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putnam_1977_a3 | 1c0300c4-24bf-544a-9243-ff1c9f1674af | train | abbrev putnam_1977_a3_solution : (β β β) β (β β β) β (β β β) := sorry
-- fun f g x β¦ g x - f (x - 3) + f (x - 1) + f (x + 1) - f (x + 3)
/--
Let $f, g, h$ be functions $\mathbb{R} \to \mathbb{R}$. Find an expression for $h(x)$ in terms of $f$ and $g$ such that $f(x) = \frac{h(x + 1) + h(x - 1)}{2}$ and $g(x) = \frac{h(x + 4) + h(x - 4)}{2}$.
-/
theorem putnam_1977_a3
(f g h : β β β)
(hf : β x, f x = (h (x + 1) + h (x - 1)) / 2)
(hg : β x, g x = (h (x + 4) + h (x - 4)) / 2) :
h = putnam_1977_a3_solution f g :=
sorry | import Mathlib
-- fun f g x β¦ g x - f (x - 3) + f (x - 1) + f (x + 1) - f (x + 3)
/--
Let $f, g, h$ be functions $\mathbb{R} \to \mathbb{R}$. Find an expression for $h(x)$ in terms of $f$ and $g$ such that $f(x) = \frac{h(x + 1) + h(x - 1)}{2}$ and $g(x) = \frac{h(x + 4) + h(x - 4)}{2}$.
-/
theorem putnam_1977_a3
(f g h : β β β)
(hf : β x, f x = (h (x + 1) + h (x - 1)) / 2)
(hg : β x, g x = (h (x + 4) + h (x - 4)) / 2) :
h = putnam_1977_a3_solution f g := by
| import Mathlib
abbrev putnam_1977_a3_solution : (β β β) β (β β β) β (β β β) := sorry
-- fun f g x β¦ g x - f (x - 3) + f (x - 1) + f (x + 1) - f (x + 3)
/--
Let $f, g, h$ be functions $\mathbb{R} \to \mathbb{R}$. Find an expression for $h(x)$ in terms of $f$ and $g$ such that $f(x) = \frac{h(x + 1) + h(x - 1)}{2}$ and $g(x) = \frac{h(x + 4) + h(x - 4)}{2}$.
-/
theorem putnam_1977_a3
(f g h : β β β)
(hf : β x, f x = (h (x + 1) + h (x - 1)) / 2)
(hg : β x, g x = (h (x + 4) + h (x - 4)) / 2) :
h = putnam_1977_a3_solution f g :=
sorry
| Let $f, g, h$ be functions $\mathbb{R} \to \mathbb{R}$. Find an expression for $h(x)$ in terms of $f$ and $g$ such that $f(x) = \frac{h(x + 1) + h(x - 1)}{2}$ and $g(x) = \frac{h(x + 4) + h(x - 4)}{2}$. | Prove that $h(x) = g(x) - f(x - 3) + f(x - 1) + f(x + 1) - f(x + 3)$ suffices. | [
"algebra"
] | null | null |
|
putnam_2022_a1 | 317edd54-89b3-5843-9616-a2fe11e4d52b | train | 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 | import Mathlib
open Polynomial
-- {(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 := by
| import Mathlib
open Polynomial
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
| 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. | Show that the solution is the set of ordered pairs $(a,b)$ which satisfy at least one of (1) $a = b = 0$, (2) $|a| \geq 1$, and (3) $0 < |a| < 1$ and $b < \log(1 + r_{-}^2) - ar_{-}$ or $b > \log(1 + r_{+}^2) - ar_{+}$ where $r_{\pm} = \frac{1 \pm \sqrt{1 - a^2}}{a}$. | [
"algebra"
] | null | null |
|
putnam_1997_b1 | bae6fe1e-8ee0-52f8-be0e-88bf1be911ff | train | 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 | import Mathlib
open Filter Topology
-- 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 := by
| import Mathlib
open Filter Topology
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
| 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 $b$.) | Show that the solution is $n$. | [
"algebra"
] | null | null |
|
putnam_1983_b2 | b79e1f31-50e1-5fcb-af9c-351e5e833cea | train | abbrev putnam_1983_b2_solution : Prop := sorry
-- True
/--
Let $f(n)$ be the number of ways of representing $n$ as a sum of powers of $2$ with no power being used more than $3$ times. For example, $f(7) = 4$ (the representations are $4 + 2 + 1$, $4 + 1 + 1 + 1$, $2 + 2 + 2 + 1$, $2 + 2 + 1 + 1 + 1$). Can we find a real polynomial $p(x)$ such that $f(n) = [p(n)]$, where $[u]$ denotes the greatest integer less than or equal to $u$?
-/
theorem putnam_1983_b2
(f : β+ β β)
(hf : f = fun (n : β+) β¦
Set.ncard {M : Multiset β |
(β m β M, β k : β, m = (2 ^ k : β€)) β§
(β m β M, M.count m β€ 3) β§
(M.sum : β€) = n}) :
putnam_1983_b2_solution β
(β p : Polynomial β, β n : β+, βp.eval (n : β)β = f n) :=
sorry | import Mathlib
open Nat Filter Topology Real
-- True
/--
Let $f(n)$ be the number of ways of representing $n$ as a sum of powers of $2$ with no power being used more than $3$ times. For example, $f(7) = 4$ (the representations are $4 + 2 + 1$, $4 + 1 + 1 + 1$, $2 + 2 + 2 + 1$, $2 + 2 + 1 + 1 + 1$). Can we find a real polynomial $p(x)$ such that $f(n) = [p(n)]$, where $[u]$ denotes the greatest integer less than or equal to $u$?
-/
theorem putnam_1983_b2
(f : β+ β β)
(hf : f = fun (n : β+) β¦
Set.ncard {M : Multiset β |
(β m β M, β k : β, m = (2 ^ k : β€)) β§
(β m β M, M.count m β€ 3) β§
(M.sum : β€) = n}) :
putnam_1983_b2_solution β
(β p : Polynomial β, β n : β+, βp.eval (n : β)β = f n) := by
| import Mathlib
open Nat Filter Topology Real
abbrev putnam_1983_b2_solution : Prop := sorry
-- True
/--
Let $f(n)$ be the number of ways of representing $n$ as a sum of powers of $2$ with no power being used more than $3$ times. For example, $f(7) = 4$ (the representations are $4 + 2 + 1$, $4 + 1 + 1 + 1$, $2 + 2 + 2 + 1$, $2 + 2 + 1 + 1 + 1$). Can we find a real polynomial $p(x)$ such that $f(n) = [p(n)]$, where $[u]$ denotes the greatest integer less than or equal to $u$?
-/
theorem putnam_1983_b2
(f : β+ β β)
(hf : f = fun (n : β+) β¦
Set.ncard {M : Multiset β |
(β m β M, β k : β, m = (2 ^ k : β€)) β§
(β m β M, M.count m β€ 3) β§
(M.sum : β€) = n}) :
putnam_1983_b2_solution β
(β p : Polynomial β, β n : β+, βp.eval (n : β)β = f n) :=
sorry
| Let $f(n)$ be the number of ways of representing $n$ as a sum of powers of $2$ with no power being used more than $3$ times. For example, $f(7) = 4$ (the representations are $4 + 2 + 1$, $4 + 1 + 1 + 1$, $2 + 2 + 2 + 1$, $2 + 2 + 1 + 1 + 1$). Can we find a real polynomial $p(x)$ such that $f(n) = [p(n)]$, where $[u]$ denotes the greatest integer less than or equal to $u$? | Prove that such a polynomial exists. | [
"algebra"
] | null | null |
|
putnam_1982_a6 | 6e3545c1-ae5d-5a39-9b78-b35fc16f4bfb | train | abbrev putnam_1982_a6_solution : Prop := sorry
-- False
/--
Let $b$ be a bijection from the positive integers to the positive integers. Also, let $x_1, x_2, x_3, \dots$ be an infinite sequence of real numbers with the following properties:
\begin{enumerate}
\item
$|x_n|$ is a strictly decreasing function of $n$;
\item
$\lim_{n \rightarrow \infty} |b(n) - n| \cdot |x_n| = 0$;
\item
$\lim_{n \rightarrow \infty}\sum_{k = 1}^{n} x_k = 1$.
\end{enumerate}
Prove or disprove: these conditions imply that $$\lim_{n \rightarrow \infty} \sum_{k = 1}^{n} x_{b(k)} = 1.$$
-/
theorem putnam_1982_a6 :
(β b : β β β,
β x : β β β,
BijOn b (Ici 1) (Ici 1) β
StrictAntiOn (fun n : β => |x n|) (Ici 1) β
Tendsto (fun n : β => |b n - (n : β€)| * |x n|) atTop (π 0) β
Tendsto (fun n : β => β k in Finset.Icc 1 n, x k) atTop (π 1) β
Tendsto (fun n : β => β k in Finset.Icc 1 n, x (b k)) atTop (π 1))
β putnam_1982_a6_solution :=
sorry | import Mathlib
open Set Function Filter Topology Polynomial Real
-- False
/--
Let $b$ be a bijection from the positive integers to the positive integers. Also, let $x_1, x_2, x_3, \dots$ be an infinite sequence of real numbers with the following properties:
\begin{enumerate}
\item
$|x_n|$ is a strictly decreasing function of $n$;
\item
$\lim_{n \rightarrow \infty} |b(n) - n| \cdot |x_n| = 0$;
\item
$\lim_{n \rightarrow \infty}\sum_{k = 1}^{n} x_k = 1$.
\end{enumerate}
Prove or disprove: these conditions imply that $$\lim_{n \rightarrow \infty} \sum_{k = 1}^{n} x_{b(k)} = 1.$$
-/
theorem putnam_1982_a6 :
(β b : β β β,
β x : β β β,
BijOn b (Ici 1) (Ici 1) β
StrictAntiOn (fun n : β => |x n|) (Ici 1) β
Tendsto (fun n : β => |b n - (n : β€)| * |x n|) atTop (π 0) β
Tendsto (fun n : β => β k in Finset.Icc 1 n, x k) atTop (π 1) β
Tendsto (fun n : β => β k in Finset.Icc 1 n, x (b k)) atTop (π 1))
β putnam_1982_a6_solution := by
| import Mathlib
open Set Function Filter Topology Polynomial Real
abbrev putnam_1982_a6_solution : Prop := sorry
-- False
/--
Let $b$ be a bijection from the positive integers to the positive integers. Also, let $x_1, x_2, x_3, \dots$ be an infinite sequence of real numbers with the following properties:
\begin{enumerate}
\item
$|x_n|$ is a strictly decreasing function of $n$;
\item
$\lim_{n \rightarrow \infty} |b(n) - n| \cdot |x_n| = 0$;
\item
$\lim_{n \rightarrow \infty}\sum_{k = 1}^{n} x_k = 1$.
\end{enumerate}
Prove or disprove: these conditions imply that $$\lim_{n \rightarrow \infty} \sum_{k = 1}^{n} x_{b(k)} = 1.$$
-/
theorem putnam_1982_a6 :
(β b : β β β,
β x : β β β,
BijOn b (Ici 1) (Ici 1) β
StrictAntiOn (fun n : β => |x n|) (Ici 1) β
Tendsto (fun n : β => |b n - (n : β€)| * |x n|) atTop (π 0) β
Tendsto (fun n : β => β k in Finset.Icc 1 n, x k) atTop (π 1) β
Tendsto (fun n : β => β k in Finset.Icc 1 n, x (b k)) atTop (π 1))
β putnam_1982_a6_solution :=
sorry
| Let $b$ be a bijection from the positive integers to the positive integers. Also, let $x_1, x_2, x_3, \dots$ be an infinite sequence of real numbers with the following properties:
\begin{enumerate}
\item
$|x_n|$ is a strictly decreasing function of $n$;
\item
$\lim_{n \rightarrow \infty} |b(n) - n| \cdot |x_n| = 0$;
\item
$\lim_{n \rightarrow \infty}\sum_{k = 1}^{n} x_k = 1$.
\end{enumerate}
Prove or disprove: these conditions imply that $$\lim_{n \rightarrow \infty} \sum_{k = 1}^{n} x_{b(k)} = 1.$$ | The limit need not equal $1$. | [
"analysis"
] | null | null |
|
putnam_1963_a2 | b6f17719-3084-5ceb-926b-9cc24bd74aa4 | train | theorem putnam_1963_a2
(f : β β β)
(hfpos : β n, f n > 0)
(hfinc : StrictMonoOn f (Set.Ici 1))
(hf2 : f 2 = 2)
(hfmn : β m n, m > 0 β n > 0 β IsRelPrime m n β f (m * n) = f m * f n)
: β n > 0, f n = n :=
sorry | import Mathlib
open Topology Filter
/--
Let $\{f(n)\}$ be a strictly increasing sequence of positive integers such that $f(2)=2$ and $f(mn)=f(m)f(n)$ for every relatively prime pair of positive integers $m$ and $n$ (the greatest common divisor of $m$ and $n$ is equal to $1$). Prove that $f(n)=n$ for every positive integer $n$.
-/
theorem putnam_1963_a2
(f : β β β)
(hfpos : β n, f n > 0)
(hfinc : StrictMonoOn f (Set.Ici 1))
(hf2 : f 2 = 2)
(hfmn : β m n, m > 0 β n > 0 β IsRelPrime m n β f (m * n) = f m * f n)
: β n > 0, f n = n := by
| import Mathlib
open Topology Filter
/--
Let $\{f(n)\}$ be a strictly increasing sequence of positive integers such that $f(2)=2$ and $f(mn)=f(m)f(n)$ for every relatively prime pair of positive integers $m$ and $n$ (the greatest common divisor of $m$ and $n$ is equal to $1$). Prove that $f(n)=n$ for every positive integer $n$.
-/
theorem putnam_1963_a2
(f : β β β)
(hfpos : β n, f n > 0)
(hfinc : StrictMonoOn f (Set.Ici 1))
(hf2 : f 2 = 2)
(hfmn : β m n, m > 0 β n > 0 β IsRelPrime m n β f (m * n) = f m * f n)
: β n > 0, f n = n :=
sorry
| Let $\{f(n)\}$ be a strictly increasing sequence of positive integers such that $f(2)=2$ and $f(mn)=f(m)f(n)$ for every relatively prime pair of positive integers $m$ and $n$ (the greatest common divisor of $m$ and $n$ is equal to $1$). Prove that $f(n)=n$ for every positive integer $n$. | null | [
"number_theory",
"algebra"
] | null | null |
|
putnam_1994_a4 | 7b6d4468-3987-59a5-abf7-079b63ea6dc8 | train | theorem putnam_1994_a4
(A B : Matrix (Fin 2) (Fin 2) β€)
(ABinv : Nonempty (Invertible A) β§
Nonempty (Invertible (A + B)) β§
Nonempty (Invertible (A + 2 * B)) β§
Nonempty (Invertible (A + 3 * B)) β§
Nonempty (Invertible (A + 4 * B)))
: Nonempty (Invertible (A + 5 * B)) :=
sorry | import Mathlib
open Filter Topology
/--
Let $A$ and $B$ be $2 \times 2$ matrices with integer entries such that $A$, $A+B$, $A+2B$, $A+3B$, and $A+4B$ are all invertible matrices whose inverses have integer entries. Show that $A+5B$ is invertible and that its inverse has integer entries.
-/
theorem putnam_1994_a4
(A B : Matrix (Fin 2) (Fin 2) β€)
(ABinv : Nonempty (Invertible A) β§
Nonempty (Invertible (A + B)) β§
Nonempty (Invertible (A + 2 * B)) β§
Nonempty (Invertible (A + 3 * B)) β§
Nonempty (Invertible (A + 4 * B)))
: Nonempty (Invertible (A + 5 * B)) := by
| import Mathlib
open Filter Topology
/--
Let $A$ and $B$ be $2 \times 2$ matrices with integer entries such that $A$, $A+B$, $A+2B$, $A+3B$, and $A+4B$ are all invertible matrices whose inverses have integer entries. Show that $A+5B$ is invertible and that its inverse has integer entries.
-/
theorem putnam_1994_a4
(A B : Matrix (Fin 2) (Fin 2) β€)
(ABinv : Nonempty (Invertible A) β§
Nonempty (Invertible (A + B)) β§
Nonempty (Invertible (A + 2 * B)) β§
Nonempty (Invertible (A + 3 * B)) β§
Nonempty (Invertible (A + 4 * B)))
: Nonempty (Invertible (A + 5 * B)) :=
sorry
| Let $A$ and $B$ be $2 \times 2$ matrices with integer entries such that $A$, $A+B$, $A+2B$, $A+3B$, and $A+4B$ are all invertible matrices whose inverses have integer entries. Show that $A+5B$ is invertible and that its inverse has integer entries. | null | [
"linear_algebra"
] | null | null |
|
putnam_1970_a1 | 5b2aff58-ea8a-5268-8b65-c814edc2abbc | train | 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 | import Mathlib
open Metric Set EuclideanGeometry
/--
Prove that, for all $a > 0$ and $b > 0$, the power series of $e^{ax} \cos (bx)$ with respect to $x$ has either zero or infinitely many zero coefficients.
-/
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 := by
| import Mathlib
open Metric Set EuclideanGeometry
/--
Prove that, for all $a > 0$ and $b > 0$, the power series of $e^{ax} \cos (bx)$ with respect to $x$ has either zero or infinitely many zero coefficients.
-/
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
| Prove that, for all $a > 0$ and $b > 0$, the power series of $e^{ax} \cos (bx)$ with respect to $x$ has either zero or infinitely many zero coefficients. | null | [
"analysis"
] | null | null |
|
putnam_1983_a5 | ae4cfbca-68db-58ef-ac02-938642808ad5 | train | 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 | import Mathlib
open Nat
-- 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 := by
| import Mathlib
open Nat
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
| 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 $x$.) | Prove that such an $\alpha$ exists. | [
"analysis"
] | null | null |
|
putnam_2011_b5 | e2d56f7d-915f-5743-8d29-954127694d2a | train | theorem putnam_2011_b5
(a : β β β)
(h : β A : β, β n : β, β« x : β, ((β i : Finset.range n, 1 / (1 + (x - a i) ^ 2)) ^ 2) β€ A * n) :
β B : β, B > 0 β§
β n : β, β' i : Finset.range n, β' j : Finset.range n, (1 + (a i - a j) ^ 2) β₯ B * n ^ 3 :=
sorry | import Mathlib
open Topology Filter Matrix
/--
Let $a_1, a_2, \dots$ be real numbers. Suppose that there is a constant $A$ such that for all $n$,
\[
\int_{-\infty}^\infty \left( \sum_{i=1}^n \frac{1}{1 + (x-a_i)^2} \right)^2\,dx \leq An.
\]
Prove there is a constant $B>0$ such that for all $n$,
\[
\sum_{i,j=1}^n (1 + (a_i - a_j)^2) \geq Bn^3.
\]
-/
theorem putnam_2011_b5
(a : β β β)
(h : β A : β, β n : β, β« x : β, ((β i : Finset.range n, 1 / (1 + (x - a i) ^ 2)) ^ 2) β€ A * n) :
β B : β, B > 0 β§
β n : β, β' i : Finset.range n, β' j : Finset.range n, (1 + (a i - a j) ^ 2) β₯ B * n ^ 3 := by
| import Mathlib
open Topology Filter Matrix
/--
Let $a_1, a_2, \dots$ be real numbers. Suppose that there is a constant $A$ such that for all $n$,
\[
\int_{-\infty}^\infty \left( \sum_{i=1}^n \frac{1}{1 + (x-a_i)^2} \right)^2\,dx \leq An.
\]
Prove there is a constant $B>0$ such that for all $n$,
\[
\sum_{i,j=1}^n (1 + (a_i - a_j)^2) \geq Bn^3.
\]
-/
theorem putnam_2011_b5
(a : β β β)
(h : β A : β, β n : β, β« x : β, ((β i : Finset.range n, 1 / (1 + (x - a i) ^ 2)) ^ 2) β€ A * n) :
β B : β, B > 0 β§
β n : β, β' i : Finset.range n, β' j : Finset.range n, (1 + (a i - a j) ^ 2) β₯ B * n ^ 3 :=
sorry
| Let $a_1, a_2, \dots$ be real numbers. Suppose that there is a constant $A$ such that for all $n$,
\[
\int_{-\infty}^\infty \left( \sum_{i=1}^n \frac{1}{1 + (x-a_i)^2} \right)^2\,dx \leq An.
\]
Prove there is a constant $B>0$ such that for all $n$,
\[
\sum_{i,j=1}^n (1 + (a_i - a_j)^2) \geq Bn^3.
\] | null | [
"analysis"
] | null | null |
|
putnam_2002_a6 | d9ad13e0-717a-5c7a-8231-aee0b0362717 | train | abbrev putnam_2002_a6_solution : Set β := sorry
-- {2}
/--
Fix an integer $b \geq 2$. Let $f(1) = 1$, $f(2) = 2$, and for each
$n \geq 3$, define $f(n) = n f(d)$, where $d$ is the number of
base-$b$ digits of $n$. For which values of $b$ does
\[
\sum_{n=1}^\infty \frac{1}{f(n)}
\]
converge?
-/
theorem putnam_2002_a6
(f : β β β β β)
(hf : β b : β, f b 1 = 1 β§ f b 2 = 2 β§ β n β Ici 3, f b n = n * f b (Nat.digits b n).length)
: {b β Ici 2 | β L : β, Tendsto (fun m : β => β n in Finset.Icc 1 m, 1/(f b n)) atTop (π L)} = putnam_2002_a6_solution :=
sorry | import Mathlib
open Nat Set Topology Filter
-- {2}
/--
Fix an integer $b \geq 2$. Let $f(1) = 1$, $f(2) = 2$, and for each
$n \geq 3$, define $f(n) = n f(d)$, where $d$ is the number of
base-$b$ digits of $n$. For which values of $b$ does
\[
\sum_{n=1}^\infty \frac{1}{f(n)}
\]
converge?
-/
theorem putnam_2002_a6
(f : β β β β β)
(hf : β b : β, f b 1 = 1 β§ f b 2 = 2 β§ β n β Ici 3, f b n = n * f b (Nat.digits b n).length)
: {b β Ici 2 | β L : β, Tendsto (fun m : β => β n in Finset.Icc 1 m, 1/(f b n)) atTop (π L)} = putnam_2002_a6_solution := by
| import Mathlib
open Nat Set Topology Filter
abbrev putnam_2002_a6_solution : Set β := sorry
-- {2}
/--
Fix an integer $b \geq 2$. Let $f(1) = 1$, $f(2) = 2$, and for each
$n \geq 3$, define $f(n) = n f(d)$, where $d$ is the number of
base-$b$ digits of $n$. For which values of $b$ does
\[
\sum_{n=1}^\infty \frac{1}{f(n)}
\]
converge?
-/
theorem putnam_2002_a6
(f : β β β β β)
(hf : β b : β, f b 1 = 1 β§ f b 2 = 2 β§ β n β Ici 3, f b n = n * f b (Nat.digits b n).length)
: {b β Ici 2 | β L : β, Tendsto (fun m : β => β n in Finset.Icc 1 m, 1/(f b n)) atTop (π L)} = putnam_2002_a6_solution :=
sorry
| Fix an integer $b \geq 2$. Let $f(1) = 1$, $f(2) = 2$, and for each
$n \geq 3$, define $f(n) = n f(d)$, where $d$ is the number of
base-$b$ digits of $n$. For which values of $b$ does
\[
\sum_{n=1}^\infty \frac{1}{f(n)}
\]
converge? | The sum converges for $b=2$ and diverges for $b \geq 3$. | [
"analysis",
"number_theory"
] | null | null |
|
putnam_2021_b2 | c1e2980d-2157-5e0d-9a1f-03395a5e0652 | train | 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 | import Mathlib
open Filter Topology
-- 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 := by
| import Mathlib
open Filter Topology
noncomputable 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
| 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 $\sum_{k=1}^\infty a_k=1$. | Show that the answer is $2/3$. | [
"analysis"
] | null | null |
|
putnam_1995_b1 | d04afa50-b58c-56be-bd9a-06391ac9fa0b | train | theorem putnam_1995_b1
(Pi : Finpartition (Finset.range 9) β (Finset.range 9) β β)
(Pi_def : β partition k, Pi partition k = (Exists.choose (Finpartition.exists_mem partition k.2)).card) :
β Pt1 Pt2,
β x y : Finset.range 9,
x β y β§
Pi Pt1 x = Pi Pt1 y β§
Pi Pt2 x = Pi Pt2 y :=
sorry | import Mathlib
open Filter Topology Real Nat
/--
For a partition $\pi$ of $\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$, let $\pi(x)$ be the number of elements in the part containing $x$. Prove that for any two partitions $\pi$ and $\pi'$, there are two distinct numbers $x$ and $y$ in $\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$ such that $\pi(x) = \pi(y)$ and $\pi'(x) = \pi'(y)$. [A {\em partition} of a set $S$ is a collection of disjoint subsets (parts) whose union is $S$.]
