = True then 0 else 1) :
M.det = putnam_2018_a2_solution n :=
sorry | 092833985e3e20a9 |
putnam_1970_b4 | theorem putnam_1970_b4
(x : β β β)
(hdiff : DifferentiableOn β x (Set.Icc 0 1) β§ DifferentiableOn β (deriv x) (Set.Icc 0 1))
(hx : x 1 - x 0 = 1)
(hv : deriv x 0 = 0 β§ deriv x 1 = 0)
(hs : β t β Set.Ioo 0 1, |deriv x t| β€ 3/2)
: β t β Set.Icc 0 1, |(deriv (deriv x)) t| β₯ 9/2 :=
sorry | 6a21dc92728d8dad |
putnam_2007_b1 | theorem putnam_2007_b1
(f : Polynomial β€)
(hf : β n : β, f.coeff n β₯ 0)
(hfnconst : β n : β, n > 0 β§ f.coeff n > 0)
(n : β€)
(hn : n > 0)
: f.eval n β£ f.eval (f.eval n + 1) β n = 1 :=
sorry | 79279b52f6d8e15f |
putnam_2017_b1 | theorem putnam_2017_b1
(lines : Set (Set (Fin 2 β β)))
(L1 L2 : Set (Fin 2 β β))
(L1L2lines : L1 β lines β§ L2 β lines)
(L1L2distinct : L1 β L2)
(hlines : lines = {L | β v w : Fin 2 β β, w β 0 β§ L = {p | β t : β, p = v + t β’ w}}) :
L1 β© L2 β β
β (β lambda : β, lambda β 0 β
β P, (P β L1 β§ P β L2) β β A1 A2, A1 β L1 β§ A2 β L2 β§ (A2 - P = lambda β’ (A1 - P))) :=
sorry | 652348a1a8a3be26 |
putnam_1973_b2 | theorem putnam_1973_b2
(z : β)
(hzrat : β q1 q2 : β, z.re = q1 β§ z.im = q2)
(hznorm : βzβ = 1)
: β n : β€, β q : β, βz^(2*n) - 1β = q :=
sorry | a1f56ebd4063f7ff |
Subsets and Splits
No saved queries yet
Save your SQL queries to embed, download, and access them later. Queries will appear here once saved.