-/
theorem putnam_1995_b1
(Pi : Finpartition (Finset.range 9) β (Finset.range 9) β β)
(Pi_def : β partition k, Pi partition k = (Exists.choose (Finpartition.exists_mem partition k.2)).card) :
β Pt1 Pt2,
β x y : Finset.range 9,
x β y β§
Pi Pt1 x = Pi Pt1 y β§
Pi Pt2 x = Pi Pt2 y := by
| import Mathlib
open Filter Topology Real Nat
/--
For a partition $\pi$ of $\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$, let $\pi(x)$ be the number of elements in the part containing $x$. Prove that for any two partitions $\pi$ and $\pi'$, there are two distinct numbers $x$ and $y$ in $\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$ such that $\pi(x) = \pi(y)$ and $\pi'(x) = \pi'(y)$. [A {\em partition} of a set $S$ is a collection of disjoint subsets (parts) whose union is $S$.]
-/
theorem putnam_1995_b1
(Pi : Finpartition (Finset.range 9) β (Finset.range 9) β β)
(Pi_def : β partition k, Pi partition k = (Exists.choose (Finpartition.exists_mem partition k.2)).card) :
β Pt1 Pt2,
β x y : Finset.range 9,
x β y β§
Pi Pt1 x = Pi Pt1 y β§
Pi Pt2 x = Pi Pt2 y :=
sorry
| For a partition $\pi$ of $\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$, let $\pi(x)$ be the number of elements in the part containing $x$. Prove that for any two partitions $\pi$ and $\pi'$, there are two distinct numbers $x$ and $y$ in $\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$ such that $\pi(x) = \pi(y)$ and $\pi'(x) = \pi'(y)$. [A {\em partition} of a set $S$ is a collection of disjoint subsets (parts) whose union is $S$.] | null | [
"combinatorics"
] | null | null |
|
putnam_1994_b2 | a58db2b1-3a4b-5016-b434-c5aa7e5950e2 | train | abbrev putnam_1994_b2_solution : Set β := sorry
-- {c : β | c < 243 / 8}
/--
For which real numbers $c$ is there a straight line that intersects the curve $x^4+9x^3+cx^2+9x+4$ in four distinct points?
-/
theorem putnam_1994_b2
(c : β) :
(β m b : β,
{x : β | m * x + b = x ^ 4 + 9 * x ^ 3 + c * x ^ 2 + 9 * x + 4}.encard = 4)
β c β putnam_1994_b2_solution :=
sorry | import Mathlib
open Filter Topology
-- {c : β | c < 243 / 8}
/--
For which real numbers $c$ is there a straight line that intersects the curve $x^4+9x^3+cx^2+9x+4$ in four distinct points?
-/
theorem putnam_1994_b2
(c : β) :
(β m b : β,
{x : β | m * x + b = x ^ 4 + 9 * x ^ 3 + c * x ^ 2 + 9 * x + 4}.encard = 4)
β c β putnam_1994_b2_solution := by
| import Mathlib
open Filter Topology
abbrev putnam_1994_b2_solution : Set β := sorry
-- {c : β | c < 243 / 8}
/--
For which real numbers $c$ is there a straight line that intersects the curve $x^4+9x^3+cx^2+9x+4$ in four distinct points?
-/
theorem putnam_1994_b2
(c : β) :
(β m b : β,
{x : β | m * x + b = x ^ 4 + 9 * x ^ 3 + c * x ^ 2 + 9 * x + 4}.encard = 4)
β c β putnam_1994_b2_solution :=
sorry
| For which real numbers $c$ is there a straight line that intersects the curve $x^4+9x^3+cx^2+9x+4$ in four distinct points? | Show that there exists such a line if and only if $c<243/8$. | [
"geometry",
"algebra"
] | null | null |
|
putnam_1963_b6 | cfe39dfb-97ad-53e7-bf5e-a605b6ace96a | train | theorem putnam_1963_b6
(d : β)
(S : Set (Fin d β β) β Set (Fin d β β))
(hS : S = fun A : Set (Fin d β β) => β p β A, β q β A, segment β p q)
(A : β β Set (Fin d β β))
(ddim : 1 β€ d β§ d β€ 3)
(hA0 : Nonempty (A 0))
(hAn : β n β₯ 1, A n = S (A (n - 1)))
: β n β₯ 2, A n = A (n + 1) :=
sorry | import Mathlib
open Topology Filter Polynomial
/--
Let $E$ be a Euclidean space of at most three dimensions. If $A$ is a nonempty subset of $E$, define $S(A)$ to be the set of all points that lie on closed segments joining pairs of points of $A$. For a given nonempty set $A_0$, define $A_n=S(A_{n-1})$ for $n=1,2,\dots$. Prove that $A_2=A_3=\cdots$. (A one-point set should be considered to be a special case of a closed segment.)
-/
theorem putnam_1963_b6
(d : β)
(S : Set (Fin d β β) β Set (Fin d β β))
(hS : S = fun A : Set (Fin d β β) => β p β A, β q β A, segment β p q)
(A : β β Set (Fin d β β))
(ddim : 1 β€ d β§ d β€ 3)
(hA0 : Nonempty (A 0))
(hAn : β n β₯ 1, A n = S (A (n - 1)))
: β n β₯ 2, A n = A (n + 1) := by
| import Mathlib
open Topology Filter Polynomial
/--
Let $E$ be a Euclidean space of at most three dimensions. If $A$ is a nonempty subset of $E$, define $S(A)$ to be the set of all points that lie on closed segments joining pairs of points of $A$. For a given nonempty set $A_0$, define $A_n=S(A_{n-1})$ for $n=1,2,\dots$. Prove that $A_2=A_3=\cdots$. (A one-point set should be considered to be a special case of a closed segment.)
-/
theorem putnam_1963_b6
(d : β)
(S : Set (Fin d β β) β Set (Fin d β β))
(hS : S = fun A : Set (Fin d β β) => β p β A, β q β A, segment β p q)
(A : β β Set (Fin d β β))
(ddim : 1 β€ d β§ d β€ 3)
(hA0 : Nonempty (A 0))
(hAn : β n β₯ 1, A n = S (A (n - 1)))
: β n β₯ 2, A n = A (n + 1) :=
sorry
| Let $E$ be a Euclidean space of at most three dimensions. If $A$ is a nonempty subset of $E$, define $S(A)$ to be the set of all points that lie on closed segments joining pairs of points of $A$. For a given nonempty set $A_0$, define $A_n=S(A_{n-1})$ for $n=1,2,\dots$. Prove that $A_2=A_3=\cdots$. (A one-point set should be considered to be a special case of a closed segment.) | null | [
"geometry",
"linear_algebra"
] | null | null |
|
putnam_2000_a1 | 2e635d87-7d56-516c-9fac-9a605c97a089 | train | 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 | import Mathlib
open Topology Filter
-- (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 := by
| import Mathlib
open Topology Filter
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
| 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 $\sum_{j=0}^\infty x_j=A$? | Show that the possible values comprise the interval $(0,A^2)$. | [
"analysis"
] | null | null |
|
putnam_2004_b1 | 84c75ff2-7140-5dbf-9415-41f12bcdf0c4 | train | theorem putnam_2004_b1
(n : β)
(P : Polynomial β€)
(r : β)
(Pdeg : P.degree = n)
(Preq0 : Polynomial.aeval r P = 0)
: β i β Finset.range n, β m : β€, m = β j in Finset.range (i + 1), (P.coeff (n - j) * r ^ (i + 1 - j)) :=
sorry | import Mathlib
open Nat Topology Filter
/--
Let $P(x)=c_nx^n+c_{n-1}x^{n-1}+\cdots+c_0$ be a polynomial with integer coefficients. Suppose that $r$ is a rational number such that $P(r)=0$. Show that the $n$ numbers $c_nr,\,c_nr^2+c_{n-1}r,\,c_nr^3+c_{n-1}r^2+c_{n-2}r,\dots,\,c_nr^n+c_{n-1}r^{n-1}+\cdots+c_1r$ are integers.
-/
theorem putnam_2004_b1
(n : β)
(P : Polynomial β€)
(r : β)
(Pdeg : P.degree = n)
(Preq0 : Polynomial.aeval r P = 0)
: β i β Finset.range n, β m : β€, m = β j in Finset.range (i + 1), (P.coeff (n - j) * r ^ (i + 1 - j)) := by
| import Mathlib
open Nat Topology Filter
/--
Let $P(x)=c_nx^n+c_{n-1}x^{n-1}+\cdots+c_0$ be a polynomial with integer coefficients. Suppose that $r$ is a rational number such that $P(r)=0$. Show that the $n$ numbers $c_nr,\,c_nr^2+c_{n-1}r,\,c_nr^3+c_{n-1}r^2+c_{n-2}r,\dots,\,c_nr^n+c_{n-1}r^{n-1}+\cdots+c_1r$ are integers.
-/
theorem putnam_2004_b1
(n : β)
(P : Polynomial β€)
(r : β)
(Pdeg : P.degree = n)
(Preq0 : Polynomial.aeval r P = 0)
: β i β Finset.range n, β m : β€, m = β j in Finset.range (i + 1), (P.coeff (n - j) * r ^ (i + 1 - j)) :=
sorry
| Let $P(x)=c_nx^n+c_{n-1}x^{n-1}+\cdots+c_0$ be a polynomial with integer coefficients. Suppose that $r$ is a rational number such that $P(r)=0$. Show that the $n$ numbers $c_nr,\,c_nr^2+c_{n-1}r,\,c_nr^3+c_{n-1}r^2+c_{n-2}r,\dots,\,c_nr^n+c_{n-1}r^{n-1}+\cdots+c_1r$ are integers. | null | [
"algebra"
] | null | null |
|
putnam_1962_a6 | df2d9bfb-812c-53d8-9051-7876ec60f149 | train | 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 | import Mathlib
/--
Let $S$ be a set of rational numbers such that whenever $a$ and $b$ are members of $S$, so are $a+b$ and $ab$, and having the property that for every rational number $r$ exactly one of the following three statements is true: \[ r \in S, -r \in S, r = 0. \] Prove that $S$ is the set of all positive rational numbers.
-/
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 } := by
| import Mathlib
/--
Let $S$ be a set of rational numbers such that whenever $a$ and $b$ are members of $S$, so are $a+b$ and $ab$, and having the property that for every rational number $r$ exactly one of the following three statements is true: \[ r \in S, -r \in S, r = 0. \] Prove that $S$ is the set of all positive rational numbers.
-/
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
| Let $S$ be a set of rational numbers such that whenever $a$ and $b$ are members of $S$, so are $a+b$ and $ab$, and having the property that for every rational number $r$ exactly one of the following three statements is true: \[ r \in S, -r \in S, r = 0. \] Prove that $S$ is the set of all positive rational numbers. | null | [
"algebra"
] | null | null |
|
putnam_2000_b1 | 3db986a5-d754-5c7e-a914-79210230855f | train | theorem putnam_2000_b1
(N : β)
(a b c : Fin N β β€)
(Nge1 : N β₯ 1)
(hodd : β j : Fin N, Odd (a j) β¨ Odd (b j) β¨ Odd (c j))
: (β r s t : β€, {j : Fin N | Odd (r * a j + s * b j + t * c j)}.ncard β₯ (4 * N : β) / 7) :=
sorry | import Mathlib
open Topology Filter
/--
Let $a_j,b_j,c_j$ be integers for $1\leq j\leq N$. Assume for each $j$, at least one of $a_j,b_j,c_j$ is odd. Show that there exist integers $r$, $s$, $t$ such that $ra_j+sb_j+tc_j$ is odd for at least $4N/7$ values of $j$, $1\leq j\leq N$.
-/
theorem putnam_2000_b1
(N : β)
(a b c : Fin N β β€)
(Nge1 : N β₯ 1)
(hodd : β j : Fin N, Odd (a j) β¨ Odd (b j) β¨ Odd (c j))
: (β r s t : β€, {j : Fin N | Odd (r * a j + s * b j + t * c j)}.ncard β₯ (4 * N : β) / 7) := by
| import Mathlib
open Topology Filter
/--
Let $a_j,b_j,c_j$ be integers for $1\leq j\leq N$. Assume for each $j$, at least one of $a_j,b_j,c_j$ is odd. Show that there exist integers $r$, $s$, $t$ such that $ra_j+sb_j+tc_j$ is odd for at least $4N/7$ values of $j$, $1\leq j\leq N$.
-/
theorem putnam_2000_b1
(N : β)
(a b c : Fin N β β€)
(Nge1 : N β₯ 1)
(hodd : β j : Fin N, Odd (a j) β¨ Odd (b j) β¨ Odd (c j))
: (β r s t : β€, {j : Fin N | Odd (r * a j + s * b j + t * c j)}.ncard β₯ (4 * N : β) / 7) :=
sorry
| Let $a_j,b_j,c_j$ be integers for $1\leq j\leq N$. Assume for each $j$, at least one of $a_j,b_j,c_j$ is odd. Show that there exist integers $r$, $s$, $t$ such that $ra_j+sb_j+tc_j$ is odd for at least $4N/7$ values of $j$, $1\leq j\leq N$. | null | [
"algebra"
] | null | null |
|
putnam_2020_a2 | 601f4365-f4ff-5e22-ac12-a75935a94cd6 | train | abbrev putnam_2020_a2_solution : β β β := sorry
-- fun k β¦ 4 ^ k
/--
Let $k$ be a nonnegative integer. Evaluate
\[
\sum_{j=0}^k 2^{k-j} \binom{k+j}{j}.
\]
-/
theorem putnam_2020_a2
(k : β)
: (β j in Finset.Icc 0 k, 2 ^ (k - j) * Nat.choose (k + j) j = putnam_2020_a2_solution k) :=
sorry | import Mathlib
-- fun k β¦ 4 ^ k
/--
Let $k$ be a nonnegative integer. Evaluate
\[
\sum_{j=0}^k 2^{k-j} \binom{k+j}{j}.
\]
-/
theorem putnam_2020_a2
(k : β)
: (β j in Finset.Icc 0 k, 2 ^ (k - j) * Nat.choose (k + j) j = putnam_2020_a2_solution k) := by
| import Mathlib
abbrev putnam_2020_a2_solution : β β β := sorry
-- fun k β¦ 4 ^ k
/--
Let $k$ be a nonnegative integer. Evaluate
\[
\sum_{j=0}^k 2^{k-j} \binom{k+j}{j}.
\]
-/
theorem putnam_2020_a2
(k : β)
: (β j in Finset.Icc 0 k, 2 ^ (k - j) * Nat.choose (k + j) j = putnam_2020_a2_solution k) :=
sorry
| Let $k$ be a nonnegative integer. Evaluate
\[
\sum_{j=0}^k 2^{k-j} \binom{k+j}{j}.
\]
| Show that the answer is $4^k$. | [
"algebra"
] | null | null |
|
putnam_1965_b3 | 4dc33823-1fc8-5886-8108-6139091a67a9 | train | theorem putnam_1965_b3
: {(a, b, c) : β€ Γ β€ Γ β€ | a > 0 β§ a β€ b β§ c > 0 β§ a^2 + b^2 = c^2 β§ a*b/(2 : β) = 2*(a + b + c)}.ncard = 3 :=
sorry | import Mathlib
open EuclideanGeometry Topology Filter Complex
/--
Prove that there are exactly three right triangles (up to orientation and translation) with integer side lengths and area equal to twice their perimeter.
-/
theorem putnam_1965_b3
: {(a, b, c) : β€ Γ β€ Γ β€ | a > 0 β§ a β€ b β§ c > 0 β§ a^2 + b^2 = c^2 β§ a*b/(2 : β) = 2*(a + b + c)}.ncard = 3 := by
| import Mathlib
open EuclideanGeometry Topology Filter Complex
/--
Prove that there are exactly three right triangles (up to orientation and translation) with integer side lengths and area equal to twice their perimeter.
-/
theorem putnam_1965_b3
: {(a, b, c) : β€ Γ β€ Γ β€ | a > 0 β§ a β€ b β§ c > 0 β§ a^2 + b^2 = c^2 β§ a*b/(2 : β) = 2*(a + b + c)}.ncard = 3 :=
sorry
| Prove that there are exactly three right triangles (up to orientation and translation) with integer side lengths and area equal to twice their perimeter. | null | [
"algebra",
"geometry"
] | null | null |
|
putnam_2017_a1 | 81ef171f-e8fe-5b8f-a242-1abecaf85a9e | train | abbrev putnam_2017_a1_solution : Set β€ := sorry
-- {x : β€ | x > 0 β§ (x = 1 β¨ 5 β£ x)}
/--
Let $S$ be the smallest set of positive integers such that (a) $2$ is in $S$, (b) $n$ is in $S$ whenever $n^2$ is in $S$, and (c) $(n+5)^2$ is in $S$ whenever $n$ is in $S$. Which positive integers are not in $S$?.
-/
theorem putnam_2017_a1
(IsQualifying : Set β€ β Prop)
(IsQualifying_def : β S, IsQualifying S β
(β n β S, 0 < n) β§
2 β S β§
(β n > 0, n ^ 2 β S β n β S) β§
(β n β S, (n + 5) ^ 2 β S))
(S : Set β€)
(hS : IsLeast IsQualifying S) :
SαΆ β© {n | 0 < n} = putnam_2017_a1_solution :=
sorry | import Mathlib
-- {x : β€ | x > 0 β§ (x = 1 β¨ 5 β£ x)}
/--
Let $S$ be the smallest set of positive integers such that (a) $2$ is in $S$, (b) $n$ is in $S$ whenever $n^2$ is in $S$, and (c) $(n+5)^2$ is in $S$ whenever $n$ is in $S$. Which positive integers are not in $S$?.
-/
theorem putnam_2017_a1
(IsQualifying : Set β€ β Prop)
(IsQualifying_def : β S, IsQualifying S β
(β n β S, 0 < n) β§
2 β S β§
(β n > 0, n ^ 2 β S β n β S) β§
(β n β S, (n + 5) ^ 2 β S))
(S : Set β€)
(hS : IsLeast IsQualifying S) :
SαΆ β© {n | 0 < n} = putnam_2017_a1_solution := by
| import Mathlib
abbrev putnam_2017_a1_solution : Set β€ := sorry
-- {x : β€ | x > 0 β§ (x = 1 β¨ 5 β£ x)}
/--
Let $S$ be the smallest set of positive integers such that (a) $2$ is in $S$, (b) $n$ is in $S$ whenever $n^2$ is in $S$, and (c) $(n+5)^2$ is in $S$ whenever $n$ is in $S$. Which positive integers are not in $S$?.
-/
theorem putnam_2017_a1
(IsQualifying : Set β€ β Prop)
(IsQualifying_def : β S, IsQualifying S β
(β n β S, 0 < n) β§
2 β S β§
(β n > 0, n ^ 2 β S β n β S) β§
(β n β S, (n + 5) ^ 2 β S))
(S : Set β€)
(hS : IsLeast IsQualifying S) :
SαΆ β© {n | 0 < n} = putnam_2017_a1_solution :=
sorry
| Let $S$ be the smallest set of positive integers such that (a) $2$ is in $S$, (b) $n$ is in $S$ whenever $n^2$ is in $S$, and (c) $(n+5)^2$ is in $S$ whenever $n$ is in $S$. Which positive integers are not in $S$?. | Show that all solutions are in the set $\{x \in \mathbb{Z}\, |\, x > 0 \land (x = 1 \lor 5 \mid x)\} | [
"number_theory"
] | null | null |
|
putnam_1980_b5 | 510e170d-737f-528f-9ece-d536e10a9a0a | train | 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 | import Mathlib
open Set
-- 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) := by
| import Mathlib
open Set
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
| 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? | $S_t$ is closed under multiplication if and only if $1 \ge t$. | [
"analysis",
"algebra"
] | null | null |
|
putnam_1976_a6 | 32e255db-22ce-5516-a3f1-c1ff63b680cc | train | theorem putnam_1976_a6
(f : β β β)
(hfdiff : ContDiff β 2 f)
(hfbd : β x : β, |f x| β€ 1)
(hf0 : (f 0)^2 + (deriv f 0)^2 = 4)
: β y : β, (f y) + (iteratedDeriv 2 f y) = 0 :=
sorry | import Mathlib
open Polynomial
/--
Suppose that $f : \mathbb{R} \to \mathbb{R}$ is a twice continuously differentiable function such that $|f(x)| \le 1$ for all real $x$ and $(f(0))^2 + (f'(0))^2 = 4$. Prove that $f(y) + f''(y) = 0$ for some real number $y$.
-/
theorem putnam_1976_a6
(f : β β β)
(hfdiff : ContDiff β 2 f)
(hfbd : β x : β, |f x| β€ 1)
(hf0 : (f 0)^2 + (deriv f 0)^2 = 4)
: β y : β, (f y) + (iteratedDeriv 2 f y) = 0 := by
| import Mathlib
open Polynomial
/--
Suppose that $f : \mathbb{R} \to \mathbb{R}$ is a twice continuously differentiable function such that $|f(x)| \le 1$ for all real $x$ and $(f(0))^2 + (f'(0))^2 = 4$. Prove that $f(y) + f''(y) = 0$ for some real number $y$.
-/
theorem putnam_1976_a6
(f : β β β)
(hfdiff : ContDiff β 2 f)
(hfbd : β x : β, |f x| β€ 1)
(hf0 : (f 0)^2 + (deriv f 0)^2 = 4)
: β y : β, (f y) + (iteratedDeriv 2 f y) = 0 :=
sorry
| Suppose that $f : \mathbb{R} \to \mathbb{R}$ is a twice continuously differentiable function such that $|f(x)| \le 1$ for all real $x$ and $(f(0))^2 + (f'(0))^2 = 4$. Prove that $f(y) + f''(y) = 0$ for some real number $y$. | null | [
"analysis",
"algebra"
] | null | null |
|
putnam_2013_b2 | e566f72b-2158-57ec-a9a3-915630438fae | train | abbrev putnam_2013_b2_solution : β := sorry
-- 3
/--
Let $C = \bigcup_{N=1}^\infty C_N$, where $C_N$ denotes the set of those `cosine polynomials' of the form
\[
f(x) = 1 + \sum_{n=1}^N a_n \cos(2 \pi n x)
\]
for which:
\begin{enumerate}
\item[(i)]
$f(x) \geq 0$ for all real $x$, and
\item[(ii)]
$a_n = 0$ whenever $n$ is a multiple of $3$.
\end{enumerate}
Determine the maximum value of $f(0)$ as $f$ ranges through $C$, and
prove that this maximum is attained.
-/
theorem putnam_2013_b2
(CN : β β Set (β β β))
(hCN : β N : β, CN N =
{f : β β β |
(β x : β, f x β₯ 0) β§
β a : List β, a.length = N + 1 β§ (β n : Fin (N + 1), 3 β£ (n : β) β a[n]! = 0) β§
β x : β, f x = 1 + β n in Finset.Icc 1 N, a[(n : β)]! * Real.cos (2*Real.pi*n*x)}) :
IsGreatest {f 0 | f β β N β Ici 1, CN N} putnam_2013_b2_solution :=
sorry | import Mathlib
open Function Set
-- 3
/--
Let $C = \bigcup_{N=1}^\infty C_N$, where $C_N$ denotes the set of those `cosine polynomials' of the form
\[
f(x) = 1 + \sum_{n=1}^N a_n \cos(2 \pi n x)
\]
for which:
\begin{enumerate}
\item[(i)]
$f(x) \geq 0$ for all real $x$, and
\item[(ii)]
$a_n = 0$ whenever $n$ is a multiple of $3$.
\end{enumerate}
Determine the maximum value of $f(0)$ as $f$ ranges through $C$, and
prove that this maximum is attained.
-/
theorem putnam_2013_b2
(CN : β β Set (β β β))
(hCN : β N : β, CN N =
{f : β β β |
(β x : β, f x β₯ 0) β§
β a : List β, a.length = N + 1 β§ (β n : Fin (N + 1), 3 β£ (n : β) β a[n]! = 0) β§
β x : β, f x = 1 + β n in Finset.Icc 1 N, a[(n : β)]! * Real.cos (2*Real.pi*n*x)}) :
IsGreatest {f 0 | f β β N β Ici 1, CN N} putnam_2013_b2_solution := by
| import Mathlib
open Function Set
abbrev putnam_2013_b2_solution : β := sorry
-- 3
/--
Let $C = \bigcup_{N=1}^\infty C_N$, where $C_N$ denotes the set of those `cosine polynomials' of the form
\[
f(x) = 1 + \sum_{n=1}^N a_n \cos(2 \pi n x)
\]
for which:
\begin{enumerate}
\item[(i)]
$f(x) \geq 0$ for all real $x$, and
\item[(ii)]
$a_n = 0$ whenever $n$ is a multiple of $3$.
\end{enumerate}
Determine the maximum value of $f(0)$ as $f$ ranges through $C$, and
prove that this maximum is attained.
-/
theorem putnam_2013_b2
(CN : β β Set (β β β))
(hCN : β N : β, CN N =
{f : β β β |
(β x : β, f x β₯ 0) β§
β a : List β, a.length = N + 1 β§ (β n : Fin (N + 1), 3 β£ (n : β) β a[n]! = 0) β§
β x : β, f x = 1 + β n in Finset.Icc 1 N, a[(n : β)]! * Real.cos (2*Real.pi*n*x)}) :
IsGreatest {f 0 | f β β N β Ici 1, CN N} putnam_2013_b2_solution :=
sorry
| Let $C = \bigcup_{N=1}^\infty C_N$, where $C_N$ denotes the set of those `cosine polynomials' of the form
\[
f(x) = 1 + \sum_{n=1}^N a_n \cos(2 \pi n x)
\]
for which:
\begin{enumerate}
\item[(i)]
$f(x) \geq 0$ for all real $x$, and
\item[(ii)]
$a_n = 0$ whenever $n$ is a multiple of $3$.
\end{enumerate}
Determine the maximum value of $f(0)$ as $f$ ranges through $C$, and
prove that this maximum is attained. | The maximum value of $f(0)$ is $3$. | [
"algebra"
] | null | null |
|
putnam_1994_a6 | c79ab4fb-c3c6-5511-850f-4a9df3ee4ee9 | train | 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 | import Mathlib
open Classical Filter Topology
-- Note: uses (β β Fin 10) instead of (Fin m β Fin 10)
/--
Let $f_1,\dots,f_{10}$ be bijections of the set of integers such that for each integer $n$, there is some composition $f_{i_1} \circ f_{i_2} \circ \cdots \circ f_{i_m}$ of these functions (allowing repetitions) which maps 0 to $n$. Consider the set of $1024$ functions $\mathcal{F}=\{f_1^{e_1} \circ f_2^{e_2} \circ \cdots \circ f_{10}^{e_{10}}\}$, $e_i=0$ or $1$ for $1 \leq i \leq 10$. ($f_i^0$ is the identity function and $f_i^1=f_i$.) Show that if $A$ is any nonempty finite set of integers, then at most $512$ of the functions in $\mathcal{F}$ map $A$ to itself.
-/
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 := by
| import Mathlib
open Classical Filter Topology
-- Note: uses (β β Fin 10) instead of (Fin m β Fin 10)
/--
Let $f_1,\dots,f_{10}$ be bijections of the set of integers such that for each integer $n$, there is some composition $f_{i_1} \circ f_{i_2} \circ \cdots \circ f_{i_m}$ of these functions (allowing repetitions) which maps 0 to $n$. Consider the set of $1024$ functions $\mathcal{F}=\{f_1^{e_1} \circ f_2^{e_2} \circ \cdots \circ f_{10}^{e_{10}}\}$, $e_i=0$ or $1$ for $1 \leq i \leq 10$. ($f_i^0$ is the identity function and $f_i^1=f_i$.) Show that if $A$ is any nonempty finite set of integers, then at most $512$ of the functions in $\mathcal{F}$ map $A$ to itself.
-/
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
| Let $f_1,\dots,f_{10}$ be bijections of the set of integers such that for each integer $n$, there is some composition $f_{i_1} \circ f_{i_2} \circ \cdots \circ f_{i_m}$ of these functions (allowing repetitions) which maps 0 to $n$. Consider the set of $1024$ functions $\mathcal{F}=\{f_1^{e_1} \circ f_2^{e_2} \circ \cdots \circ f_{10}^{e_{10}}\}$, $e_i=0$ or $1$ for $1 \leq i \leq 10$. ($f_i^0$ is the identity function and $f_i^1=f_i$.) Show that if $A$ is any nonempty finite set of integers, then at most $512$ of the functions in $\mathcal{F}$ map $A$ to itself. | null | [
"algebra"
] | null | null |
|
putnam_1994_a5 | b4ccaf2b-0086-58af-b1cf-98b373e105bd | train | 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 | import Mathlib
open Filter Topology
/--
Let $(r_n)_{n \geq 0}$ be a sequence of positive real numbers such that $\lim_{n \to \infty} r_n=0$. Let $S$ be the set of numbers representable as a sum $r_{i_1}+r_{i_2}+\cdots+r_{i_{1994}}$, with $i_1<i_2<\cdots<i_{1994}$. Show that every nonempty interval $(a,b)$ contains a nonempty subinterval $(c,d)$ that does not intersect $S$.
-/
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
) := by
| import Mathlib
open Filter Topology
/--
Let $(r_n)_{n \geq 0}$ be a sequence of positive real numbers such that $\lim_{n \to \infty} r_n=0$. Let $S$ be the set of numbers representable as a sum $r_{i_1}+r_{i_2}+\cdots+r_{i_{1994}}$, with $i_1<i_2<\cdots<i_{1994}$. Show that every nonempty interval $(a,b)$ contains a nonempty subinterval $(c,d)$ that does not intersect $S$.
-/
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
| Let $(r_n)_{n \geq 0}$ be a sequence of positive real numbers such that $\lim_{n \to \infty} r_n=0$. Let $S$ be the set of numbers representable as a sum $r_{i_1}+r_{i_2}+\cdots+r_{i_{1994}}$, with $i_1<i_2<\cdots<i_{1994}$. Show that every nonempty interval $(a,b)$ contains a nonempty subinterval $(c,d)$ that does not intersect $S$. | null | [
"analysis"
] | null | null |
|
putnam_2021_a3 | 72446904-edc4-59aa-8e80-4f458505874a | train | abbrev putnam_2021_a3_solution : Set β := sorry
-- {3 * m ^ 2 | m > 0}
/--
Determine all positive integers $N$ for which the sphere $x^2+y^2+z^2=N$ has an inscribed regular tetrahedron whose vertices have integer coordinates.
-/
theorem putnam_2021_a3
(N : β)
(Nsphere : Set (EuclideanSpace β (Fin 3)))
(hNsphere : Nsphere = {p | (p 0) ^ 2 + (p 1) ^ 2 + (p 2) ^ 2 = (N : β)})
(intcoords : (EuclideanSpace β (Fin 3)) β Prop)
(intcoords_def : β p, intcoords p β β i : Fin 3, p i = round (p i)) :
(0 < N β§ β A B C D : EuclideanSpace β (Fin 3),
A β Nsphere β§ B β Nsphere β§ C β Nsphere β§ D β Nsphere β§
intcoords A β§ intcoords B β§ intcoords C β§ intcoords D β§
(β s > 0, dist A B = s β§ dist A C = s β§ dist A D = s β§ dist B C = s β§ dist B D = s β§ dist C D = s))
β N β putnam_2021_a3_solution :=
sorry | import Mathlib
open Filter Topology
-- {3 * m ^ 2 | m > 0}
/--
Determine all positive integers $N$ for which the sphere $x^2+y^2+z^2=N$ has an inscribed regular tetrahedron whose vertices have integer coordinates.
-/
theorem putnam_2021_a3
(N : β)
(Nsphere : Set (EuclideanSpace β (Fin 3)))
(hNsphere : Nsphere = {p | (p 0) ^ 2 + (p 1) ^ 2 + (p 2) ^ 2 = (N : β)})
(intcoords : (EuclideanSpace β (Fin 3)) β Prop)
(intcoords_def : β p, intcoords p β β i : Fin 3, p i = round (p i)) :
(0 < N β§ β A B C D : EuclideanSpace β (Fin 3),
A β Nsphere β§ B β Nsphere β§ C β Nsphere β§ D β Nsphere β§
intcoords A β§ intcoords B β§ intcoords C β§ intcoords D β§
(β s > 0, dist A B = s β§ dist A C = s β§ dist A D = s β§ dist B C = s β§ dist B D = s β§ dist C D = s))
β N β putnam_2021_a3_solution := by
| import Mathlib
open Filter Topology
abbrev putnam_2021_a3_solution : Set β := sorry
-- {3 * m ^ 2 | m > 0}
/--
Determine all positive integers $N$ for which the sphere $x^2+y^2+z^2=N$ has an inscribed regular tetrahedron whose vertices have integer coordinates.
-/
theorem putnam_2021_a3
(N : β)
(Nsphere : Set (EuclideanSpace β (Fin 3)))
(hNsphere : Nsphere = {p | (p 0) ^ 2 + (p 1) ^ 2 + (p 2) ^ 2 = (N : β)})
(intcoords : (EuclideanSpace β (Fin 3)) β Prop)
(intcoords_def : β p, intcoords p β β i : Fin 3, p i = round (p i)) :
(0 < N β§ β A B C D : EuclideanSpace β (Fin 3),
A β Nsphere β§ B β Nsphere β§ C β Nsphere β§ D β Nsphere β§
intcoords A β§ intcoords B β§ intcoords C β§ intcoords D β§
(β s > 0, dist A B = s β§ dist A C = s β§ dist A D = s β§ dist B C = s β§ dist B D = s β§ dist C D = s))
β N β putnam_2021_a3_solution :=
sorry
| Determine all positive integers $N$ for which the sphere $x^2+y^2+z^2=N$ has an inscribed regular tetrahedron whose vertices have integer coordinates. | Show that the integers $N$ with this property are those of the form $3m^2$ for some positive integer $m$. | [
"geometry"
] | null | null |
|
putnam_2018_b5 | 5ca74870-acdd-5d88-885c-91fb94e3ba0c | train | theorem putnam_2018_b5
(f : (Fin 2 β β) β (Fin 2 β β))
(hβ : ContDiff β 1 f)
(hβ : β x i j, 0 < fderiv β f x (Pi.single i 1) j)
(hβ : β x, 0 < fderiv β f x ![1, 0] 0 * fderiv β f x ![0, 1] 1 -
(1 / 4) * (fderiv β f x ![1, 0] 1 + fderiv β f x ![0, 1] 0) ^ 2) :
Injective f :=
sorry | import Mathlib
open Function
/--
Let $f=(f_1,f_2)$ be a function from $\mathbb{R}^2$ to $\mathbb{R}^2$ with continuous partial derivatives $\frac{\partial f_i}{\partial x_j}$ that are positive everywhere. Suppose that $\frac{\partial f_1}{\partial x_1} \frac{\partial f_2}{\partial x_2}-\frac{1}{4}\left(\frac{\partial f_1}{\partial x_2}+\frac{\partial f_2}{\partial x_1}\right)^2>0$ everywhere. Prove that $f$ is one-to-one.
-/
theorem putnam_2018_b5
(f : (Fin 2 β β) β (Fin 2 β β))
(hβ : ContDiff β 1 f)
(hβ : β x i j, 0 < fderiv β f x (Pi.single i 1) j)
(hβ : β x, 0 < fderiv β f x ![1, 0] 0 * fderiv β f x ![0, 1] 1 -
(1 / 4) * (fderiv β f x ![1, 0] 1 + fderiv β f x ![0, 1] 0) ^ 2) :
Injective f := by
| import Mathlib
open Function
/--
Let $f=(f_1,f_2)$ be a function from $\mathbb{R}^2$ to $\mathbb{R}^2$ with continuous partial derivatives $\frac{\partial f_i}{\partial x_j}$ that are positive everywhere. Suppose that $\frac{\partial f_1}{\partial x_1} \frac{\partial f_2}{\partial x_2}-\frac{1}{4}\left(\frac{\partial f_1}{\partial x_2}+\frac{\partial f_2}{\partial x_1}\right)^2>0$ everywhere. Prove that $f$ is one-to-one.
-/
theorem putnam_2018_b5
(f : (Fin 2 β β) β (Fin 2 β β))
(hβ : ContDiff β 1 f)
(hβ : β x i j, 0 < fderiv β f x (Pi.single i 1) j)
(hβ : β x, 0 < fderiv β f x ![1, 0] 0 * fderiv β f x ![0, 1] 1 -
(1 / 4) * (fderiv β f x ![1, 0] 1 + fderiv β f x ![0, 1] 0) ^ 2) :
Injective f :=
sorry
| Let $f=(f_1,f_2)$ be a function from $\mathbb{R}^2$ to $\mathbb{R}^2$ with continuous partial derivatives $\frac{\partial f_i}{\partial x_j}$ that are positive everywhere. Suppose that $\frac{\partial f_1}{\partial x_1} \frac{\partial f_2}{\partial x_2}-\frac{1}{4}\left(\frac{\partial f_1}{\partial x_2}+\frac{\partial f_2}{\partial x_1}\right)^2>0$ everywhere. Prove that $f$ is one-to-one. | null | [
"analysis"
] | null | null |
|
putnam_1989_b2 | d61fa2a5-0cfa-57b2-a83c-ea18440e2081 | train | abbrev putnam_1989_b2_solution : Prop := sorry
-- True
/--
Let $S$ be a non-empty set with an associative operation that is left and right cancellative ($xy=xz$ implies $y=z$, and $yx=zx$ implies $y=z$). Assume that for every $a$ in $S$ the set $\{a^n:\,n=1, 2, 3, \ldots\}$ is finite. Must $S$ be a group?
-/
theorem putnam_1989_b2 :
(β (S : Type) [Nonempty S] [Semigroup S] [IsCancelMul S]
(h_fin : β a : S, {(a * Β·)^[n] a | n : β}.Finite),
β e : S, β x, e * x = x β§ x * e = x β§ β y, x * y = e β§ y * x = e) β
putnam_1989_b2_solution :=
sorry | import Mathlib
open Nat
-- Note: This formalization uses "Type" rather than "Type*" as the underlying structure for a semigroup.
-- True
/--
Let $S$ be a non-empty set with an associative operation that is left and right cancellative ($xy=xz$ implies $y=z$, and $yx=zx$ implies $y=z$). Assume that for every $a$ in $S$ the set $\{a^n:\,n=1, 2, 3, \ldots\}$ is finite. Must $S$ be a group?
-/
theorem putnam_1989_b2 :
(β (S : Type) [Nonempty S] [Semigroup S] [IsCancelMul S]
(h_fin : β a : S, {(a * Β·)^[n] a | n : β}.Finite),
β e : S, β x, e * x = x β§ x * e = x β§ β y, x * y = e β§ y * x = e) β
putnam_1989_b2_solution := by
| import Mathlib
open Nat
-- Note: This formalization uses "Type" rather than "Type*" as the underlying structure for a semigroup.
abbrev putnam_1989_b2_solution : Prop := sorry
-- True
/--
Let $S$ be a non-empty set with an associative operation that is left and right cancellative ($xy=xz$ implies $y=z$, and $yx=zx$ implies $y=z$). Assume that for every $a$ in $S$ the set $\{a^n:\,n=1, 2, 3, \ldots\}$ is finite. Must $S$ be a group?
-/
theorem putnam_1989_b2 :
(β (S : Type) [Nonempty S] [Semigroup S] [IsCancelMul S]
(h_fin : β a : S, {(a * Β·)^[n] a | n : β}.Finite),
β e : S, β x, e * x = x β§ x * e = x β§ β y, x * y = e β§ y * x = e) β
putnam_1989_b2_solution :=
sorry
| Let $S$ be a non-empty set with an associative operation that is left and right cancellative ($xy=xz$ implies $y=z$, and $yx=zx$ implies $y=z$). Assume that for every $a$ in $S$ the set $\{a^n:\,n=1, 2, 3, \ldots\}$ is finite. Must $S$ be a group? | Prove that $S$ must be a group. | [
"abstract_algebra"
] | null | null |
|
putnam_1965_a2 | d535878c-0dbc-56c9-be7e-54c156f6c28b | train | 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 | import Mathlib
open EuclideanGeometry
/--
Prove that $$\sum_{r=0}^{\lfloor\frac{n-1}{2}\rfloor} \left(\frac{n - 2r}{n} {n \choose r}\right)^2 = \frac{1}{n} {{2n - 2} \choose {n - 1}}$$ for every positive integer $n$.
-/
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) := by
| import Mathlib
open EuclideanGeometry
/--
Prove that $$\sum_{r=0}^{\lfloor\frac{n-1}{2}\rfloor} \left(\frac{n - 2r}{n} {n \choose r}\right)^2 = \frac{1}{n} {{2n - 2} \choose {n - 1}}$$ for every positive integer $n$.
-/
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
| Prove that $$\sum_{r=0}^{\lfloor\frac{n-1}{2}\rfloor} \left(\frac{n - 2r}{n} {n \choose r}\right)^2 = \frac{1}{n} {{2n - 2} \choose {n - 1}}$$ for every positive integer $n$. | null | [
"algebra"
] | null | null |
|
putnam_1975_a5 | 3d4c6d5c-5bc0-54a1-bfd9-f414854b14d3 | train | theorem putnam_1975_a5
(I : Interval β)
(f y1 y2 : β β β)
(hfcont : Continuous f)
(hycdiff : ContDiff β 2 y1 β§ ContDiff β 2 y2)
(hy1sol : β x : β, iteratedDeriv 2 y1 x = (f x) * (y1 x))
(hy2sol : β x : β, iteratedDeriv 2 y2 x = (f x) * (y2 x))
(hylinindep : β c1 c2 : β, (β x : β, c1 * y1 x + c2 * y2 x = 0) β (c1 = 0 β§ c2 = 0))
(hyI : β x β I, y1 x > 0 β§ y2 x > 0)
: β c : β, c > 0 β§ (let z : β β β := fun x => c * Real.sqrt ((y1 x) * (y2 x)); β x β I, iteratedDeriv 2 z x + 1/(z x)^3 = (f x) * (z x)) :=
sorry | import Mathlib
open Polynomial Real Complex
-- Note: Only the part showing a constant exists
/--
On some interval $I$ of the real line, let $y_1(x), y_2(x)$ be linearly independent solutions of the differential equation \[y'' = f(x)y\], where $f(x)$ is a continuous real-valued function. Suppose that $y_1(x) > 0$ and $y_2(x) > 0$ on $I$. Show that there exists a positive constant $c$ such that, on $I$, the function \[z(x) = c \sqrt{y_1(x)y_2(x)}\] satisfies the equation \[z'' + 1/z^3 = f(x)z.\]
-/
theorem putnam_1975_a5
(I : Interval β)
(f y1 y2 : β β β)
(hfcont : Continuous f)
(hycdiff : ContDiff β 2 y1 β§ ContDiff β 2 y2)
(hy1sol : β x : β, iteratedDeriv 2 y1 x = (f x) * (y1 x))
(hy2sol : β x : β, iteratedDeriv 2 y2 x = (f x) * (y2 x))
(hylinindep : β c1 c2 : β, (β x : β, c1 * y1 x + c2 * y2 x = 0) β (c1 = 0 β§ c2 = 0))
(hyI : β x β I, y1 x > 0 β§ y2 x > 0)
: β c : β, c > 0 β§ (let z : β β β := fun x => c * Real.sqrt ((y1 x) * (y2 x)); β x β I, iteratedDeriv 2 z x + 1/(z x)^3 = (f x) * (z x)) := by
| import Mathlib
open Polynomial Real Complex
-- Note: Only the part showing a constant exists
/--
On some interval $I$ of the real line, let $y_1(x), y_2(x)$ be linearly independent solutions of the differential equation \[y'' = f(x)y\], where $f(x)$ is a continuous real-valued function. Suppose that $y_1(x) > 0$ and $y_2(x) > 0$ on $I$. Show that there exists a positive constant $c$ such that, on $I$, the function \[z(x) = c \sqrt{y_1(x)y_2(x)}\] satisfies the equation \[z'' + 1/z^3 = f(x)z.\]
-/
theorem putnam_1975_a5
(I : Interval β)
(f y1 y2 : β β β)
(hfcont : Continuous f)
(hycdiff : ContDiff β 2 y1 β§ ContDiff β 2 y2)
(hy1sol : β x : β, iteratedDeriv 2 y1 x = (f x) * (y1 x))
(hy2sol : β x : β, iteratedDeriv 2 y2 x = (f x) * (y2 x))
(hylinindep : β c1 c2 : β, (β x : β, c1 * y1 x + c2 * y2 x = 0) β (c1 = 0 β§ c2 = 0))
(hyI : β x β I, y1 x > 0 β§ y2 x > 0)
: β c : β, c > 0 β§ (let z : β β β := fun x => c * Real.sqrt ((y1 x) * (y2 x)); β x β I, iteratedDeriv 2 z x + 1/(z x)^3 = (f x) * (z x)) :=
sorry
| On some interval $I$ of the real line, let $y_1(x), y_2(x)$ be linearly independent solutions of the differential equation \[y'' = f(x)y\], where $f(x)$ is a continuous real-valued function. Suppose that $y_1(x) > 0$ and $y_2(x) > 0$ on $I$. Show that there exists a positive constant $c$ such that, on $I$, the function \[z(x) = c \sqrt{y_1(x)y_2(x)}\] satisfies the equation \[z'' + 1/z^3 = f(x)z.\] | null | [
"analysis"
] | null | null |
|
putnam_2017_b2 | 014874b9-023b-58bc-84c1-ef8efda78333 | train | 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 | import Mathlib
open Topology Filter
-- 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 := by
| import Mathlib
open Topology Filter
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
| 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? | Prove that the smallest value of $a$ is $16$. | [
"algebra"
] | null | null |
|
putnam_1994_b5 | b42ddbf1-bc61-5c19-b3d0-898a77dba00c | train | theorem putnam_1994_b5
(n : β) (hn : n > 0)
(f : β β β€ β β€)
(hf : β (Ξ± : β) (x : β€), f Ξ± x = Int.floor (Ξ± * x)) :
β Ξ± : β,
β k β Set.Icc 1 n,
((f Ξ±)^[k] (n ^ 2) = n ^ 2 - k) β§
(f (Ξ± ^ k) (n ^ 2) = n ^ 2 - k) :=
sorry | import Mathlib
open Filter Topology
/--
For any real number $\alpha$, define the function $f_\alpha(x)=\lfloor \alpha x \rfloor$. Let $n$ be a positive integer. Show that there exists an $\alpha$ such that for $1 \leq k \leq n$, $f_\alpha^k(n^2)=n^2-k=f_{\alpha^k}(n^2)$.
-/
theorem putnam_1994_b5
(n : β) (hn : n > 0)
(f : β β β€ β β€)
(hf : β (Ξ± : β) (x : β€), f Ξ± x = Int.floor (Ξ± * x)) :
β Ξ± : β,
β k β Set.Icc 1 n,
((f Ξ±)^[k] (n ^ 2) = n ^ 2 - k) β§
(f (Ξ± ^ k) (n ^ 2) = n ^ 2 - k) := by
| import Mathlib
open Filter Topology
/--
For any real number $\alpha$, define the function $f_\alpha(x)=\lfloor \alpha x \rfloor$. Let $n$ be a positive integer. Show that there exists an $\alpha$ such that for $1 \leq k \leq n$, $f_\alpha^k(n^2)=n^2-k=f_{\alpha^k}(n^2)$.
-/
theorem putnam_1994_b5
(n : β) (hn : n > 0)
(f : β β β€ β β€)
(hf : β (Ξ± : β) (x : β€), f Ξ± x = Int.floor (Ξ± * x)) :
β Ξ± : β,
β k β Set.Icc 1 n,
((f Ξ±)^[k] (n ^ 2) = n ^ 2 - k) β§
(f (Ξ± ^ k) (n ^ 2) = n ^ 2 - k) :=
sorry
| For any real number $\alpha$, define the function $f_\alpha(x)=\lfloor \alpha x \rfloor$. Let $n$ be a positive integer. Show that there exists an $\alpha$ such that for $1 \leq k \leq n$, $f_\alpha^k(n^2)=n^2-k=f_{\alpha^k}(n^2)$. | null | [
"algebra"
] | null | null |
|
putnam_2000_a2 | 716a6cae-3df1-5300-a762-d7a5fbffb941 | train | theorem putnam_2000_a2 :
β n : β,
β N : β€,
β i : Fin 6 β β,
N > n β§
N = (i 0)^2 + (i 1)^2 β§
N + 1 = (i 2)^2 + (i 3)^2 β§
N + 2 = (i 4)^2 + (i 5)^2 :=
sorry | import Mathlib
open Topology Filter
/--
Prove that there exist infinitely many integers $n$ such that $n,n+1,n+2$ are each the sum of the squares of two integers.
-/
theorem putnam_2000_a2 :
β n : β,
β N : β€,
β i : Fin 6 β β,
N > n β§
N = (i 0)^2 + (i 1)^2 β§
N + 1 = (i 2)^2 + (i 3)^2 β§
N + 2 = (i 4)^2 + (i 5)^2 := by
| import Mathlib
open Topology Filter
/--
Prove that there exist infinitely many integers $n$ such that $n,n+1,n+2$ are each the sum of the squares of two integers.
-/
theorem putnam_2000_a2 :
β n : β,
β N : β€,
β i : Fin 6 β β,
N > n β§
N = (i 0)^2 + (i 1)^2 β§
N + 1 = (i 2)^2 + (i 3)^2 β§
N + 2 = (i 4)^2 + (i 5)^2 :=
sorry
| Prove that there exist infinitely many integers $n$ such that $n,n+1,n+2$ are each the sum of the squares of two integers. | null | [
"number_theory"
] | null | null |
|
putnam_1994_b1 | ae5e2d32-9510-52ea-8fb2-1f88fb3abaad | train | abbrev putnam_1994_b1_solution : Set β€ := sorry
-- {n : β€ | (315 β€ n β§ n β€ 325) β¨ (332 β€ n β§ n β€ 350)}
/--
Find all positive integers $n$ that are within $250$ of exactly $15$ perfect squares.
-/
theorem putnam_1994_b1
(n : β€) :
n β putnam_1994_b1_solution β
(0 < n β§ {m : β | |n - m ^ 2| β€ 250}.encard = 15) :=
sorry | import Mathlib
open Filter Topology
-- {n : β€ | (315 β€ n β§ n β€ 325) β¨ (332 β€ n β§ n β€ 350)}
/--
Find all positive integers $n$ that are within $250$ of exactly $15$ perfect squares.
-/
theorem putnam_1994_b1
(n : β€) :
n β putnam_1994_b1_solution β
(0 < n β§ {m : β | |n - m ^ 2| β€ 250}.encard = 15) := by
| import Mathlib
open Filter Topology
abbrev putnam_1994_b1_solution : Set β€ := sorry
-- {n : β€ | (315 β€ n β§ n β€ 325) β¨ (332 β€ n β§ n β€ 350)}
/--
Find all positive integers $n$ that are within $250$ of exactly $15$ perfect squares.
-/
theorem putnam_1994_b1
(n : β€) :
n β putnam_1994_b1_solution β
(0 < n β§ {m : β | |n - m ^ 2| β€ 250}.encard = 15) :=
sorry
| Find all positive integers $n$ that are within $250$ of exactly $15$ perfect squares. | Show that an integer $n$ is within $250$ of exactly $15$ perfect squares if and only if either $315 \leq n \leq 325$ or $332 \leq n \leq 350$. | [
"algebra"
] | null | null |
|
putnam_2003_a5 | adc92473-9211-50ab-8bfb-b31f7f0544c0 | train | theorem putnam_2003_a5
(n : β) (npos : n > 0)
(dyckpath : (m : β) β Set ((Fin (2 * m)) β β€))
(hdyckpath : dyckpath = fun m β¦ {p |
range p β {-1, 1} β§ β k, p k = 0 β§ β j, β k, ite (k β€ j) (p k) 0 β₯ 0})
(noevenreturn : (m : β) β Set ((Fin (2 * m)) β β€))
(hnoevenreturn : noevenreturn = fun m β¦ {p |
Β¬β i j, i < j β§ p i = 1 β§ (β k β Ioc i j, p i = -1) β§
Even (j.1 - i.1) β§ β k, ite (k β€ j) (p k) 0 = 0})
: β f : ((Fin (2 * n)) β β€) β (Fin (2 * (n - 1)) β β€),
β y β dyckpath (n - 1), β! x, x β dyckpath n β© noevenreturn n β§ f x = y :=
sorry | import Mathlib
open MvPolynomial Set
/--
A Dyck $n$-path is a lattice path of $n$ upsteps $(1,1)$ and $n$ downsteps $(1,-1)$ that starts at the origin $O$ and never dips below the $x$-axis. A return is a maximal sequence of contiguous downsteps that terminates on the $x$-axis. Show that there is a one-to-one correspondence between the Dyck $n$-paths with no return of even length and the Dyck $(n-1)$-paths.
-/
theorem putnam_2003_a5
(n : β) (npos : n > 0)
(dyckpath : (m : β) β Set ((Fin (2 * m)) β β€))
(hdyckpath : dyckpath = fun m β¦ {p |
range p β {-1, 1} β§ β k, p k = 0 β§ β j, β k, ite (k β€ j) (p k) 0 β₯ 0})
(noevenreturn : (m : β) β Set ((Fin (2 * m)) β β€))
(hnoevenreturn : noevenreturn = fun m β¦ {p |
Β¬β i j, i < j β§ p i = 1 β§ (β k β Ioc i j, p i = -1) β§
Even (j.1 - i.1) β§ β k, ite (k β€ j) (p k) 0 = 0})
: β f : ((Fin (2 * n)) β β€) β (Fin (2 * (n - 1)) β β€),
β y β dyckpath (n - 1), β! x, x β dyckpath n β© noevenreturn n β§ f x = y := by
| import Mathlib
open MvPolynomial Set
/--
A Dyck $n$-path is a lattice path of $n$ upsteps $(1,1)$ and $n$ downsteps $(1,-1)$ that starts at the origin $O$ and never dips below the $x$-axis. A return is a maximal sequence of contiguous downsteps that terminates on the $x$-axis. Show that there is a one-to-one correspondence between the Dyck $n$-paths with no return of even length and the Dyck $(n-1)$-paths.
-/
theorem putnam_2003_a5
(n : β) (npos : n > 0)
(dyckpath : (m : β) β Set ((Fin (2 * m)) β β€))
(hdyckpath : dyckpath = fun m β¦ {p |
range p β {-1, 1} β§ β k, p k = 0 β§ β j, β k, ite (k β€ j) (p k) 0 β₯ 0})
(noevenreturn : (m : β) β Set ((Fin (2 * m)) β β€))
(hnoevenreturn : noevenreturn = fun m β¦ {p |
Β¬β i j, i < j β§ p i = 1 β§ (β k β Ioc i j, p i = -1) β§
Even (j.1 - i.1) β§ β k, ite (k β€ j) (p k) 0 = 0})
: β f : ((Fin (2 * n)) β β€) β (Fin (2 * (n - 1)) β β€),
β y β dyckpath (n - 1), β! x, x β dyckpath n β© noevenreturn n β§ f x = y :=
sorry
| A Dyck $n$-path is a lattice path of $n$ upsteps $(1,1)$ and $n$ downsteps $(1,-1)$ that starts at the origin $O$ and never dips below the $x$-axis. A return is a maximal sequence of contiguous downsteps that terminates on the $x$-axis. Show that there is a one-to-one correspondence between the Dyck $n$-paths with no return of even length and the Dyck $(n-1)$-paths. | null | [
"combinatorics"
] | null | null |
|
putnam_1987_a4 | 9694ad1e-b55e-511c-97d2-b9f3d9a183be | train | abbrev putnam_1987_a4_solution : β := sorry
-- (5 / 3) * sqrt 30
/--
Let $P$ be a polynomial, with real coefficients, in three variables and $F$ be a function of two variables such that
\[
P(ux, uy, uz) = u^2 F(y-x,z-x) \quad \mbox{for all real $x,y,z,u$},
\]
and such that $P(1,0,0)=4$, $P(0,1,0)=5$, and $P(0,0,1)=6$. Also let $A,B,C$ be complex numbers with $P(A,B,C)=0$ and $|B-A|=10$. Find $|C-A|$.
-/
theorem putnam_1987_a4
(P : MvPolynomial (Fin 3) β)
(hPreal : β i : Fin 3 ββ β, (coeff i P).im = 0)
(F : β β β β β)
(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))
(h : β x y z u : β, eval (vars (u * x) (u * y) (u * z)) P = u ^ 2 * F (y - x) (z - x))
(hPval : eval (vars 1 0 0) P = 4 β§ eval (vars 0 1 0) P = 5 β§ eval (vars 0 0 1) P = 6)
(A B C : β)
(hPABC : eval (vars A B C) P = 0)
(habs : βB - Aβ = 10)
: (βC - Aβ = putnam_1987_a4_solution) :=
sorry | import Mathlib
open MvPolynomial Real
-- (5 / 3) * sqrt 30
/--
Let $P$ be a polynomial, with real coefficients, in three variables and $F$ be a function of two variables such that
\[
P(ux, uy, uz) = u^2 F(y-x,z-x) \quad \mbox{for all real $x,y,z,u$},
\]
and such that $P(1,0,0)=4$, $P(0,1,0)=5$, and $P(0,0,1)=6$. Also let $A,B,C$ be complex numbers with $P(A,B,C)=0$ and $|B-A|=10$. Find $|C-A|$.
-/
theorem putnam_1987_a4
(P : MvPolynomial (Fin 3) β)
(hPreal : β i : Fin 3 ββ β, (coeff i P).im = 0)
(F : β β β β β)
(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))
(h : β x y z u : β, eval (vars (u * x) (u * y) (u * z)) P = u ^ 2 * F (y - x) (z - x))
(hPval : eval (vars 1 0 0) P = 4 β§ eval (vars 0 1 0) P = 5 β§ eval (vars 0 0 1) P = 6)
(A B C : β)
(hPABC : eval (vars A B C) P = 0)
(habs : βB - Aβ = 10)
: (βC - Aβ = putnam_1987_a4_solution) := by
| import Mathlib
open MvPolynomial Real
noncomputable abbrev putnam_1987_a4_solution : β := sorry
-- (5 / 3) * sqrt 30
/--
Let $P$ be a polynomial, with real coefficients, in three variables and $F$ be a function of two variables such that
\[
P(ux, uy, uz) = u^2 F(y-x,z-x) \quad \mbox{for all real $x,y,z,u$},
\]
and such that $P(1,0,0)=4$, $P(0,1,0)=5$, and $P(0,0,1)=6$. Also let $A,B,C$ be complex numbers with $P(A,B,C)=0$ and $|B-A|=10$. Find $|C-A|$.
-/
theorem putnam_1987_a4
(P : MvPolynomial (Fin 3) β)
(hPreal : β i : Fin 3 ββ β, (coeff i P).im = 0)
(F : β β β β β)
(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))
(h : β x y z u : β, eval (vars (u * x) (u * y) (u * z)) P = u ^ 2 * F (y - x) (z - x))
(hPval : eval (vars 1 0 0) P = 4 β§ eval (vars 0 1 0) P = 5 β§ eval (vars 0 0 1) P = 6)
(A B C : β)
(hPABC : eval (vars A B C) P = 0)
(habs : βB - Aβ = 10)
: (βC - Aβ = putnam_1987_a4_solution) :=
sorry
| Let $P$ be a polynomial, with real coefficients, in three variables and $F$ be a function of two variables such that
\[
P(ux, uy, uz) = u^2 F(y-x,z-x) \quad \mbox{for all real $x,y,z,u$},
\]
and such that $P(1,0,0)=4$, $P(0,1,0)=5$, and $P(0,0,1)=6$. Also let $A,B,C$ be complex numbers with $P(A,B,C)=0$ and $|B-A|=10$. Find $|C-A|$. | Prove that $|C - A| = \frac{5}{3}\sqrt{30}$. | [
"algebra"
] | null | null |
|
putnam_2003_b2 | 3c4da28e-d7e7-5606-a3d6-a3313c2b48e1 | train | theorem putnam_2003_b2
(n : β)
(hn : n > 0)
(seq : β β β β β)
(hinit : β j β Icc 1 n, seq 0 j = 1 / j)
(havg : βα΅ (k β Icc 1 (n - 1)) (j β Icc 1 (n - k)),
seq k j = (seq (k - 1) j + seq (k - 1) (j + 1)) / 2) :
(seq (n - 1) 1 < 2 / n) :=
sorry | import Mathlib
open MvPolynomial Set
-- uses (β β β β β) instead of (Fin n β Icc 1 n β β)
/--
Let $n$ be a positive integer. Starting with the sequence $$1, \frac{1}{2}, \frac{1}{3}, \dots, \frac{1}{n},$$ form a new sequence of $n-1$ entries $$\frac{3}{4}, \frac{5}{12}, \dots, \frac{2n-1}{2n(n-1)}$$ by taking the averages of two consecutive entries in the first sequence. Repeat the averaging of neighbors on the second sequence to obtain a third sequence of $n-2$ entries, and continue until the final sequence produced consists of a single number $x_n$. Show that $x_n < 2/n$.
-/
theorem putnam_2003_b2
(n : β)
(hn : n > 0)
(seq : β β β β β)
(hinit : β j β Icc 1 n, seq 0 j = 1 / j)
(havg : βα΅ (k β Icc 1 (n - 1)) (j β Icc 1 (n - k)),
seq k j = (seq (k - 1) j + seq (k - 1) (j + 1)) / 2) :
(seq (n - 1) 1 < 2 / n) := by
| import Mathlib
open MvPolynomial Set
-- uses (β β β β β) instead of (Fin n β Icc 1 n β β)
/--
Let $n$ be a positive integer. Starting with the sequence $$1, \frac{1}{2}, \frac{1}{3}, \dots, \frac{1}{n},$$ form a new sequence of $n-1$ entries $$\frac{3}{4}, \frac{5}{12}, \dots, \frac{2n-1}{2n(n-1)}$$ by taking the averages of two consecutive entries in the first sequence. Repeat the averaging of neighbors on the second sequence to obtain a third sequence of $n-2$ entries, and continue until the final sequence produced consists of a single number $x_n$. Show that $x_n < 2/n$.
-/
theorem putnam_2003_b2
(n : β)
(hn : n > 0)
(seq : β β β β β)
(hinit : β j β Icc 1 n, seq 0 j = 1 / j)
(havg : βα΅ (k β Icc 1 (n - 1)) (j β Icc 1 (n - k)),
seq k j = (seq (k - 1) j + seq (k - 1) (j + 1)) / 2) :
(seq (n - 1) 1 < 2 / n) :=
sorry
| Let $n$ be a positive integer. Starting with the sequence $$1, \frac{1}{2}, \frac{1}{3}, \dots, \frac{1}{n},$$ form a new sequence of $n-1$ entries $$\frac{3}{4}, \frac{5}{12}, \dots, \frac{2n-1}{2n(n-1)}$$ by taking the averages of two consecutive entries in the first sequence. Repeat the averaging of neighbors on the second sequence to obtain a third sequence of $n-2$ entries, and continue until the final sequence produced consists of a single number $x_n$. Show that $x_n < 2/n$. | null | [
"algebra"
] | null | null |
|
putnam_1976_b6 | 52ff0ebf-c11c-558a-b920-c07b0d28b341 | train | 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 | import Mathlib
open Polynomial Filter Topology ProbabilityTheory MeasureTheory
/--
Let $\sigma(N)$ denote the sum of all positive integer divisors of $N$, including $1$ and $N$. Call a positive integer $N$ \textit{quasiperfect} if $\sigma(N) = 2N + 1$. Prove that every quasiperfect number is the square of an odd integer.
-/
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 := by
| import Mathlib
open Polynomial Filter Topology ProbabilityTheory MeasureTheory
/--
Let $\sigma(N)$ denote the sum of all positive integer divisors of $N$, including $1$ and $N$. Call a positive integer $N$ \textit{quasiperfect} if $\sigma(N) = 2N + 1$. Prove that every quasiperfect number is the square of an odd integer.
-/
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
| Let $\sigma(N)$ denote the sum of all positive integer divisors of $N$, including $1$ and $N$. Call a positive integer $N$ \textit{quasiperfect} if $\sigma(N) = 2N + 1$. Prove that every quasiperfect number is the square of an odd integer. | null | [
"number_theory"
] | null | null |
|
putnam_1993_b1 | 209b3777-7622-5169-a58a-e98f6de2a62d | train | abbrev putnam_1993_b1_solution : β := sorry
-- 3987
/--
Find the smallest positive integer $n$ such that for every integer $m$ with $0<m<1993$, there exists an integer $k$ for which $\frac{m}{1993}<\frac{k}{n}<\frac{m+1}{1994}$.
-/
theorem putnam_1993_b1 :
IsLeast
{n : β | 0 < n β§
β m β Set.Ioo (0 : β€) (1993), β k : β€,
(m / 1993 < (k : β) / n) β§ ((k : β) / n < (m + 1) / 1994) }
putnam_1993_b1_solution :=
sorry | import Mathlib
-- 3987
/--
Find the smallest positive integer $n$ such that for every integer $m$ with $0<m<1993$, there exists an integer $k$ for which $\frac{m}{1993}<\frac{k}{n}<\frac{m+1}{1994}$.
-/
theorem putnam_1993_b1 :
IsLeast
{n : β | 0 < n β§
β m β Set.Ioo (0 : β€) (1993), β k : β€,
(m / 1993 < (k : β) / n) β§ ((k : β) / n < (m + 1) / 1994) }
putnam_1993_b1_solution := by
| import Mathlib
abbrev putnam_1993_b1_solution : β := sorry
-- 3987
/--
Find the smallest positive integer $n$ such that for every integer $m$ with $0<m<1993$, there exists an integer $k$ for which $\frac{m}{1993}<\frac{k}{n}<\frac{m+1}{1994}$.
-/
theorem putnam_1993_b1 :
IsLeast
{n : β | 0 < n β§
β m β Set.Ioo (0 : β€) (1993), β k : β€,
(m / 1993 < (k : β) / n) β§ ((k : β) / n < (m + 1) / 1994) }
putnam_1993_b1_solution :=
sorry
| Find the smallest positive integer $n$ such that for every integer $m$ with $0<m<1993$, there exists an integer $k$ for which $\frac{m}{1993}<\frac{k}{n}<\frac{m+1}{1994}$. | Show that the smallest positive integer $n$ satisfying the condition is $n=3987$. | [
"algebra"
] | null | null |
|
putnam_1976_b1 | ff1bc99c-639b-5273-9d22-c5ec9d651e21 | train | 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 | import Mathlib
open Polynomial Filter Topology
-- (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)) := by
| import Mathlib
open Polynomial Filter Topology
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
| 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. | The limit equals $\ln(4) - 1$, so $a = 4$ and $b = 1$. | [
"analysis"
] | null | null |
|
putnam_1995_a4 | 867b7765-c326-5ed7-ba7b-be8ab8327665 | train | 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 | import Mathlib
open Filter Topology Real
/--
Suppose we have a necklace of $n$ beads. Each bead is labeled with an integer and the sum of all these labels is $n-1$. Prove that we can cut the necklace to form a string whose consecutive labels $x_{1},x\_{2},\dots,x_{n}$ satisfy \[\sum_{i=1}^{k} x_{i} \leq k-1 \qquad \mbox{for} \quad k=1,2,\dots,n.\]
-/
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 := by
| import Mathlib
open Filter Topology Real
/--
Suppose we have a necklace of $n$ beads. Each bead is labeled with an integer and the sum of all these labels is $n-1$. Prove that we can cut the necklace to form a string whose consecutive labels $x_{1},x\_{2},\dots,x_{n}$ satisfy \[\sum_{i=1}^{k} x_{i} \leq k-1 \qquad \mbox{for} \quad k=1,2,\dots,n.\]
-/
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
| Suppose we have a necklace of $n$ beads. Each bead is labeled with an integer and the sum of all these labels is $n-1$. Prove that we can cut the necklace to form a string whose consecutive labels $x_{1},x\_{2},\dots,x_{n}$ satisfy \[\sum_{i=1}^{k} x_{i} \leq k-1 \qquad \mbox{for} \quad k=1,2,\dots,n.\] | null | [
"combinatorics"
] | null | null |
|
putnam_1962_b1 | 10e25587-ee1f-5eae-b59c-ae5bd6ce2cec | train | theorem putnam_1962_b1
(p : β β β β β)
(x y : β)
(n : β)
(h0 : p 0 = fun x : β => 1)
(hp : β n > 0, p n = fun x : β => β i in Finset.range n, (x - i))
: p n (x+y) = β k in Finset.range (n+1), Nat.choose n k * (p k x) * (p (n - k) y) :=
sorry | import Mathlib
/--
Let $x^{(n)} = x(x-1)\cdots(x-n+1)$ for $n$ a positive integer and let $x^{(0)} = 1.$ Prove that \[ (x+y)^{(n)} = \sum_{k=0}^n {n \choose k} x^{(k)} y^{(n-k)}. \]
-/
theorem putnam_1962_b1
(p : β β β β β)
(x y : β)
(n : β)
(h0 : p 0 = fun x : β => 1)
(hp : β n > 0, p n = fun x : β => β i in Finset.range n, (x - i))
: p n (x+y) = β k in Finset.range (n+1), Nat.choose n k * (p k x) * (p (n - k) y) := by
| import Mathlib
/--
Let $x^{(n)} = x(x-1)\cdots(x-n+1)$ for $n$ a positive integer and let $x^{(0)} = 1.$ Prove that \[ (x+y)^{(n)} = \sum_{k=0}^n {n \choose k} x^{(k)} y^{(n-k)}. \]
-/
theorem putnam_1962_b1
(p : β β β β β)
(x y : β)
(n : β)
(h0 : p 0 = fun x : β => 1)
(hp : β n > 0, p n = fun x : β => β i in Finset.range n, (x - i))
: p n (x+y) = β k in Finset.range (n+1), Nat.choose n k * (p k x) * (p (n - k) y) :=
sorry
| Let $x^{(n)} = x(x-1)\cdots(x-n+1)$ for $n$ a positive integer and let $x^{(0)} = 1.$ Prove that \[ (x+y)^{(n)} = \sum_{k=0}^n {n \choose k} x^{(k)} y^{(n-k)}. \] | null | [
"algebra",
"combinatorics"
] | null | null |
|
putnam_2009_b4 | 79d709a8-73fe-5871-92f2-b864c4f9160a | train | abbrev putnam_2009_b4_solution : β := sorry
-- 2020050
/--
Say that a polynomial with real coefficients in two variables, $x,y$, is \emph{balanced} if the average value of the polynomial on each circle centered at the origin is $0$. The balanced polynomials of degree at most $2009$ form a vector space $V$ over $\mathbb{R}$. Find the dimension of $V$.
-/
theorem putnam_2009_b4
(IsBalanced : MvPolynomial (Fin 2) β β Prop)
(IsBalanced_def : β P, IsBalanced P β β r > 0,
(β« t in (0 : β)..(2 * Ο), eval ![r * cos t, r * sin t] P) / (2 * Ο * r) = 0)
(V : Submodule β (MvPolynomial (Fin 2) β))
(V_def : β P, P β V β IsBalanced P β§ P.totalDegree β€ 2009) :
Module.rank β V = putnam_2009_b4_solution :=
sorry | import Mathlib
open intervalIntegral MvPolynomial Real
-- 2020050
/--
Say that a polynomial with real coefficients in two variables, $x,y$, is \emph{balanced} if the average value of the polynomial on each circle centered at the origin is $0$. The balanced polynomials of degree at most $2009$ form a vector space $V$ over $\mathbb{R}$. Find the dimension of $V$.
-/
theorem putnam_2009_b4
(IsBalanced : MvPolynomial (Fin 2) β β Prop)
(IsBalanced_def : β P, IsBalanced P β β r > 0,
(β« t in (0 : β)..(2 * Ο), eval ![r * cos t, r * sin t] P) / (2 * Ο * r) = 0)
(V : Submodule β (MvPolynomial (Fin 2) β))
(V_def : β P, P β V β IsBalanced P β§ P.totalDegree β€ 2009) :
Module.rank β V = putnam_2009_b4_solution := by
| import Mathlib
open intervalIntegral MvPolynomial Real
abbrev putnam_2009_b4_solution : β := sorry
-- 2020050
/--
Say that a polynomial with real coefficients in two variables, $x,y$, is \emph{balanced} if the average value of the polynomial on each circle centered at the origin is $0$. The balanced polynomials of degree at most $2009$ form a vector space $V$ over $\mathbb{R}$. Find the dimension of $V$.
-/
theorem putnam_2009_b4
(IsBalanced : MvPolynomial (Fin 2) β β Prop)
(IsBalanced_def : β P, IsBalanced P β β r > 0,
(β« t in (0 : β)..(2 * Ο), eval ![r * cos t, r * sin t] P) / (2 * Ο * r) = 0)
(V : Submodule β (MvPolynomial (Fin 2) β))
(V_def : β P, P β V β IsBalanced P β§ P.totalDegree β€ 2009) :
Module.rank β V = putnam_2009_b4_solution :=
sorry
| Say that a polynomial with real coefficients in two variables, $x,y$, is \emph{balanced} if the average value of the polynomial on each circle centered at the origin is $0$. The balanced polynomials of degree at most $2009$ form a vector space $V$ over $\mathbb{R}$. Find the dimension of $V$. | Prove that the dimension of $V$ is $2020050$. | [
"algebra",
"linear_algebra"
] | null | null |
|
putnam_2009_a1 | 176cf516-6159-5111-a70e-a0d8141c1fab | train | abbrev putnam_2009_a1_solution : Prop := sorry
-- True
/--
Let $f$ be a real-valued function on the plane such that for every square $ABCD$ in the plane, $f(A)+f(B)+f(C)+f(D)=0$. Does it follow that $f(P)=0$ for all points $P$ in the plane?
-/
theorem putnam_2009_a1
: ((β f : (β Γ β) β β, (β O v : β Γ β, v β (0, 0) β f (O.1, O.2) + f (O.1 + v.1, O.2 + v.2) + f (O.1 + v.1 - v.2, O.2 + v.2 + v.1) + f (O.1 - v.2, O.2 + v.1) = 0) β β P : β Γ β, f P = 0) β putnam_2009_a1_solution) :=
sorry | import Mathlib
open Topology MvPolynomial Filter
-- True
/--
Let $f$ be a real-valued function on the plane such that for every square $ABCD$ in the plane, $f(A)+f(B)+f(C)+f(D)=0$. Does it follow that $f(P)=0$ for all points $P$ in the plane?
-/
theorem putnam_2009_a1
: ((β f : (β Γ β) β β, (β O v : β Γ β, v β (0, 0) β f (O.1, O.2) + f (O.1 + v.1, O.2 + v.2) + f (O.1 + v.1 - v.2, O.2 + v.2 + v.1) + f (O.1 - v.2, O.2 + v.1) = 0) β β P : β Γ β, f P = 0) β putnam_2009_a1_solution) := by
| import Mathlib
open Topology MvPolynomial Filter
abbrev putnam_2009_a1_solution : Prop := sorry
-- True
/--
Let $f$ be a real-valued function on the plane such that for every square $ABCD$ in the plane, $f(A)+f(B)+f(C)+f(D)=0$. Does it follow that $f(P)=0$ for all points $P$ in the plane?
-/
theorem putnam_2009_a1
: ((β f : (β Γ β) β β, (β O v : β Γ β, v β (0, 0) β f (O.1, O.2) + f (O.1 + v.1, O.2 + v.2) + f (O.1 + v.1 - v.2, O.2 + v.2 + v.1) + f (O.1 - v.2, O.2 + v.1) = 0) β β P : β Γ β, f P = 0) β putnam_2009_a1_solution) :=
sorry
| Let $f$ be a real-valued function on the plane such that for every square $ABCD$ in the plane, $f(A)+f(B)+f(C)+f(D)=0$. Does it follow that $f(P)=0$ for all points $P$ in the plane? | Prove that $f$ is identically $0$. | [
"geometry",
"algebra"
] | null | null |
|
putnam_2004_b5 | c06547e1-185a-5326-97df-7bf303253ce7 | train | abbrev putnam_2004_b5_solution : β := sorry
-- 2 / Real.exp 1
/--
Evaluate $\lim_{x \to 1^-} \prod_{n=0}^\infty \left(\frac{1+x^{n+1}}{1+x^n}\right)^{x^n}$.
-/
theorem putnam_2004_b5
(xprod : β β β)
(hxprod : β x β Set.Ioo 0 1,
Tendsto (fun N β¦ β n in Finset.range N, ((1 + x ^ (n + 1)) / (1 + x ^ n)) ^ (x ^ n))
atTop (π (xprod x))) :
Tendsto xprod (π[<] 1) (π putnam_2004_b5_solution) :=
sorry | import Mathlib
open Nat Topology Filter
-- 2 / Real.exp 1
/--
Evaluate $\lim_{x \to 1^-} \prod_{n=0}^\infty \left(\frac{1+x^{n+1}}{1+x^n}\right)^{x^n}$.
-/
theorem putnam_2004_b5
(xprod : β β β)
(hxprod : β x β Set.Ioo 0 1,
Tendsto (fun N β¦ β n in Finset.range N, ((1 + x ^ (n + 1)) / (1 + x ^ n)) ^ (x ^ n))
atTop (π (xprod x))) :
Tendsto xprod (π[<] 1) (π putnam_2004_b5_solution) := by
| import Mathlib
open Nat Topology Filter
abbrev putnam_2004_b5_solution : β := sorry
-- 2 / Real.exp 1
/--
Evaluate $\lim_{x \to 1^-} \prod_{n=0}^\infty \left(\frac{1+x^{n+1}}{1+x^n}\right)^{x^n}$.
-/
theorem putnam_2004_b5
(xprod : β β β)
(hxprod : β x β Set.Ioo 0 1,
Tendsto (fun N β¦ β n in Finset.range N, ((1 + x ^ (n + 1)) / (1 + x ^ n)) ^ (x ^ n))
atTop (π (xprod x))) :
Tendsto xprod (π[<] 1) (π putnam_2004_b5_solution) :=
sorry
| Evaluate $\lim_{x \to 1^-} \prod_{n=0}^\infty \left(\frac{1+x^{n+1}}{1+x^n}\right)^{x^n}$. | Show that the desired limit is $2/e$. | [
"analysis"
] | null | null |
|
putnam_1990_a4 | 30cd724f-8850-5a3d-b682-b9496965c134 | train | abbrev putnam_1990_a4_solution : β := sorry
-- 3
/--
Consider a paper punch that can be centered at any point of the plane and that, when operated, removes from the plane precisely those points whose distance from the center is irrational. How many punches are needed to remove every point?
-/
theorem putnam_1990_a4
: sInf {n : β | β S : Set (EuclideanSpace β (Fin 2)), S.encard = n β§ β Q : EuclideanSpace β (Fin 2), β P β S, Irrational (dist P Q)} = putnam_1990_a4_solution :=
sorry | import Mathlib
open Filter Topology Nat
-- 3
/--
Consider a paper punch that can be centered at any point of the plane and that, when operated, removes from the plane precisely those points whose distance from the center is irrational. How many punches are needed to remove every point?
-/
theorem putnam_1990_a4
: sInf {n : β | β S : Set (EuclideanSpace β (Fin 2)), S.encard = n β§ β Q : EuclideanSpace β (Fin 2), β P β S, Irrational (dist P Q)} = putnam_1990_a4_solution := by
| import Mathlib
open Filter Topology Nat
abbrev putnam_1990_a4_solution : β := sorry
-- 3
/--
Consider a paper punch that can be centered at any point of the plane and that, when operated, removes from the plane precisely those points whose distance from the center is irrational. How many punches are needed to remove every point?
-/
theorem putnam_1990_a4
: sInf {n : β | β S : Set (EuclideanSpace β (Fin 2)), S.encard = n β§ β Q : EuclideanSpace β (Fin 2), β P β S, Irrational (dist P Q)} = putnam_1990_a4_solution :=
sorry
| Consider a paper punch that can be centered at any point of the plane and that, when operated, removes from the plane precisely those points whose distance from the center is irrational. How many punches are needed to remove every point? | Show that three punches are needed. | [
"set_theory",
"number_theory"
] | null | null |
|
putnam_2009_a3 | 264121ff-30a2-56b2-b0dd-d007caa02263 | train | 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 | import Mathlib
open Topology MvPolynomial Filter Set
-- 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) := by
| import Mathlib
open Topology MvPolynomial Filter Set
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
| 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 $\lim_{n\to\infty} d_n$. | Show that the limit is 0. | [
"linear_algebra",
"analysis"
] | null | null |
|
putnam_2023_a2 | 0e22f201-2a7c-52e5-968c-fd8c3bb8a5f3 | train | abbrev putnam_2023_a2_solution : β β Set β := sorry
-- fun n => {(1 : β)/(factorial n), -(1 : β)/(factorial n)}
/--
Let $n$ be an even positive integer. Let $p$ be a monic, real polynomial of degree $2n$; that is to say, $p(x) = x^{2n} + a_{2n-1} x^{2n-1} + \cdots + a_1 x + a_0$ for some real coefficients $a_0, \dots, a_{2n-1}$. Suppose that $p(1/k) = k^2$ for all integers $k$ such that $1 \leq |k| \leq n$. Find all other real numbers $x$ for which $p(1/x) = x^2$.
-/
theorem putnam_2023_a2
(n : β)
(hn : n > 0 β§ Even n)
(p : Polynomial β)
(hp : Polynomial.Monic p β§ p.degree = 2*n)
(S : Set β)
(hS : S = {x : β | β k : β€, x = k β§ 1 β€ |k| β§ |k| β€ n})
(hpinv : β k β S, p.eval (1/k) = k^2)
: {x : β | p.eval (1/x) = x^2} \ S = putnam_2023_a2_solution n :=
sorry | import Mathlib
open Nat
-- fun n => {(1 : β)/(factorial n), -(1 : β)/(factorial n)}
/--
Let $n$ be an even positive integer. Let $p$ be a monic, real polynomial of degree $2n$; that is to say, $p(x) = x^{2n} + a_{2n-1} x^{2n-1} + \cdots + a_1 x + a_0$ for some real coefficients $a_0, \dots, a_{2n-1}$. Suppose that $p(1/k) = k^2$ for all integers $k$ such that $1 \leq |k| \leq n$. Find all other real numbers $x$ for which $p(1/x) = x^2$.
-/
theorem putnam_2023_a2
(n : β)
(hn : n > 0 β§ Even n)
(p : Polynomial β)
(hp : Polynomial.Monic p β§ p.degree = 2*n)
(S : Set β)
(hS : S = {x : β | β k : β€, x = k β§ 1 β€ |k| β§ |k| β€ n})
(hpinv : β k β S, p.eval (1/k) = k^2)
: {x : β | p.eval (1/x) = x^2} \ S = putnam_2023_a2_solution n := by
| import Mathlib
open Nat
abbrev putnam_2023_a2_solution : β β Set β := sorry
-- fun n => {(1 : β)/(factorial n), -(1 : β)/(factorial n)}
/--
Let $n$ be an even positive integer. Let $p$ be a monic, real polynomial of degree $2n$; that is to say, $p(x) = x^{2n} + a_{2n-1} x^{2n-1} + \cdots + a_1 x + a_0$ for some real coefficients $a_0, \dots, a_{2n-1}$. Suppose that $p(1/k) = k^2$ for all integers $k$ such that $1 \leq |k| \leq n$. Find all other real numbers $x$ for which $p(1/x) = x^2$.
-/
theorem putnam_2023_a2
(n : β)
(hn : n > 0 β§ Even n)
(p : Polynomial β)
(hp : Polynomial.Monic p β§ p.degree = 2*n)
(S : Set β)
(hS : S = {x : β | β k : β€, x = k β§ 1 β€ |k| β§ |k| β€ n})
(hpinv : β k β S, p.eval (1/k) = k^2)
: {x : β | p.eval (1/x) = x^2} \ S = putnam_2023_a2_solution n :=
sorry
| Let $n$ be an even positive integer. Let $p$ be a monic, real polynomial of degree $2n$; that is to say, $p(x) = x^{2n} + a_{2n-1} x^{2n-1} + \cdots + a_1 x + a_0$ for some real coefficients $a_0, \dots, a_{2n-1}$. Suppose that $p(1/k) = k^2$ for all integers $k$ such that $1 \leq |k| \leq n$. Find all other real numbers $x$ for which $p(1/x) = x^2$. | Show that the other real numbers satisfying $p(1/x) = x^2$ are $\pm \frac{1}{n!}.$ | [
"algebra"
] | null | null |
|
putnam_1967_a3 | 48c563a8-d844-5032-b38c-e312319ca2ba | train | abbrev putnam_1967_a3_solution : β := sorry
-- 5
/--
Consider polynomial forms $ax^2-bx+c$ with integer coefficients which have two distinct zeros in the open interval $0<x<1$. Exhibit with a proof the least positive integer value of $a$ for which such a polynomial exists.
-/
theorem putnam_1967_a3 :
IsLeast
{a | β P : Polynomial β€,
P.degree = 2 β§
(β z1 z2 : Set.Ioo (0 : β) 1, z1 β z2 β§ aeval (z1 : β) P = 0 β§ aeval (z2 : β) P = 0) β§
P.coeff 2 = a β§ a > 0}
putnam_1967_a3_solution :=
sorry | import Mathlib
open Polynomial
-- 5
/--
Consider polynomial forms $ax^2-bx+c$ with integer coefficients which have two distinct zeros in the open interval $0<x<1$. Exhibit with a proof the least positive integer value of $a$ for which such a polynomial exists.
-/
theorem putnam_1967_a3 :
IsLeast
{a | β P : Polynomial β€,
P.degree = 2 β§
(β z1 z2 : Set.Ioo (0 : β) 1, z1 β z2 β§ aeval (z1 : β) P = 0 β§ aeval (z2 : β) P = 0) β§
P.coeff 2 = a β§ a > 0}
putnam_1967_a3_solution := by
| import Mathlib
open Polynomial
abbrev putnam_1967_a3_solution : β := sorry
-- 5
/--
Consider polynomial forms $ax^2-bx+c$ with integer coefficients which have two distinct zeros in the open interval $0<x<1$. Exhibit with a proof the least positive integer value of $a$ for which such a polynomial exists.
-/
theorem putnam_1967_a3 :
IsLeast
{a | β P : Polynomial β€,
P.degree = 2 β§
(β z1 z2 : Set.Ioo (0 : β) 1, z1 β z2 β§ aeval (z1 : β) P = 0 β§ aeval (z2 : β) P = 0) β§
P.coeff 2 = a β§ a > 0}
putnam_1967_a3_solution :=
sorry
| Consider polynomial forms $ax^2-bx+c$ with integer coefficients which have two distinct zeros in the open interval $0<x<1$. Exhibit with a proof the least positive integer value of $a$ for which such a polynomial exists. | Show that the minimum possible value for $a$ is $5$. | [
"algebra"
] | null | null |
|
putnam_2017_a2 | ce793b5b-1fa0-5d36-96d8-b558e05a6f02 | train | theorem putnam_2017_a2
(Q : β β RatFunc β)
(hQbase : Q 0 = 1 β§ Q 1 = (X : β[X]))
(hQn : β n, Q (n + 2) = (Q (n + 1) ^ 2 - 1) / Q n)
(n : β) (hn : 0 < n) :
β P : β€[X], Q n = P.map (Int.castRingHom β) :=
sorry | import Mathlib
open Polynomial
/--
Let $Q_0(x)=1$, $Q_1(x)=x$, and $Q_n(x)=\frac{(Q_{n-1}(x))^2-1}{Q_{n-2}(x)}$ for all $n \geq 2$. Show that, whenever $n$ is a positive integer, $Q_n(x)$ is equal to a polynomial with integer coefficients.
-/
theorem putnam_2017_a2
(Q : β β RatFunc β)
(hQbase : Q 0 = 1 β§ Q 1 = (X : β[X]))
(hQn : β n, Q (n + 2) = (Q (n + 1) ^ 2 - 1) / Q n)
(n : β) (hn : 0 < n) :
β P : β€[X], Q n = P.map (Int.castRingHom β) := by
| import Mathlib
open Polynomial
/--
Let $Q_0(x)=1$, $Q_1(x)=x$, and $Q_n(x)=\frac{(Q_{n-1}(x))^2-1}{Q_{n-2}(x)}$ for all $n \geq 2$. Show that, whenever $n$ is a positive integer, $Q_n(x)$ is equal to a polynomial with integer coefficients.
-/
theorem putnam_2017_a2
(Q : β β RatFunc β)
(hQbase : Q 0 = 1 β§ Q 1 = (X : β[X]))
(hQn : β n, Q (n + 2) = (Q (n + 1) ^ 2 - 1) / Q n)
(n : β) (hn : 0 < n) :
β P : β€[X], Q n = P.map (Int.castRingHom β) :=
sorry
| Let $Q_0(x)=1$, $Q_1(x)=x$, and $Q_n(x)=\frac{(Q_{n-1}(x))^2-1}{Q_{n-2}(x)}$ for all $n \geq 2$. Show that, whenever $n$ is a positive integer, $Q_n(x)$ is equal to a polynomial with integer coefficients. | null | [
"algebra"
] | null | null |
|
putnam_1985_a4 | a1ff1e68-5fae-547a-9d50-c717a50f78c6 | train | 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 | import Mathlib
open Set Filter Topology Real
-- {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 := by
| import Mathlib
open Set Filter Topology Real
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
| 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 $a_i$? | Prove that the only number that occurs infinitely often is $87$. | [
"number_theory"
] | null | null |
|
putnam_1987_b1 | fdfad460-2c51-56b4-b538-1469dfec4945 | train | 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 | import Mathlib
open MvPolynomial Real Nat
-- 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) := by
| import Mathlib
open MvPolynomial Real Nat
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
| Evaluate
\[
\int_2^4 \frac{\sqrt{\ln(9-x)}\,dx}{\sqrt{\ln(9-x)}+\sqrt{\ln(x+3)}}.
\] | Prove that the integral evaluates to $1$. | [
"analysis"
] | null | null |
|
putnam_1993_a1 | 93921199-8b22-528c-af3e-dd302d85b1b9 | train | 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 | import Mathlib
-- 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) := by
| import Mathlib
noncomputable 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
| 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.] | Show that the area of the two regions are equal when $c=4/9$. | [
"analysis",
"algebra"
] | null | null |
|
putnam_2022_a3 | 5888ca0b-823e-5132-8cb8-2c8f171ec9c5 | train | theorem putnam_2022_a3
(p f : β)
(hp : Nat.Prime p β§ p > 5)
(hf : f = {a : β β (ZMod p) | β n : β, a n β 0 β§ a n * a (n + 2) = 1 + a (n + 1)}.ncard)
: f β‘ 0 [MOD 5] β¨ f β‘ 2 [MOD 5] :=
sorry | import Mathlib
open Polynomial
/--
Let $p$ be a prime number greater than 5. Let $f(p)$ denote the number of infinite sequences $a_1, a_2, a_3, \dots$ such that $a_n \in \{1, 2, \dots, p-1\}$ and $a_n a_{n+2} \equiv 1 + a_{n+1} \pmod{p}$ for all $n \geq 1$. Prove that $f(p)$ is congruent to 0 or 2 $\pmod{5}$.
-/
theorem putnam_2022_a3
(p f : β)
(hp : Nat.Prime p β§ p > 5)
(hf : f = {a : β β (ZMod p) | β n : β, a n β 0 β§ a n * a (n + 2) = 1 + a (n + 1)}.ncard)
: f β‘ 0 [MOD 5] β¨ f β‘ 2 [MOD 5] := by
| import Mathlib
open Polynomial
/--
Let $p$ be a prime number greater than 5. Let $f(p)$ denote the number of infinite sequences $a_1, a_2, a_3, \dots$ such that $a_n \in \{1, 2, \dots, p-1\}$ and $a_n a_{n+2} \equiv 1 + a_{n+1} \pmod{p}$ for all $n \geq 1$. Prove that $f(p)$ is congruent to 0 or 2 $\pmod{5}$.
-/
theorem putnam_2022_a3
(p f : β)
(hp : Nat.Prime p β§ p > 5)
(hf : f = {a : β β (ZMod p) | β n : β, a n β 0 β§ a n * a (n + 2) = 1 + a (n + 1)}.ncard)
: f β‘ 0 [MOD 5] β¨ f β‘ 2 [MOD 5] :=
sorry
| Let $p$ be a prime number greater than 5. Let $f(p)$ denote the number of infinite sequences $a_1, a_2, a_3, \dots$ such that $a_n \in \{1, 2, \dots, p-1\}$ and $a_n a_{n+2} \equiv 1 + a_{n+1} \pmod{p}$ for all $n \geq 1$. Prove that $f(p)$ is congruent to 0 or 2 $\pmod{5}$. | null | [
"number_theory"
] | null | null |
|
putnam_2018_a5 | 6cb66dc8-8296-58de-86f9-22e9d854dc74 | train | theorem putnam_2018_a5
(f : β β β)
(h0 : f 0 = 0)
(h1 : f 1 = 1)
(hpos : β x : β, f x β₯ 0)
(hf : ContDiff β β€ f)
: β n > 0, β x : β, iteratedDeriv n f x < 0 :=
sorry | import Mathlib
/--
Let $f: \mathbb{R} \to \mathbb{R}$ be an infinitely differentiable function satisfying $f(0) = 0$, $f(1)= 1$, and $f(x) \geq 0$ for all $x \in \mathbb{R}$. Show that there exist a positive integer $n$ and a real number $x$ such that $f^{(n)}(x) < 0$.
-/
theorem putnam_2018_a5
(f : β β β)
(h0 : f 0 = 0)
(h1 : f 1 = 1)
(hpos : β x : β, f x β₯ 0)
(hf : ContDiff β β€ f)
: β n > 0, β x : β, iteratedDeriv n f x < 0 := by
| import Mathlib
/--
Let $f: \mathbb{R} \to \mathbb{R}$ be an infinitely differentiable function satisfying $f(0) = 0$, $f(1)= 1$, and $f(x) \geq 0$ for all $x \in \mathbb{R}$. Show that there exist a positive integer $n$ and a real number $x$ such that $f^{(n)}(x) < 0$.
-/
theorem putnam_2018_a5
(f : β β β)
(h0 : f 0 = 0)
(h1 : f 1 = 1)
(hpos : β x : β, f x β₯ 0)
(hf : ContDiff β β€ f)
: β n > 0, β x : β, iteratedDeriv n f x < 0 :=
sorry
| Let $f: \mathbb{R} \to \mathbb{R}$ be an infinitely differentiable function satisfying $f(0) = 0$, $f(1)= 1$, and $f(x) \geq 0$ for all $x \in \mathbb{R}$. Show that there exist a positive integer $n$ and a real number $x$ such that $f^{(n)}(x) < 0$. | null | [
"analysis"
] | null | null |
|
putnam_1996_b4 | 37936eda-c1e3-59a2-b626-7b29da1a9117 | train | abbrev putnam_1996_b4_solution : Prop := sorry
-- False
/--
For any square matrix $A$, we can define $\sin A$ by the usual power series: $\sin A=\sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!}A^{2n+1}$. Prove or disprove: there exists a $2 \times 2$ matrix $A$ with real entries such that $\sin A=\begin{pmatrix} 1 & 1996 \\ 0 & 1 \end{pmatrix}$.
-/
theorem putnam_1996_b4
(matsin : Matrix (Fin 2) (Fin 2) β β Matrix (Fin 2) (Fin 2) β)
(mat1996 : Matrix (Fin 2) (Fin 2) β)
(hmatsin : β A, matsin A = β' n : β, ((-(1 : β)) ^ n / (2 * n + 1)!) β’ A ^ (2 * n + 1))
(hmat1996 : mat1996 0 0 = 1 β§ mat1996 0 1 = 1996 β§ mat1996 1 0 = 0 β§ mat1996 1 1 = 1)
: (β A, matsin A = mat1996) β putnam_1996_b4_solution :=
sorry | import Mathlib
open Function Nat
-- False
/--
For any square matrix $A$, we can define $\sin A$ by the usual power series: $\sin A=\sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!}A^{2n+1}$. Prove or disprove: there exists a $2 \times 2$ matrix $A$ with real entries such that $\sin A=\begin{pmatrix} 1 & 1996 \\ 0 & 1 \end{pmatrix}$.
-/
theorem putnam_1996_b4
(matsin : Matrix (Fin 2) (Fin 2) β β Matrix (Fin 2) (Fin 2) β)
(mat1996 : Matrix (Fin 2) (Fin 2) β)
(hmatsin : β A, matsin A = β' n : β, ((-(1 : β)) ^ n / (2 * n + 1)!) β’ A ^ (2 * n + 1))
(hmat1996 : mat1996 0 0 = 1 β§ mat1996 0 1 = 1996 β§ mat1996 1 0 = 0 β§ mat1996 1 1 = 1)
: (β A, matsin A = mat1996) β putnam_1996_b4_solution := by
| import Mathlib
open Function Nat
abbrev putnam_1996_b4_solution : Prop := sorry
-- False
/--
For any square matrix $A$, we can define $\sin A$ by the usual power series: $\sin A=\sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!}A^{2n+1}$. Prove or disprove: there exists a $2 \times 2$ matrix $A$ with real entries such that $\sin A=\begin{pmatrix} 1 & 1996 \\ 0 & 1 \end{pmatrix}$.
-/
theorem putnam_1996_b4
(matsin : Matrix (Fin 2) (Fin 2) β β Matrix (Fin 2) (Fin 2) β)
(mat1996 : Matrix (Fin 2) (Fin 2) β)
(hmatsin : β A, matsin A = β' n : β, ((-(1 : β)) ^ n / (2 * n + 1)!) β’ A ^ (2 * n + 1))
(hmat1996 : mat1996 0 0 = 1 β§ mat1996 0 1 = 1996 β§ mat1996 1 0 = 0 β§ mat1996 1 1 = 1)
: (β A, matsin A = mat1996) β putnam_1996_b4_solution :=
sorry
| For any square matrix $A$, we can define $\sin A$ by the usual power series: $\sin A=\sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!}A^{2n+1}$. Prove or disprove: there exists a $2 \times 2$ matrix $A$ with real entries such that $\sin A=\begin{pmatrix} 1 & 1996 \\ 0 & 1 \end{pmatrix}$. | Show that there does not exist such a matrix $A$. | [
"linear_algebra"
] | null | null |
|
putnam_2009_b6 | e8a0abf9-a237-519c-aa8b-e4148ed98f2b | train | 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 | import Mathlib
open Topology MvPolynomial Filter Set Metric
/--
Prove that for every positive integer $n$, there is a sequence of integers $a_0, a_1, \dots, a_{2009}$ with $a_0 = 0$ and $a_{2009} = n$ such that each term after $a_0$ is either an earlier term plus $2^k$ for some nonnegative integer $k$, or of the form $b\,\mathrm{mod}\,c$ for some earlier positive terms $b$ and $c$. [Here $b\,\mathrm{mod}\,c$ denotes the remainder when $b$ is divided by $c$, so $0 \leq (b\,\mathrm{mod}\,c) < c$.]
-/
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))) := by
| import Mathlib
open Topology MvPolynomial Filter Set Metric
/--
Prove that for every positive integer $n$, there is a sequence of integers $a_0, a_1, \dots, a_{2009}$ with $a_0 = 0$ and $a_{2009} = n$ such that each term after $a_0$ is either an earlier term plus $2^k$ for some nonnegative integer $k$, or of the form $b\,\mathrm{mod}\,c$ for some earlier positive terms $b$ and $c$. [Here $b\,\mathrm{mod}\,c$ denotes the remainder when $b$ is divided by $c$, so $0 \leq (b\,\mathrm{mod}\,c) < c$.]
-/
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
| Prove that for every positive integer $n$, there is a sequence of integers $a_0, a_1, \dots, a_{2009}$ with $a_0 = 0$ and $a_{2009} = n$ such that each term after $a_0$ is either an earlier term plus $2^k$ for some nonnegative integer $k$, or of the form $b\,\mathrm{mod}\,c$ for some earlier positive terms $b$ and $c$. [Here $b\,\mathrm{mod}\,c$ denotes the remainder when $b$ is divided by $c$, so $0 \leq (b\,\mathrm{mod}\,c) < c$.] | null | [
"number_theory"
] | null | null |
|
putnam_1996_b2 | a15cc369-900e-5ef0-9418-17899cb380f6 | train | theorem putnam_1996_b2
(n : β)
(prododd : β)
(npos : n > 0)
(hprododd : prododd = β i in Finset.range (2 * n), if Odd i then i else 1)
: ((2 * n - 1 : β) / Real.exp 1) ^ ((2 * n - 1 : β) / 2) < prododd β§ prododd < ((2 * n + 1 : β) / Real.exp 1) ^ ((2 * n + 1 : β) / 2) :=
sorry | import Mathlib
open Function
/--
Show that for every positive integer $n$, $(\frac{2n-1}{e})^{\frac{2n-1}{2}}<1 \cdot 3 \cdot 5 \cdots (2n-1)<(\frac{2n+1}{e})^{\frac{2n+1}{2}}$.
-/
theorem putnam_1996_b2
(n : β)
(prododd : β)
(npos : n > 0)
(hprododd : prododd = β i in Finset.range (2 * n), if Odd i then i else 1)
: ((2 * n - 1 : β) / Real.exp 1) ^ ((2 * n - 1 : β) / 2) < prododd β§ prododd < ((2 * n + 1 : β) / Real.exp 1) ^ ((2 * n + 1 : β) / 2) := by
| import Mathlib
open Function
/--
Show that for every positive integer $n$, $(\frac{2n-1}{e})^{\frac{2n-1}{2}}<1 \cdot 3 \cdot 5 \cdots (2n-1)<(\frac{2n+1}{e})^{\frac{2n+1}{2}}$.
-/
theorem putnam_1996_b2
(n : β)
(prododd : β)
(npos : n > 0)
(hprododd : prododd = β i in Finset.range (2 * n), if Odd i then i else 1)
: ((2 * n - 1 : β) / Real.exp 1) ^ ((2 * n - 1 : β) / 2) < prododd β§ prododd < ((2 * n + 1 : β) / Real.exp 1) ^ ((2 * n + 1 : β) / 2) :=
sorry
| Show that for every positive integer $n$, $(\frac{2n-1}{e})^{\frac{2n-1}{2}}<1 \cdot 3 \cdot 5 \cdots (2n-1)<(\frac{2n+1}{e})^{\frac{2n+1}{2}}$. | null | [
"analysis"
] | null | null |
|
putnam_2012_b5 | ca3eb972-f6a7-52bf-b3a6-c33ca06be2ee | train | 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 | import Mathlib
open Matrix Function Real Topology Filter
/--
Prove that, for any two bounded functions $g_1, g_2: \RR \to [1, \infty)$, there exist functions $h_1, h_2: \RR \to \RR$ such that, for every $x \in \RR$, \[ \sup_{s \in \RR} (g_1(s)^x g_2(s)) = \max_{t \in \RR} (x h_1(t) + h_2(t)).\]
-/
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} := by
| import Mathlib
open Matrix Function Real Topology Filter
/--
Prove that, for any two bounded functions $g_1, g_2: \RR \to [1, \infty)$, there exist functions $h_1, h_2: \RR \to \RR$ such that, for every $x \in \RR$, \[ \sup_{s \in \RR} (g_1(s)^x g_2(s)) = \max_{t \in \RR} (x h_1(t) + h_2(t)).\]
-/
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
| Prove that, for any two bounded functions $g_1, g_2: \RR \to [1, \infty)$, there exist functions $h_1, h_2: \RR \to \RR$ such that, for every $x \in \RR$, \[ \sup_{s \in \RR} (g_1(s)^x g_2(s)) = \max_{t \in \RR} (x h_1(t) + h_2(t)).\] | null | [
"analysis"
] | null | null |
|
putnam_1999_a2 | 2f78f9f8-d714-559e-80ef-1064cdf8cfc7 | train | theorem putnam_1999_a2
(p : Polynomial β)
(hp : β x, p.eval x β₯ 0)
: βα΅ (k) (f : Fin k β Polynomial β),
k > 0 β§
β x : β, p.eval x = β j : Fin k, ((f j).eval x) ^ 2 :=
sorry | import Mathlib
/--
Let $p(x)$ be a polynomial that is nonnegative for all real $x$. Prove that for some $k$, there are polynomials $f_1(x),\dots,f_k(x$) such that \[p(x) = \sum_{j=1}^k (f_j(x))^2.\]
-/
theorem putnam_1999_a2
(p : Polynomial β)
(hp : β x, p.eval x β₯ 0)
: βα΅ (k) (f : Fin k β Polynomial β),
k > 0 β§
β x : β, p.eval x = β j : Fin k, ((f j).eval x) ^ 2 := by
| import Mathlib
/--
Let $p(x)$ be a polynomial that is nonnegative for all real $x$. Prove that for some $k$, there are polynomials $f_1(x),\dots,f_k(x$) such that \[p(x) = \sum_{j=1}^k (f_j(x))^2.\]
-/
theorem putnam_1999_a2
(p : Polynomial β)
(hp : β x, p.eval x β₯ 0)
: βα΅ (k) (f : Fin k β Polynomial β),
k > 0 β§
β x : β, p.eval x = β j : Fin k, ((f j).eval x) ^ 2 :=
sorry
| Let $p(x)$ be a polynomial that is nonnegative for all real $x$. Prove that for some $k$, there are polynomials $f_1(x),\dots,f_k(x$) such that \[p(x) = \sum_{j=1}^k (f_j(x))^2.\] | null | [
"algebra"
] | null | null |
|
putnam_2016_a5 | 0c948bdf-2ca5-5d9f-8e78-cb7fc554c28d | train | theorem putnam_2016_a5
(G : Type*) [Group G]
(Gfin : Fintype G)
(g h : G)
(ghgen : Group.closure {g, h} = G β§ Β¬Group.closure {g} = G β§ Β¬Group.closure {h} = G)
(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 | import Mathlib
open Polynomial Filter Topology Real Set Nat List
/--
Suppose that $G$ is a finite group generated by the two elements $g$ and $h$, where the order of $g$ is odd. Show that every element of $G$ can be written in the form
\[
g^{m_1} h^{n_1} g^{m_2} h^{n_2} \cdots g^{m_r} h^{n_r}
\]
with $1 \leq r \leq |G|$ and $m_1, n_1, m_2, n_2, \ldots, m_r, n_r \in \{-1, 1\}$.
(Here $|G|$ is the number of elements of $G$.)
-/
theorem putnam_2016_a5
(G : Type*) [Group G]
(Gfin : Fintype G)
(g h : G)
(ghgen : Group.closure {g, h} = G β§ Β¬Group.closure {g} = G β§ Β¬Group.closure {h} = G)
(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) := by
| import Mathlib
open Polynomial Filter Topology Real Set Nat List
/--
Suppose that $G$ is a finite group generated by the two elements $g$ and $h$, where the order of $g$ is odd. Show that every element of $G$ can be written in the form
\[
g^{m_1} h^{n_1} g^{m_2} h^{n_2} \cdots g^{m_r} h^{n_r}
\]
with $1 \leq r \leq |G|$ and $m_1, n_1, m_2, n_2, \ldots, m_r, n_r \in \{-1, 1\}$.
(Here $|G|$ is the number of elements of $G$.)
-/
theorem putnam_2016_a5
(G : Type*) [Group G]
(Gfin : Fintype G)
(g h : G)
(ghgen : Group.closure {g, h} = G β§ Β¬Group.closure {g} = G β§ Β¬Group.closure {h} = G)
(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
| Suppose that $G$ is a finite group generated by the two elements $g$ and $h$, where the order of $g$ is odd. Show that every element of $G$ can be written in the form
\[
g^{m_1} h^{n_1} g^{m_2} h^{n_2} \cdots g^{m_r} h^{n_r}
\]
with $1 \leq r \leq |G|$ and $m_1, n_1, m_2, n_2, \ldots, m_r, n_r \in \{-1, 1\}$.
(Here $|G|$ is the number of elements of $G$.) | null | [
"abstract_algebra"
] | null | null |
|
putnam_1997_a6 | 7b85c311-a4ff-5854-b6e5-e99afc5ef7fb | train | abbrev putnam_1997_a6_solution : β€ β β€ β β := sorry
-- fun n k => Nat.choose (n.toNat-1) (k.toNat-1)
/--
For a positive integer $n$ and any real number $c$, define $x_k$ recursively by $x_0=0$, $x_1=1$, and for $k\geq 0$, \[x_{k+2}=\frac{cx_{k+1}-(n-k)x_k}{k+1}.\] Fix $n$ and then take $c$ to be the largest value for which $x_{n+1}=0$. Find $x_k$ in terms of $n$ and $k$, $1\leq k\leq n$.
-/
theorem putnam_1997_a6
(n : β€)
(hn : n > 0)
(x : β β (β€ β β))
(hx0 : β c, x c 0 = 0)
(hx1 : β c, x c 1 = 1)
(hxk : β c, β k β₯ 0, x c (k + 2) = (c*(x c (k + 1)) - (n - k)*(x c k))/(k + 1))
(S : Set β)
(hS : S = {c : β | x c (n + 1) = 0}) :
β k : Set.Icc 1 n, x (sSup S) k = putnam_1997_a6_solution n k :=
sorry | import Mathlib
open Filter Topology
-- fun n k => Nat.choose (n.toNat-1) (k.toNat-1)
/--
For a positive integer $n$ and any real number $c$, define $x_k$ recursively by $x_0=0$, $x_1=1$, and for $k\geq 0$, \[x_{k+2}=\frac{cx_{k+1}-(n-k)x_k}{k+1}.\] Fix $n$ and then take $c$ to be the largest value for which $x_{n+1}=0$. Find $x_k$ in terms of $n$ and $k$, $1\leq k\leq n$.
-/
theorem putnam_1997_a6
(n : β€)
(hn : n > 0)
(x : β β (β€ β β))
(hx0 : β c, x c 0 = 0)
(hx1 : β c, x c 1 = 1)
(hxk : β c, β k β₯ 0, x c (k + 2) = (c*(x c (k + 1)) - (n - k)*(x c k))/(k + 1))
(S : Set β)
(hS : S = {c : β | x c (n + 1) = 0}) :
β k : Set.Icc 1 n, x (sSup S) k = putnam_1997_a6_solution n k := by
| import Mathlib
open Filter Topology
abbrev putnam_1997_a6_solution : β€ β β€ β β := sorry
-- fun n k => Nat.choose (n.toNat-1) (k.toNat-1)
/--
For a positive integer $n$ and any real number $c$, define $x_k$ recursively by $x_0=0$, $x_1=1$, and for $k\geq 0$, \[x_{k+2}=\frac{cx_{k+1}-(n-k)x_k}{k+1}.\] Fix $n$ and then take $c$ to be the largest value for which $x_{n+1}=0$. Find $x_k$ in terms of $n$ and $k$, $1\leq k\leq n$.
-/
theorem putnam_1997_a6
(n : β€)
(hn : n > 0)
(x : β β (β€ β β))
(hx0 : β c, x c 0 = 0)
(hx1 : β c, x c 1 = 1)
(hxk : β c, β k β₯ 0, x c (k + 2) = (c*(x c (k + 1)) - (n - k)*(x c k))/(k + 1))
(S : Set β)
(hS : S = {c : β | x c (n + 1) = 0}) :
β k : Set.Icc 1 n, x (sSup S) k = putnam_1997_a6_solution n k :=
sorry
| For a positive integer $n$ and any real number $c$, define $x_k$ recursively by $x_0=0$, $x_1=1$, and for $k\geq 0$, \[x_{k+2}=\frac{cx_{k+1}-(n-k)x_k}{k+1}.\] Fix $n$ and then take $c$ to be the largest value for which $x_{n+1}=0$. Find $x_k$ in terms of $n$ and $k$, $1\leq k\leq n$. | Show that the solution is that $x_k = {n - 1 \choose k - 1}$. | [
"algebra"
] | null | null |
|
putnam_1992_b6 | d43f1421-71e8-5f65-a1e0-10850b1a488d | train | theorem putnam_1992_b6
(n : β)
(npos : 0 < n)
(M : Set (Matrix (Fin n) (Fin n) β))
(h1 : 1 β M)
(h2 : β A β M, β B β M, Xor' (A * B β M) (-A * B β M))
(h3 : β A β M, β B β M, (A * B = B * A) β¨ (A * B = -B * A))
(h4 : β A β M, A β 1 β β B β M, A * B = -B * A) :
M.encard β€ n ^ 2 :=
sorry | import Mathlib
open Topology Filter Nat Function Polynomial
/--
Let $M$ be a set of real $n \times n$ matrices such that
\begin{itemize}
\item[(i)] $I \in M$, where $I$ is the $n \times n$ identity matrix;
\item[(ii)] if $A \in M$ and $B \in M$, then either $AB \in M$ or $-AB \in M$, but not both;
\item[(iii)] if $A \in M$ and $B \in M$, then either $AB = BA$ or $AB = -BA$;
\item[(iv)] if $A \in M$ and $A \neq I$, there is at least one $B \in M$ such that $AB = -BA$.
\end{itemize}
Prove that $M$ contains at most $n^2$ matrices.
-/
theorem putnam_1992_b6
(n : β)
(npos : 0 < n)
(M : Set (Matrix (Fin n) (Fin n) β))
(h1 : 1 β M)
(h2 : β A β M, β B β M, Xor' (A * B β M) (-A * B β M))
(h3 : β A β M, β B β M, (A * B = B * A) β¨ (A * B = -B * A))
(h4 : β A β M, A β 1 β β B β M, A * B = -B * A) :
M.encard β€ n ^ 2 := by
| import Mathlib
open Topology Filter Nat Function Polynomial
/--
Let $M$ be a set of real $n \times n$ matrices such that
\begin{itemize}
\item[(i)] $I \in M$, where $I$ is the $n \times n$ identity matrix;
\item[(ii)] if $A \in M$ and $B \in M$, then either $AB \in M$ or $-AB \in M$, but not both;
\item[(iii)] if $A \in M$ and $B \in M$, then either $AB = BA$ or $AB = -BA$;
\item[(iv)] if $A \in M$ and $A \neq I$, there is at least one $B \in M$ such that $AB = -BA$.
\end{itemize}
Prove that $M$ contains at most $n^2$ matrices.
-/
theorem putnam_1992_b6
(n : β)
(npos : 0 < n)
(M : Set (Matrix (Fin n) (Fin n) β))
(h1 : 1 β M)
(h2 : β A β M, β B β M, Xor' (A * B β M) (-A * B β M))
(h3 : β A β M, β B β M, (A * B = B * A) β¨ (A * B = -B * A))
(h4 : β A β M, A β 1 β β B β M, A * B = -B * A) :
M.encard β€ n ^ 2 :=
sorry
| Let $M$ be a set of real $n \times n$ matrices such that
\begin{itemize}
\item[(i)] $I \in M$, where $I$ is the $n \times n$ identity matrix;
\item[(ii)] if $A \in M$ and $B \in M$, then either $AB \in M$ or $-AB \in M$, but not both;
\item[(iii)] if $A \in M$ and $B \in M$, then either $AB = BA$ or $AB = -BA$;
\item[(iv)] if $A \in M$ and $A \neq I$, there is at least one $B \in M$ such that $AB = -BA$.
\end{itemize}
Prove that $M$ contains at most $n^2$ matrices. | null | [
"linear_algebra"
] | null | null |
|
putnam_1962_a5 | af498095-2197-562b-8abc-adef92ef34bc | train | abbrev putnam_1962_a5_solution : β β β := sorry
-- fun n : β => n * (n + 1) * 2^(n - 2)
/--
Evaluate in closed form \[ \sum_{k=1}^n {n \choose k} k^2. \]
-/
theorem putnam_1962_a5
: β n β₯ 2, putnam_1962_a5_solution n = β k in Finset.Icc 1 n, Nat.choose n k * k^2 :=
sorry | import Mathlib
-- fun n : β => n * (n + 1) * 2^(n - 2)
/--
Evaluate in closed form \[ \sum_{k=1}^n {n \choose k} k^2. \]
-/
theorem putnam_1962_a5
: β n β₯ 2, putnam_1962_a5_solution n = β k in Finset.Icc 1 n, Nat.choose n k * k^2 := by
| import Mathlib
abbrev putnam_1962_a5_solution : β β β := sorry
-- fun n : β => n * (n + 1) * 2^(n - 2)
/--
Evaluate in closed form \[ \sum_{k=1}^n {n \choose k} k^2. \]
-/
theorem putnam_1962_a5
: β n β₯ 2, putnam_1962_a5_solution n = β k in Finset.Icc 1 n, Nat.choose n k * k^2 :=
sorry
| Evaluate in closed form \[ \sum_{k=1}^n {n \choose k} k^2. \] | Show that the expression equals $n(n+1)2^{n-2}$. | [
"algebra",
"combinatorics"
] | null | null |
|
putnam_2012_a4 | db715975-6bf7-53f3-a76a-b838a2c45046 | train | theorem putnam_2012_a4
(q r : β€)
(A B : Fin 2 β β)
(T : Set β)
(S : Set β€)
(qpos : q > 0)
(ABlt : A 0 < A 1 β§ B 0 < B 1)
(hT : T = {x : β | β b m : β€, ((b : β) β Set.Icc (B 0) (B 1)) β§ (x = b + m * q)})
(hS : S = {a : β€ | ((a : β) β Set.Icc (A 0) (A 1)) β§ (β t β T, r * a = t)})
: ((A 1 - A 0) * (B 1 - B 0) < q) β (β n : β, β a1 d : β, n > 2 β§ {s : β | s = round s β§ round s β S} = (Set.Icc (A 0) (A 1)) β© {x : β | β i : Fin n, x = a1 + i * d}) :=
sorry | import Mathlib
open Matrix Function
/--
Let $q$ and $r$ be integers with $q>0$, and let $A$ and $B$ be intervals on the real line. Let $T$ be the set of all $b+mq$ where $b$ and $m$ are integers with $b$ in $B$, and let $S$ be the set of all integers $a$ in $A$ such that $ra$ is in $T$. Show that if the product of the lengths of $A$ and $B$ is less than $q$, then $S$ is the intersection of $A$ with some arithmetic progression.
-/
theorem putnam_2012_a4
(q r : β€)
(A B : Fin 2 β β)
(T : Set β)
(S : Set β€)
(qpos : q > 0)
(ABlt : A 0 < A 1 β§ B 0 < B 1)
(hT : T = {x : β | β b m : β€, ((b : β) β Set.Icc (B 0) (B 1)) β§ (x = b + m * q)})
(hS : S = {a : β€ | ((a : β) β Set.Icc (A 0) (A 1)) β§ (β t β T, r * a = t)})
: ((A 1 - A 0) * (B 1 - B 0) < q) β (β n : β, β a1 d : β, n > 2 β§ {s : β | s = round s β§ round s β S} = (Set.Icc (A 0) (A 1)) β© {x : β | β i : Fin n, x = a1 + i * d}) := by
| import Mathlib
open Matrix Function
/--
Let $q$ and $r$ be integers with $q>0$, and let $A$ and $B$ be intervals on the real line. Let $T$ be the set of all $b+mq$ where $b$ and $m$ are integers with $b$ in $B$, and let $S$ be the set of all integers $a$ in $A$ such that $ra$ is in $T$. Show that if the product of the lengths of $A$ and $B$ is less than $q$, then $S$ is the intersection of $A$ with some arithmetic progression.
-/
theorem putnam_2012_a4
(q r : β€)
(A B : Fin 2 β β)
(T : Set β)
(S : Set β€)
(qpos : q > 0)
(ABlt : A 0 < A 1 β§ B 0 < B 1)
(hT : T = {x : β | β b m : β€, ((b : β) β Set.Icc (B 0) (B 1)) β§ (x = b + m * q)})
(hS : S = {a : β€ | ((a : β) β Set.Icc (A 0) (A 1)) β§ (β t β T, r * a = t)})
: ((A 1 - A 0) * (B 1 - B 0) < q) β (β n : β, β a1 d : β, n > 2 β§ {s : β | s = round s β§ round s β S} = (Set.Icc (A 0) (A 1)) β© {x : β | β i : Fin n, x = a1 + i * d}) :=
sorry
| Let $q$ and $r$ be integers with $q>0$, and let $A$ and $B$ be intervals on the real line. Let $T$ be the set of all $b+mq$ where $b$ and $m$ are integers with $b$ in $B$, and let $S$ be the set of all integers $a$ in $A$ such that $ra$ is in $T$. Show that if the product of the lengths of $A$ and $B$ is less than $q$, then $S$ is the intersection of $A$ with some arithmetic progression. | null | [
"algebra"
] | null | null |
|
putnam_1976_a4 | dbdd51d0-cbf8-59e0-ac9b-9d5c5b891ec6 | train | 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 | import Mathlib
open Polynomial
-- (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) := by
| import Mathlib
open Polynomial
noncomputable 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
| 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 $d$. | The possible answers are $s = -\frac{1}{r + 1}$ and $s = -\frac{r + 1}{r}$. | [
"algebra"
] | null | null |
|
putnam_1985_b1 | 636964ba-b29f-5ef2-b24a-86603db3e126 | train | abbrev putnam_1985_b1_solution : Fin 5 β β€ := sorry
-- fun i β¦ i - 2
/--
Let $k$ be the smallest positive integer for which there exist distinct integers $m_1, m_2, m_3, m_4, m_5$ such that the polynomial
\[
p(x) = (x-m_1)(x-m_2)(x-m_3)(x-m_4)(x-m_5)
\]
has exactly $k$ nonzero coefficients. Find, with proof, a set of integers $m_1, m_2, m_3, m_4, m_5$ for which this minimum $k$ is achieved.
-/
theorem putnam_1985_b1
(p : (Fin 5 β β€) β (Polynomial β))
(hp : p = fun m β¦ β i : Fin 5, ((X : Polynomial β) - m i))
(numnzcoeff : Polynomial β β β)
(hnumnzcoeff : numnzcoeff = fun p β¦ {j β Finset.range (p.natDegree + 1) | coeff p j β 0}.card)
: (Injective putnam_1985_b1_solution β§ β m : Fin 5 β β€, Injective m β numnzcoeff (p putnam_1985_b1_solution) β€ numnzcoeff (p m)) :=
sorry | import Mathlib
open Set Filter Topology Real Polynomial Function
-- fun i β¦ i - 2
/--
Let $k$ be the smallest positive integer for which there exist distinct integers $m_1, m_2, m_3, m_4, m_5$ such that the polynomial
\[
p(x) = (x-m_1)(x-m_2)(x-m_3)(x-m_4)(x-m_5)
\]
has exactly $k$ nonzero coefficients. Find, with proof, a set of integers $m_1, m_2, m_3, m_4, m_5$ for which this minimum $k$ is achieved.
-/
theorem putnam_1985_b1
(p : (Fin 5 β β€) β (Polynomial β))
(hp : p = fun m β¦ β i : Fin 5, ((X : Polynomial β) - m i))
(numnzcoeff : Polynomial β β β)
(hnumnzcoeff : numnzcoeff = fun p β¦ {j β Finset.range (p.natDegree + 1) | coeff p j β 0}.card)
: (Injective putnam_1985_b1_solution β§ β m : Fin 5 β β€, Injective m β numnzcoeff (p putnam_1985_b1_solution) β€ numnzcoeff (p m)) := by
| import Mathlib
open Set Filter Topology Real Polynomial Function
abbrev putnam_1985_b1_solution : Fin 5 β β€ := sorry
-- fun i β¦ i - 2
/--
Let $k$ be the smallest positive integer for which there exist distinct integers $m_1, m_2, m_3, m_4, m_5$ such that the polynomial
\[
p(x) = (x-m_1)(x-m_2)(x-m_3)(x-m_4)(x-m_5)
\]
has exactly $k$ nonzero coefficients. Find, with proof, a set of integers $m_1, m_2, m_3, m_4, m_5$ for which this minimum $k$ is achieved.
-/
theorem putnam_1985_b1
(p : (Fin 5 β β€) β (Polynomial β))
(hp : p = fun m β¦ β i : Fin 5, ((X : Polynomial β) - m i))
(numnzcoeff : Polynomial β β β)
(hnumnzcoeff : numnzcoeff = fun p β¦ {j β Finset.range (p.natDegree + 1) | coeff p j β 0}.card)
: (Injective putnam_1985_b1_solution β§ β m : Fin 5 β β€, Injective m β numnzcoeff (p putnam_1985_b1_solution) β€ numnzcoeff (p m)) :=
sorry
| Let $k$ be the smallest positive integer for which there exist distinct integers $m_1, m_2, m_3, m_4, m_5$ such that the polynomial
\[
p(x) = (x-m_1)(x-m_2)(x-m_3)(x-m_4)(x-m_5)
\]
has exactly $k$ nonzero coefficients. Find, with proof, a set of integers $m_1, m_2, m_3, m_4, m_5$ for which this minimum $k$ is achieved. | Show that the minimum $k = 3$ is obtained for $\{m_1, m_2, m_3, m_4, m_5\} = \{-2, -1, 0, 1, 2\}$. | [
"algebra"
] | null | null |
|
putnam_1989_a1 | 62b77870-657f-5f4c-a9d9-b0d6f484d294 | train | abbrev putnam_1989_a1_solution : ββ := sorry
-- 1
/--
How many primes among the positive integers, written as usual in base $10$, are alternating $1$'s and $0$'s, beginning and ending with $1$?
-/
theorem putnam_1989_a1
(pdigalt : List β β Prop)
(hpdigalt : β l, pdigalt l β Odd l.length β§ (β i, l.get i = if Even (i : β) then 1 else 0)) :
{p : β | p.Prime β§ pdigalt (Nat.digits 10 p)}.encard = putnam_1989_a1_solution :=
sorry | import Mathlib
-- 1
/--
How many primes among the positive integers, written as usual in base $10$, are alternating $1$'s and $0$'s, beginning and ending with $1$?
-/
theorem putnam_1989_a1
(pdigalt : List β β Prop)
(hpdigalt : β l, pdigalt l β Odd l.length β§ (β i, l.get i = if Even (i : β) then 1 else 0)) :
{p : β | p.Prime β§ pdigalt (Nat.digits 10 p)}.encard = putnam_1989_a1_solution := by
| import Mathlib
abbrev putnam_1989_a1_solution : ββ := sorry
-- 1
/--
How many primes among the positive integers, written as usual in base $10$, are alternating $1$'s and $0$'s, beginning and ending with $1$?
-/
theorem putnam_1989_a1
(pdigalt : List β β Prop)
(hpdigalt : β l, pdigalt l β Odd l.length β§ (β i, l.get i = if Even (i : β) then 1 else 0)) :
{p : β | p.Prime β§ pdigalt (Nat.digits 10 p)}.encard = putnam_1989_a1_solution :=
sorry
| How many primes among the positive integers, written as usual in base $10$, are alternating $1$'s and $0$'s, beginning and ending with $1$? | Show that there is only one such prime. | [
"algebra",
"number_theory"
] | null | null |
|
putnam_2012_a1 | aeda67c9-ad8e-58da-9b42-db9f4aa0b907 | train | theorem putnam_2012_a1
(d : Fin 12 β β)
(hd : β i, d i β Set.Ioo 1 12)
: β i j k, i β j β§ i β k β§ j β k β§ d k β₯ d i β§ d k β₯ d j β§ (d i) ^ 2 + (d j) ^ 2 > (d k) ^ 2 :=
sorry | import Mathlib
open Matrix
-- Note: This proof statement strays a bit from the problem statement since it manually checks a condition for acuteness based on side lengths.
/--
Let $d_1,d_2,\dots,d_{12}$ be real numbers in the open interval $(1,12)$. Show that there exist distinct indices $i,j,k$ such that $d_i,d_j,d_k$ are the side lengths of an acute triangle.
-/
theorem putnam_2012_a1
(d : Fin 12 β β)
(hd : β i, d i β Set.Ioo 1 12)
: β i j k, i β j β§ i β k β§ j β k β§ d k β₯ d i β§ d k β₯ d j β§ (d i) ^ 2 + (d j) ^ 2 > (d k) ^ 2 := by
| import Mathlib
open Matrix
-- Note: This proof statement strays a bit from the problem statement since it manually checks a condition for acuteness based on side lengths.
/--
Let $d_1,d_2,\dots,d_{12}$ be real numbers in the open interval $(1,12)$. Show that there exist distinct indices $i,j,k$ such that $d_i,d_j,d_k$ are the side lengths of an acute triangle.
-/
theorem putnam_2012_a1
(d : Fin 12 β β)
(hd : β i, d i β Set.Ioo 1 12)
: β i j k, i β j β§ i β k β§ j β k β§ d k β₯ d i β§ d k β₯ d j β§ (d i) ^ 2 + (d j) ^ 2 > (d k) ^ 2 :=
sorry
| Let $d_1,d_2,\dots,d_{12}$ be real numbers in the open interval $(1,12)$. Show that there exist distinct indices $i,j,k$ such that $d_i,d_j,d_k$ are the side lengths of an acute triangle. | null | [
"geometry",
"algebra"
] | null | null |
|
putnam_1976_a2 | b587a7a8-d190-5bf0-a590-6f7221d4e32f | train | theorem putnam_1976_a2
(P Q : MvPolynomial (Fin 2) β€)
(hP : P = X 0 ^ 2 * X 1 + X 0 * X 1 ^ 2)
(hQ : Q = X 0 ^ 2 + X 0 * X 1 + X 1 ^ 2)
(F G : β β MvPolynomial (Fin 2) β€)
(hF : β n, F n = (X 0 + X 1) ^ n - X 0 ^ n - X 1 ^ n)
(hG : β n, G n = (X 0 + X 1) ^ n + X 0 ^ n + X 1 ^ n)
(n : β)
(hn : 0 < n) :
β A : MvPolynomial (Fin 2) β€,
F n = aeval ![P, Q] A β¨
G n = aeval ![P, Q] A :=
sorry | import Mathlib
open MvPolynomial
/--
Let $P(x, y) = x^2y + xy^2$, $Q(x, y) = x^2 + xy + y^2$, $F_n(x, y) = (x + y)^n - x^n - y^n$, and $G_n(x, y) = (x + y)^n + x^n + y^n$. Prove that for all positive integers $n$, either $F_n$ or $G_n$ can be represented as a polynomial in $P$ and $Q$ with integer coefficients.
-/
theorem putnam_1976_a2
(P Q : MvPolynomial (Fin 2) β€)
(hP : P = X 0 ^ 2 * X 1 + X 0 * X 1 ^ 2)
(hQ : Q = X 0 ^ 2 + X 0 * X 1 + X 1 ^ 2)
(F G : β β MvPolynomial (Fin 2) β€)
(hF : β n, F n = (X 0 + X 1) ^ n - X 0 ^ n - X 1 ^ n)
(hG : β n, G n = (X 0 + X 1) ^ n + X 0 ^ n + X 1 ^ n)
(n : β)
(hn : 0 < n) :
β A : MvPolynomial (Fin 2) β€,
F n = aeval ![P, Q] A β¨
G n = aeval ![P, Q] A := by
| import Mathlib
open MvPolynomial
/--
Let $P(x, y) = x^2y + xy^2$, $Q(x, y) = x^2 + xy + y^2$, $F_n(x, y) = (x + y)^n - x^n - y^n$, and $G_n(x, y) = (x + y)^n + x^n + y^n$. Prove that for all positive integers $n$, either $F_n$ or $G_n$ can be represented as a polynomial in $P$ and $Q$ with integer coefficients.
-/
theorem putnam_1976_a2
(P Q : MvPolynomial (Fin 2) β€)
(hP : P = X 0 ^ 2 * X 1 + X 0 * X 1 ^ 2)
(hQ : Q = X 0 ^ 2 + X 0 * X 1 + X 1 ^ 2)
(F G : β β MvPolynomial (Fin 2) β€)
(hF : β n, F n = (X 0 + X 1) ^ n - X 0 ^ n - X 1 ^ n)
(hG : β n, G n = (X 0 + X 1) ^ n + X 0 ^ n + X 1 ^ n)
(n : β)
(hn : 0 < n) :
β A : MvPolynomial (Fin 2) β€,
F n = aeval ![P, Q] A β¨
G n = aeval ![P, Q] A :=
sorry
| Let $P(x, y) = x^2y + xy^2$, $Q(x, y) = x^2 + xy + y^2$, $F_n(x, y) = (x + y)^n - x^n - y^n$, and $G_n(x, y) = (x + y)^n + x^n + y^n$. Prove that for all positive integers $n$, either $F_n$ or $G_n$ can be represented as a polynomial in $P$ and $Q$ with integer coefficients. | null | [
"algebra"
] | null | null |
|
putnam_2006_a3 | d93eee95-de7e-56fc-be15-c32fb434756e | train | theorem putnam_2006_a3
(x : β€ β β€)
(hxlo : β k : β€, 0 β€ k β§ k β€ 2006 β x k = k)
(hxhi : β k : β€, k β₯ 2006 β x (k + 1) = x k + x (k - 2005))
: (β i : β, i > 0 β§ β j : Finset.range 2005, 2006 β£ x (i + j)) :=
sorry | import Mathlib
/--
Let $1, 2, 3, \dots, 2005, 2006, 2007, 2009, 2012, 2016, \dots$ be a sequence defined by $x_k = k$ for $k=1, 2, \dots, 2006$ and $x_{k+1} = x_k + x_{k-2005}$ for $k \geq 2006$. Show that the sequence has $2005$ consecutive terms each divisible by $2006$.
-/
theorem putnam_2006_a3
(x : β€ β β€)
(hxlo : β k : β€, 0 β€ k β§ k β€ 2006 β x k = k)
(hxhi : β k : β€, k β₯ 2006 β x (k + 1) = x k + x (k - 2005))
: (β i : β, i > 0 β§ β j : Finset.range 2005, 2006 β£ x (i + j)) := by
| import Mathlib
/--
Let $1, 2, 3, \dots, 2005, 2006, 2007, 2009, 2012, 2016, \dots$ be a sequence defined by $x_k = k$ for $k=1, 2, \dots, 2006$ and $x_{k+1} = x_k + x_{k-2005}$ for $k \geq 2006$. Show that the sequence has $2005$ consecutive terms each divisible by $2006$.
-/
theorem putnam_2006_a3
(x : β€ β β€)
(hxlo : β k : β€, 0 β€ k β§ k β€ 2006 β x k = k)
(hxhi : β k : β€, k β₯ 2006 β x (k + 1) = x k + x (k - 2005))
: (β i : β, i > 0 β§ β j : Finset.range 2005, 2006 β£ x (i + j)) :=
sorry
| Let $1, 2, 3, \dots, 2005, 2006, 2007, 2009, 2012, 2016, \dots$ be a sequence defined by $x_k = k$ for $k=1, 2, \dots, 2006$ and $x_{k+1} = x_k + x_{k-2005}$ for $k \geq 2006$. Show that the sequence has $2005$ consecutive terms each divisible by $2006$. | null | [
"algebra"
] | null | null |
|
putnam_2016_b4 | 420f9c74-a107-5cb8-b6f0-cc88482e6b59 | train | abbrev putnam_2016_b4_solution : β β β := sorry
-- (fun n : β => (2 * n)! / (4 ^ n * (n)!))
/--
Let $A$ be a $2n \times 2n$ matrix, with entries chosen independently at random. Every entry is chosen to be $0$ or $1$, each with probability $1/2$. Find the expected value of $\det(A-A^t)$ (as a function of $n$), where $A^t$ is the transpose of $A$.
-/
theorem putnam_2016_b4
(n : β) (npos : n β₯ 1)
(mats01 : Set (Matrix (Fin (2 * n)) (Fin (2 * n)) β))
(hmats01 : mats01 = {A | β i j : Fin (2 * n), A i j = 0 β¨ A i j = 1}) :
(β' A : mats01, (A.1 - (Matrix.transpose A)).det) / mats01.ncard = putnam_2016_b4_solution n :=
sorry | import Mathlib
open Real Set Nat
-- (fun n : β => (2 * n)! / (4 ^ n * (n)!))
/--
Let $A$ be a $2n \times 2n$ matrix, with entries chosen independently at random. Every entry is chosen to be $0$ or $1$, each with probability $1/2$. Find the expected value of $\det(A-A^t)$ (as a function of $n$), where $A^t$ is the transpose of $A$.
-/
theorem putnam_2016_b4
(n : β) (npos : n β₯ 1)
(mats01 : Set (Matrix (Fin (2 * n)) (Fin (2 * n)) β))
(hmats01 : mats01 = {A | β i j : Fin (2 * n), A i j = 0 β¨ A i j = 1}) :
(β' A : mats01, (A.1 - (Matrix.transpose A)).det) / mats01.ncard = putnam_2016_b4_solution n := by
| import Mathlib
open Real Set Nat
noncomputable abbrev putnam_2016_b4_solution : β β β := sorry
-- (fun n : β => (2 * n)! / (4 ^ n * (n)!))
/--
Let $A$ be a $2n \times 2n$ matrix, with entries chosen independently at random. Every entry is chosen to be $0$ or $1$, each with probability $1/2$. Find the expected value of $\det(A-A^t)$ (as a function of $n$), where $A^t$ is the transpose of $A$.
-/
theorem putnam_2016_b4
(n : β) (npos : n β₯ 1)
(mats01 : Set (Matrix (Fin (2 * n)) (Fin (2 * n)) β))
(hmats01 : mats01 = {A | β i j : Fin (2 * n), A i j = 0 β¨ A i j = 1}) :
(β' A : mats01, (A.1 - (Matrix.transpose A)).det) / mats01.ncard = putnam_2016_b4_solution n :=
sorry
| Let $A$ be a $2n \times 2n$ matrix, with entries chosen independently at random. Every entry is chosen to be $0$ or $1$, each with probability $1/2$. Find the expected value of $\det(A-A^t)$ (as a function of $n$), where $A^t$ is the transpose of $A$. | Show that the expected value equals $\frac{(2n)!}{4^nn!}$. | [
"linear_algebra",
"probability"
] | null | null |
|
putnam_2001_b3 | 453dc01a-6bff-5349-9567-39f62810f3a7 | train | 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 | import Mathlib
open Topology Filter Polynomial Set
-- 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 := by
| import Mathlib
open Topology Filter Polynomial Set
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
| For any positive integer $n$, let $\langle n \rangle$ denote the closest integer to $\sqrt{n}$. Evaluate $\sum_{n=1}^\infty \frac{2^{\langle n \rangle}+2^{-\langle n \rangle}}{2^n}$. | Show that the sum is $3$. | [
"analysis"
] | null | null |
|
putnam_1968_a5 | 8c711c8b-97c0-5c73-90a9-95add2098e0a | train | abbrev putnam_1968_a5_solution : β := sorry
-- 8
/--
Let $V$ be the set of all quadratic polynomials with real coefficients such that $|P(x)| \le 1$ for all $x \in [0, 1]$. Find the supremum of $|P'(0)|$ across all $P \in V$.
-/
theorem putnam_1968_a5
(V : Set β[X])
(V_def : V = {P : β[X] | P.degree = 2 β§ β x β Set.Icc 0 1, |P.eval x| β€ 1})
: sSup {|(derivative P).eval 0| | P β V} = putnam_1968_a5_solution :=
sorry | import Mathlib
open Finset Polynomial
-- 8
/--
Let $V$ be the set of all quadratic polynomials with real coefficients such that $|P(x)| \le 1$ for all $x \in [0, 1]$. Find the supremum of $|P'(0)|$ across all $P \in V$.
-/
theorem putnam_1968_a5
(V : Set β[X])
(V_def : V = {P : β[X] | P.degree = 2 β§ β x β Set.Icc 0 1, |P.eval x| β€ 1})
: sSup {|(derivative P).eval 0| | P β V} = putnam_1968_a5_solution := by
| import Mathlib
open Finset Polynomial
abbrev putnam_1968_a5_solution : β := sorry
-- 8
/--
Let $V$ be the set of all quadratic polynomials with real coefficients such that $|P(x)| \le 1$ for all $x \in [0, 1]$. Find the supremum of $|P'(0)|$ across all $P \in V$.
-/
theorem putnam_1968_a5
(V : Set β[X])
(V_def : V = {P : β[X] | P.degree = 2 β§ β x β Set.Icc 0 1, |P.eval x| β€ 1})
: sSup {|(derivative P).eval 0| | P β V} = putnam_1968_a5_solution :=
sorry
| Let $V$ be the set of all quadratic polynomials with real coefficients such that $|P(x)| \le 1$ for all $x \in [0, 1]$. Find the supremum of $|P'(0)|$ across all $P \in V$. | The supremum is $8$. | [
"algebra"
] | null | null |
|
putnam_1966_a6 | 8b4943c3-e687-5849-9ab8-03a22dcf3a08 | train | theorem putnam_1966_a6
(a : β β (β β β))
(ha : β n β₯ 1, a n n = n β§ β m β₯ 1, m < n β a n m = m * Real.sqrt (1 + a n (m + 1)))
: Tendsto (fun n => a n 1) atTop (π 3) :=
sorry | import Mathlib
open Topology Filter
/--
Prove that $$\sqrt {1 + 2 \sqrt {1 + 3 \sqrt {1 + 4 \sqrt {1 + 5 \sqrt {\dots}}}}} = 3.$$
-/
theorem putnam_1966_a6
(a : β β (β β β))
(ha : β n β₯ 1, a n n = n β§ β m β₯ 1, m < n β a n m = m * Real.sqrt (1 + a n (m + 1)))
: Tendsto (fun n => a n 1) atTop (π 3) := by
| import Mathlib
open Topology Filter
/--
Prove that $$\sqrt {1 + 2 \sqrt {1 + 3 \sqrt {1 + 4 \sqrt {1 + 5 \sqrt {\dots}}}}} = 3.$$
-/
theorem putnam_1966_a6
(a : β β (β β β))
(ha : β n β₯ 1, a n n = n β§ β m β₯ 1, m < n β a n m = m * Real.sqrt (1 + a n (m + 1)))
: Tendsto (fun n => a n 1) atTop (π 3) :=
sorry
| Prove that $$\sqrt {1 + 2 \sqrt {1 + 3 \sqrt {1 + 4 \sqrt {1 + 5 \sqrt {\dots}}}}} = 3.$$ | null | [
"analysis"
] | null | null |
|
putnam_2007_b6 | 0aca389b-b967-58e6-92e5-f0b074583fac | train | theorem putnam_2007_b6
(f : β β β)
(hf : f = fun n β¦ {M : Multiset β | M.sum = (n)! β§ β m β M, β k β Icc 1 n, m = (k)!}.ncard)
: (β C : β, β n : β, n β₯ 2 β n ^ (n ^ 2 / 2 - C * n) * Real.exp (-(n ^ 2) / 4) β€ f n β§ f n β€ n ^ (n ^ 2 / 2 + C * n) * Real.exp (-(n ^ 2) / 4)) :=
sorry | import Mathlib
open Set Nat Function
/--
For each positive integer $n$, let $f(n)$ be the number of ways to make $n!$ cents using an unordered collection of coins, each worth $k!$ cents for some $k$, $1 \leq k \leq n$. Prove that for some constant $C$, independent of $n$,
\[
n^{n^2/2 - Cn} e^{-n^2/4} \leq f(n) \leq n^{n^2/2 + Cn}e^{-n^2/4}.
\]
-/
theorem putnam_2007_b6
(f : β β β)
(hf : f = fun n β¦ {M : Multiset β | M.sum = (n)! β§ β m β M, β k β Icc 1 n, m = (k)!}.ncard)
: (β C : β, β n : β, n β₯ 2 β n ^ (n ^ 2 / 2 - C * n) * Real.exp (-(n ^ 2) / 4) β€ f n β§ f n β€ n ^ (n ^ 2 / 2 + C * n) * Real.exp (-(n ^ 2) / 4)) := by
| import Mathlib
open Set Nat Function
/--
For each positive integer $n$, let $f(n)$ be the number of ways to make $n!$ cents using an unordered collection of coins, each worth $k!$ cents for some $k$, $1 \leq k \leq n$. Prove that for some constant $C$, independent of $n$,
\[
n^{n^2/2 - Cn} e^{-n^2/4} \leq f(n) \leq n^{n^2/2 + Cn}e^{-n^2/4}.
\]
-/
theorem putnam_2007_b6
(f : β β β)
(hf : f = fun n β¦ {M : Multiset β | M.sum = (n)! β§ β m β M, β k β Icc 1 n, m = (k)!}.ncard)
: (β C : β, β n : β, n β₯ 2 β n ^ (n ^ 2 / 2 - C * n) * Real.exp (-(n ^ 2) / 4) β€ f n β§ f n β€ n ^ (n ^ 2 / 2 + C * n) * Real.exp (-(n ^ 2) / 4)) :=
sorry
| For each positive integer $n$, let $f(n)$ be the number of ways to make $n!$ cents using an unordered collection of coins, each worth $k!$ cents for some $k$, $1 \leq k \leq n$. Prove that for some constant $C$, independent of $n$,
\[
n^{n^2/2 - Cn} e^{-n^2/4} \leq f(n) \leq n^{n^2/2 + Cn}e^{-n^2/4}.
\] | null | [
"combinatorics",
"analysis"
] | null | null |
|
putnam_1973_b3 | 7656da08-00ea-57cf-916a-5c7cf2a5e77b | train | theorem putnam_1973_b3
(p : β)
(pgt1 : p > 1)
(hprime : β x β Set.Ico 0 p, Nat.Prime (x^2 - x + p))
: β! triple : β€ Γ β€ Γ β€, let (a,b,c) := triple; b^2 - 4*a*c = 1 - 4*p β§ 0 < a β§ a β€ c β§ -a β€ b β§ b < a :=
sorry | import Mathlib
open Nat Set MeasureTheory Topology Filter
/--
Let $p > 1$ be an integer with the property that $x^2 - x + p$ is prime for all $x$ in the range $0 < x < p$. Show there exists exactly one triple of integers $a,b,c$ satisfying $b^2 - 4ac = 1 - 4p$, $0 < a \leq c$, and $-a \leq b < a$.
-/
theorem putnam_1973_b3
(p : β)
(pgt1 : p > 1)
(hprime : β x β Set.Ico 0 p, Nat.Prime (x^2 - x + p))
: β! triple : β€ Γ β€ Γ β€, let (a,b,c) := triple; b^2 - 4*a*c = 1 - 4*p β§ 0 < a β§ a β€ c β§ -a β€ b β§ b < a := by
| import Mathlib
open Nat Set MeasureTheory Topology Filter
/--
Let $p > 1$ be an integer with the property that $x^2 - x + p$ is prime for all $x$ in the range $0 < x < p$. Show there exists exactly one triple of integers $a,b,c$ satisfying $b^2 - 4ac = 1 - 4p$, $0 < a \leq c$, and $-a \leq b < a$.
-/
theorem putnam_1973_b3
(p : β)
(pgt1 : p > 1)
(hprime : β x β Set.Ico 0 p, Nat.Prime (x^2 - x + p))
: β! triple : β€ Γ β€ Γ β€, let (a,b,c) := triple; b^2 - 4*a*c = 1 - 4*p β§ 0 < a β§ a β€ c β§ -a β€ b β§ b < a :=
sorry
| Let $p > 1$ be an integer with the property that $x^2 - x + p$ is prime for all $x$ in the range $0 < x < p$. Show there exists exactly one triple of integers $a,b,c$ satisfying $b^2 - 4ac = 1 - 4p$, $0 < a \leq c$, and $-a \leq b < a$. | null | [
"number_theory",
"algebra"
] | null | null |
|
putnam_1970_a3 | 38d7ec32-1545-578e-8b6e-f79a236c379e | train | abbrev putnam_1970_a3_solution : β Γ β := sorry
-- (3, 1444)
/--
Find the length of the longest possible sequence of equal nonzero digits (in base 10) in which a perfect square can terminate. Also, find the smallest square that attains this length.
-/
theorem putnam_1970_a3
(L : β β β)
(hL : β n : β, L n β€ (Nat.digits 10 n).length β§
(β k : β, k < L n β (Nat.digits 10 n)[k]! = (Nat.digits 10 n)[0]!) β§
(L n β (Nat.digits 10 n).length β (Nat.digits 10 n)[L n]! β (Nat.digits 10 n)[0]!))
: (β n : β, (Nat.digits 10 (n^2))[0]! β 0 β§ L (n^2) = putnam_1970_a3_solution.1) β§
(β n : β, (Nat.digits 10 (n^2))[0]! β 0 β L (n^2) β€ putnam_1970_a3_solution.1) β§
(β m : β, m^2 = putnam_1970_a3_solution.2) β§
L (putnam_1970_a3_solution.2) = putnam_1970_a3_solution.1 β§
(Nat.digits 10 putnam_1970_a3_solution.2)[0]! β 0 β§
β n : β, (Nat.digits 10 (n^2))[0]! β 0 β§ L (n^2) = putnam_1970_a3_solution.1 β n^2 β₯ putnam_1970_a3_solution.2 :=
sorry | import Mathlib
open Metric Set EuclideanGeometry
-- (3, 1444)
/--
Find the length of the longest possible sequence of equal nonzero digits (in base 10) in which a perfect square can terminate. Also, find the smallest square that attains this length.
-/
theorem putnam_1970_a3
(L : β β β)
(hL : β n : β, L n β€ (Nat.digits 10 n).length β§
(β k : β, k < L n β (Nat.digits 10 n)[k]! = (Nat.digits 10 n)[0]!) β§
(L n β (Nat.digits 10 n).length β (Nat.digits 10 n)[L n]! β (Nat.digits 10 n)[0]!))
: (β n : β, (Nat.digits 10 (n^2))[0]! β 0 β§ L (n^2) = putnam_1970_a3_solution.1) β§
(β n : β, (Nat.digits 10 (n^2))[0]! β 0 β L (n^2) β€ putnam_1970_a3_solution.1) β§
(β m : β, m^2 = putnam_1970_a3_solution.2) β§
L (putnam_1970_a3_solution.2) = putnam_1970_a3_solution.1 β§
(Nat.digits 10 putnam_1970_a3_solution.2)[0]! β 0 β§
β n : β, (Nat.digits 10 (n^2))[0]! β 0 β§ L (n^2) = putnam_1970_a3_solution.1 β n^2 β₯ putnam_1970_a3_solution.2 := by
| import Mathlib
open Metric Set EuclideanGeometry
abbrev putnam_1970_a3_solution : β Γ β := sorry
-- (3, 1444)
/--
Find the length of the longest possible sequence of equal nonzero digits (in base 10) in which a perfect square can terminate. Also, find the smallest square that attains this length.
-/
theorem putnam_1970_a3
(L : β β β)
(hL : β n : β, L n β€ (Nat.digits 10 n).length β§
(β k : β, k < L n β (Nat.digits 10 n)[k]! = (Nat.digits 10 n)[0]!) β§
(L n β (Nat.digits 10 n).length β (Nat.digits 10 n)[L n]! β (Nat.digits 10 n)[0]!))
: (β n : β, (Nat.digits 10 (n^2))[0]! β 0 β§ L (n^2) = putnam_1970_a3_solution.1) β§
(β n : β, (Nat.digits 10 (n^2))[0]! β 0 β L (n^2) β€ putnam_1970_a3_solution.1) β§
(β m : β, m^2 = putnam_1970_a3_solution.2) β§
L (putnam_1970_a3_solution.2) = putnam_1970_a3_solution.1 β§
(Nat.digits 10 putnam_1970_a3_solution.2)[0]! β 0 β§
β n : β, (Nat.digits 10 (n^2))[0]! β 0 β§ L (n^2) = putnam_1970_a3_solution.1 β n^2 β₯ putnam_1970_a3_solution.2 :=
sorry
| Find the length of the longest possible sequence of equal nonzero digits (in base 10) in which a perfect square can terminate. Also, find the smallest square that attains this length. | The maximum attainable length is $3$; the smallest such square is $38^2 = 1444$. | [
"number_theory"
] | null | null |
|
putnam_2003_b6 | 6d4fcfe3-4a9d-5b0e-b4d4-2ff7eb5efa6f | train | theorem putnam_2003_b6
(f : β β β)
(hf : Continuous f)
: (β« x in (0 : β)..1, (β« y in (0 : β)..1, |f x + f y|)) β₯ (β« x in (0 : β)..1, |f x|) :=
sorry | import Mathlib
open MvPolynomial Set Nat
/--
Let $f(x)$ be a continuous real-valued function defined on the interval $[0,1]$. Show that \[ \int_0^1 \int_0^1 | f(x) + f(y) |\,dx\,dy \geq \int_0^1 |f(x)|\,dx. \]
-/
theorem putnam_2003_b6
(f : β β β)
(hf : Continuous f)
: (β« x in (0 : β)..1, (β« y in (0 : β)..1, |f x + f y|)) β₯ (β« x in (0 : β)..1, |f x|) := by
| import Mathlib
open MvPolynomial Set Nat
/--
Let $f(x)$ be a continuous real-valued function defined on the interval $[0,1]$. Show that \[ \int_0^1 \int_0^1 | f(x) + f(y) |\,dx\,dy \geq \int_0^1 |f(x)|\,dx. \]
-/
theorem putnam_2003_b6
(f : β β β)
(hf : Continuous f)
: (β« x in (0 : β)..1, (β« y in (0 : β)..1, |f x + f y|)) β₯ (β« x in (0 : β)..1, |f x|) :=
sorry
| Let $f(x)$ be a continuous real-valued function defined on the interval $[0,1]$. Show that \[ \int_0^1 \int_0^1 | f(x) + f(y) |\,dx\,dy \geq \int_0^1 |f(x)|\,dx. \] | null | [
"analysis"
] | null | null |
|
putnam_2010_a6 | 352ed91e-4818-59f5-b6b4-e2b5b8931b37 | train | theorem putnam_2010_a6
(f : β β β)
(hf : (StrictAntiOn f (Ici 0) β§ ContinuousOn f (Ici 0) β§ Tendsto f atTop (π 0)))
: Β¬β y : β, Tendsto (fun z : β => β« x in Ico 0 z, ((f x) - f (x + 1))/(f x)) atTop (π y) :=
sorry | import Mathlib
open Filter Topology Set
-- Note: uses (β β β) instead of (Ici 0 β β)
/--
Let $f:[0,\infty)\to \mathbb{R}$ be a strictly decreasing continuous function
such that $\lim_{x\to\infty} f(x) = 0$. Prove that
$\int_0^\infty \frac{f(x)-f(x+1)}{f(x)}\,dx$ diverges.
-/
theorem putnam_2010_a6
(f : β β β)
(hf : (StrictAntiOn f (Ici 0) β§ ContinuousOn f (Ici 0) β§ Tendsto f atTop (π 0)))
: Β¬β y : β, Tendsto (fun z : β => β« x in Ico 0 z, ((f x) - f (x + 1))/(f x)) atTop (π y) := by
| import Mathlib
open Filter Topology Set
-- Note: uses (β β β) instead of (Ici 0 β β)
/--
Let $f:[0,\infty)\to \mathbb{R}$ be a strictly decreasing continuous function
such that $\lim_{x\to\infty} f(x) = 0$. Prove that
$\int_0^\infty \frac{f(x)-f(x+1)}{f(x)}\,dx$ diverges.
-/
theorem putnam_2010_a6
(f : β β β)
(hf : (StrictAntiOn f (Ici 0) β§ ContinuousOn f (Ici 0) β§ Tendsto f atTop (π 0)))
: Β¬β y : β, Tendsto (fun z : β => β« x in Ico 0 z, ((f x) - f (x + 1))/(f x)) atTop (π y) :=
sorry
| Let $f:[0,\infty)\to \mathbb{R}$ be a strictly decreasing continuous function
such that $\lim_{x\to\infty} f(x) = 0$. Prove that
$\int_0^\infty \frac{f(x)-f(x+1)}{f(x)}\,dx$ diverges. | null | [
"analysis"
] | null | null |
|
putnam_1986_a1 | 1aed03bd-28ff-5e12-8f56-272a95070bd8 | train | abbrev putnam_1986_a1_solution : β := sorry
-- 18
/--
Find, with explanation, the maximum value of $f(x)=x^3-3x$ on the set of all real numbers $x$ satisfying $x^4+36 \leq 13x^2$.
-/
theorem putnam_1986_a1
(S : Set β) (f : β β β)
(hS : S = {x : β | x ^ 4 + 36 β€ 13 * x ^ 2})
(hf : f = fun x β¦ x ^ 3 - 3 * x) :
IsGreatest
{f x | x β S}
putnam_1986_a1_solution :=
sorry | import Mathlib
-- 18
/--
Find, with explanation, the maximum value of $f(x)=x^3-3x$ on the set of all real numbers $x$ satisfying $x^4+36 \leq 13x^2$.
-/
theorem putnam_1986_a1
(S : Set β) (f : β β β)
(hS : S = {x : β | x ^ 4 + 36 β€ 13 * x ^ 2})
(hf : f = fun x β¦ x ^ 3 - 3 * x) :
IsGreatest
{f x | x β S}
putnam_1986_a1_solution := by
| import Mathlib
abbrev putnam_1986_a1_solution : β := sorry
-- 18
/--
Find, with explanation, the maximum value of $f(x)=x^3-3x$ on the set of all real numbers $x$ satisfying $x^4+36 \leq 13x^2$.
-/
theorem putnam_1986_a1
(S : Set β) (f : β β β)
(hS : S = {x : β | x ^ 4 + 36 β€ 13 * x ^ 2})
(hf : f = fun x β¦ x ^ 3 - 3 * x) :
IsGreatest
{f x | x β S}
putnam_1986_a1_solution :=
sorry
| Find, with explanation, the maximum value of $f(x)=x^3-3x$ on the set of all real numbers $x$ satisfying $x^4+36 \leq 13x^2$. | Show that the maximum value is $18$. | [
"algebra",
"analysis"
] | null | null |
|
putnam_2013_a6 | f5da2363-6022-5ce5-b611-042c210263d1 | train | 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 | import Mathlib
open Function Set
/--
Define a function $w:\mathbb{Z} \times \mathbb{Z} \to \mathbb{Z}$ as follows. For $|a|,|b| \leq 2$, let $w(a,b)$ be as in the table shown; otherwise, let $w(a,b)=0$.
\begin{center}
\begin{tabular}{|cc|r|r|r|r|r|}
\hline
\multicolumn{2}{|c|}{\multirow{2}{*}{$w(a,b)$}} & \multicolumn{5}{|c|}{$b$} \\
& & -2 & -1 & 0 & 1 & 2 \\
\hline
& -2 & -1 & -2 & 2 & -2 & -1 \\
& -1 & -2 & 4 & -4 & 4 & -2 \\
$a$ & 0 & 2 & -4 & 12 & -4 & 2 \\
& 1 & -2 & 4 & -4 & 4 & -2 \\
& 2 & -1 & -2 & 2 & -2 & -1 \\
\hline
\end{tabular}
\end{center}
For every finite subset $S$ of $\mathbb{Z} \times \mathbb{Z}$, define $A(S)=\sum_{(\mathbf{s},\mathbf{s}') \in S \times S} w(\mathbf{s}-\mathbf{s}')$. Prove that if $S$ is any finite nonempty subset of $\mathbb{Z} \times \mathbb{Z}$, then $A(S)>0$. (For example, if $S=\{(0,1),(0,2),(2,0),(3,1)\}$, then the terms in $A(S)$ are $12,12,12,12,4,4,0,0,0,0,-1,-1,-2,-2,-4,-4$.)
-/
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 := by
| import Mathlib
open Function Set
/--
Define a function $w:\mathbb{Z} \times \mathbb{Z} \to \mathbb{Z}$ as follows. For $|a|,|b| \leq 2$, let $w(a,b)$ be as in the table shown; otherwise, let $w(a,b)=0$.
\begin{center}
\begin{tabular}{|cc|r|r|r|r|r|}
\hline
\multicolumn{2}{|c|}{\multirow{2}{*}{$w(a,b)$}} & \multicolumn{5}{|c|}{$b$} \\
& & -2 & -1 & 0 & 1 & 2 \\
\hline
& -2 & -1 & -2 & 2 & -2 & -1 \\
& -1 & -2 & 4 & -4 & 4 & -2 \\
$a$ & 0 & 2 & -4 & 12 & -4 & 2 \\
& 1 & -2 & 4 & -4 & 4 & -2 \\
& 2 & -1 & -2 & 2 & -2 & -1 \\
\hline
\end{tabular}
\end{center}
For every finite subset $S$ of $\mathbb{Z} \times \mathbb{Z}$, define $A(S)=\sum_{(\mathbf{s},\mathbf{s}') \in S \times S} w(\mathbf{s}-\mathbf{s}')$. Prove that if $S$ is any finite nonempty subset of $\mathbb{Z} \times \mathbb{Z}$, then $A(S)>0$. (For example, if $S=\{(0,1),(0,2),(2,0),(3,1)\}$, then the terms in $A(S)$ are $12,12,12,12,4,4,0,0,0,0,-1,-1,-2,-2,-4,-4$.)
-/
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
| Define a function $w:\mathbb{Z} \times \mathbb{Z} \to \mathbb{Z}$ as follows. For $|a|,|b| \leq 2$, let $w(a,b)$ be as in the table shown; otherwise, let $w(a,b)=0$.
\begin{center}
\begin{tabular}{|cc|r|r|r|r|r|}
\hline
\multicolumn{2}{|c|}{\multirow{2}{*}{$w(a,b)$}} & \multicolumn{5}{|c|}{$b$} \\
& & -2 & -1 & 0 & 1 & 2 \\
\hline
& -2 & -1 & -2 & 2 & -2 & -1 \\
& -1 & -2 & 4 & -4 & 4 & -2 \\
$a$ & 0 & 2 & -4 & 12 & -4 & 2 \\
& 1 & -2 & 4 & -4 & 4 & -2 \\
& 2 & -1 & -2 & 2 & -2 & -1 \\
\hline
\end{tabular}
\end{center}
For every finite subset $S$ of $\mathbb{Z} \times \mathbb{Z}$, define $A(S)=\sum_{(\mathbf{s},\mathbf{s}') \in S \times S} w(\mathbf{s}-\mathbf{s}')$. Prove that if $S$ is any finite nonempty subset of $\mathbb{Z} \times \mathbb{Z}$, then $A(S)>0$. (For example, if $S=\{(0,1),(0,2),(2,0),(3,1)\}$, then the terms in $A(S)$ are $12,12,12,12,4,4,0,0,0,0,-1,-1,-2,-2,-4,-4$.) | null | [
"algebra"
] | null | null |
|
putnam_2011_b1 | 9061b5ef-cd70-5bb6-b47a-f81cb8a30901 | train | theorem putnam_2011_b1
(h k : β€)
(hkpos : h > 0 β§ k > 0)
: β Ξ΅ > 0, β m n : β€, m > 0 β§ n > 0 β§ Ξ΅ < |h * Real.sqrt m - k * Real.sqrt n| β§ |h * Real.sqrt m - k * Real.sqrt n| < 2 * Ξ΅ :=
sorry | import Mathlib
open Topology Filter Matrix
/--
Let $h$ and $k$ be positive integers. Prove that for every $\epsilon>0$, there are positive integers $m$ and $n$ such that $\epsilon<|h\sqrt{m}-k\sqrt{n}|<2\epsilon$.
-/
theorem putnam_2011_b1
(h k : β€)
(hkpos : h > 0 β§ k > 0)
: β Ξ΅ > 0, β m n : β€, m > 0 β§ n > 0 β§ Ξ΅ < |h * Real.sqrt m - k * Real.sqrt n| β§ |h * Real.sqrt m - k * Real.sqrt n| < 2 * Ξ΅ := by
| import Mathlib
open Topology Filter Matrix
/--
Let $h$ and $k$ be positive integers. Prove that for every $\epsilon>0$, there are positive integers $m$ and $n$ such that $\epsilon<|h\sqrt{m}-k\sqrt{n}|<2\epsilon$.
-/
theorem putnam_2011_b1
(h k : β€)
(hkpos : h > 0 β§ k > 0)
: β Ξ΅ > 0, β m n : β€, m > 0 β§ n > 0 β§ Ξ΅ < |h * Real.sqrt m - k * Real.sqrt n| β§ |h * Real.sqrt m - k * Real.sqrt n| < 2 * Ξ΅ :=
sorry
| Let $h$ and $k$ be positive integers. Prove that for every $\epsilon>0$, there are positive integers $m$ and $n$ such that $\epsilon<|h\sqrt{m}-k\sqrt{n}|<2\epsilon$. | null | [
"algebra"
] | null | null |
|
putnam_2020_a6 | 6de361fe-8e94-502a-8e10-f9a87200f9e9 | train | 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 | import Mathlib
open Filter Topology Set
-- 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} := by
| import Mathlib
open Filter Topology Set
noncomputable 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
| 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 $x$. | The smallest constant $M$ is $\pi/4$. | [
"algebra"
] | null | null |
|
putnam_1999_a1 | a46cf9d2-0644-5937-8568-ece643ac8375 | train | abbrev putnam_1999_a1_solution : Prop := sorry
-- True
/--
Find polynomials $f(x)$,$g(x)$, and $h(x)$, if they exist, such that for all $x$, \[|f(x)|-|g(x)|+h(x) = \begin{cases} -1 & \mbox{if $x<-1$} \\3x+2 & \mbox{if $-1 \leq x \leq 0$} \\-2x+2 & \mbox{if $x>0$.}\end{cases}\]?
-/
theorem putnam_1999_a1
: putnam_1999_a1_solution β β f g h : Polynomial β, β x : β, |f.eval x| - |g.eval x| + h.eval x = if x < -1 then -1 else (if (x β€ 0) then 3 * x + 2 else -2 * x + 2) :=
sorry | import Mathlib
-- Note: The actual problem asks to "find" such polynomials as well - but the solution does not give a set of all possible solutions. Hence, we would need to do the analysis ourselves, the following formalization should work.
-- True
/--
Find polynomials $f(x)$,$g(x)$, and $h(x)$, if they exist, such that for all $x$, \[|f(x)|-|g(x)|+h(x) = \begin{cases} -1 & \mbox{if $x<-1$} \\3x+2 & \mbox{if $-1 \leq x \leq 0$} \\-2x+2 & \mbox{if $x>0$.}\end{cases}\]?
-/
theorem putnam_1999_a1
: putnam_1999_a1_solution β β f g h : Polynomial β, β x : β, |f.eval x| - |g.eval x| + h.eval x = if x < -1 then -1 else (if (x β€ 0) then 3 * x + 2 else -2 * x + 2) := by
| import Mathlib
-- Note: The actual problem asks to "find" such polynomials as well - but the solution does not give a set of all possible solutions. Hence, we would need to do the analysis ourselves, the following formalization should work.
abbrev putnam_1999_a1_solution : Prop := sorry
-- True
/--
Find polynomials $f(x)$,$g(x)$, and $h(x)$, if they exist, such that for all $x$, \[|f(x)|-|g(x)|+h(x) = \begin{cases} -1 & \mbox{if $x<-1$} \\3x+2 & \mbox{if $-1 \leq x \leq 0$} \\-2x+2 & \mbox{if $x>0$.}\end{cases}\]?
-/
theorem putnam_1999_a1
: putnam_1999_a1_solution β β f g h : Polynomial β, β x : β, |f.eval x| - |g.eval x| + h.eval x = if x < -1 then -1 else (if (x β€ 0) then 3 * x + 2 else -2 * x + 2) :=
sorry
| Find polynomials $f(x)$,$g(x)$, and $h(x)$, if they exist, such that for all $x$, \[|f(x)|-|g(x)|+h(x) = \begin{cases} -1 & \mbox{if $x<-1$} \\3x+2 & \mbox{if $-1 \leq x \leq 0$} \\-2x+2 & \mbox{if $x>0$.}\end{cases}\]? | Show that the answer is such functions do exist. | [
"algebra"
] | null | null |
|
putnam_1994_a1 | 1fb96b60-adf3-587b-b94b-c0944031c32b | train | 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 | import Mathlib
open Filter Topology
/--
Suppose that a sequence $a_1,a_2,a_3,\dots$ satisfies $0<a_n \leq a_{2n}+a_{2n+1}$ for all $n \geq 1$. Prove that the series $\sum_{n=1}^\infty a_n$ diverges.
-/
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 := by
| import Mathlib
open Filter Topology
/--
Suppose that a sequence $a_1,a_2,a_3,\dots$ satisfies $0<a_n \leq a_{2n}+a_{2n+1}$ for all $n \geq 1$. Prove that the series $\sum_{n=1}^\infty a_n$ diverges.
-/
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
| Suppose that a sequence $a_1,a_2,a_3,\dots$ satisfies $0<a_n \leq a_{2n}+a_{2n+1}$ for all $n \geq 1$. Prove that the series $\sum_{n=1}^\infty a_n$ diverges. | null | [
"analysis"
] | null | null |
|
putnam_2001_b6 | e4e0768c-ff79-5f03-86b0-3e04b549e84a | train | abbrev putnam_2001_b6_solution : Prop := sorry
-- True
/--
Assume that $(a_n)_{n \geq 1}$ is an increasing sequence of positive real numbers such that $\lim a_n/n=0$. Must there exist infinitely many positive integers $n$ such that $a_{n-i}+a_{n+i}<2a_n$ for $i=1,2,\ldots,n-1$?
-/
theorem putnam_2001_b6 :
(β a : β€ β β,
(β n β₯ 1, a n > 0 β§ a n < a (n + 1)) β
(Tendsto (fun n : β€ => a (n + 1) / (n + 1)) atTop (π 0)) β
{n : β€ | n > 0 β§ (β i β Set.Icc 1 (n - 1), a (n - i) + a (n + i) < 2 * a n)}.Infinite)
β putnam_2001_b6_solution :=
sorry | import Mathlib
open Topology Filter Polynomial Set
-- Note: uses (β€ β β) instead of (Set.Ici 1 β β)
-- True
/--
Assume that $(a_n)_{n \geq 1}$ is an increasing sequence of positive real numbers such that $\lim a_n/n=0$. Must there exist infinitely many positive integers $n$ such that $a_{n-i}+a_{n+i}<2a_n$ for $i=1,2,\ldots,n-1$?
-/
theorem putnam_2001_b6 :
(β a : β€ β β,
(β n β₯ 1, a n > 0 β§ a n < a (n + 1)) β
(Tendsto (fun n : β€ => a (n + 1) / (n + 1)) atTop (π 0)) β
{n : β€ | n > 0 β§ (β i β Set.Icc 1 (n - 1), a (n - i) + a (n + i) < 2 * a n)}.Infinite)
β putnam_2001_b6_solution := by
| import Mathlib
open Topology Filter Polynomial Set
-- Note: uses (β€ β β) instead of (Set.Ici 1 β β)
abbrev putnam_2001_b6_solution : Prop := sorry
-- True
/--
Assume that $(a_n)_{n \geq 1}$ is an increasing sequence of positive real numbers such that $\lim a_n/n=0$. Must there exist infinitely many positive integers $n$ such that $a_{n-i}+a_{n+i}<2a_n$ for $i=1,2,\ldots,n-1$?
-/
theorem putnam_2001_b6 :
(β a : β€ β β,
(β n β₯ 1, a n > 0 β§ a n < a (n + 1)) β
(Tendsto (fun n : β€ => a (n + 1) / (n + 1)) atTop (π 0)) β
{n : β€ | n > 0 β§ (β i β Set.Icc 1 (n - 1), a (n - i) + a (n + i) < 2 * a n)}.Infinite)
β putnam_2001_b6_solution :=
sorry
| Assume that $(a_n)_{n \geq 1}$ is an increasing sequence of positive real numbers such that $\lim a_n/n=0$. Must there exist infinitely many positive integers $n$ such that $a_{n-i}+a_{n+i}<2a_n$ for $i=1,2,\ldots,n-1$? | Show that the answer is yes, there must exist infinitely many such $n$. | [
"analysis"
] | null | null |
|
putnam_1986_a3 | a0b055ec-7576-54d1-bc5d-148c26307f97 | train | abbrev putnam_1986_a3_solution : β := sorry
-- Real.pi / 2
/--
Evaluate $\sum_{n=0}^\infty \mathrm{Arccot}(n^2+n+1)$, where $\mathrm{Arccot}\,t$ for $t \geq 0$ denotes the number $\theta$ in the interval $0 < \theta \leq \pi/2$ with $\cot \theta = t$.
-/
theorem putnam_1986_a3
(cot : β β β)
(fcot : cot = fun ΞΈ β¦ cos ΞΈ / sin ΞΈ)
(arccot : β β β)
(harccot : β t : β, t β₯ 0 β arccot t β Set.Ioc 0 (Real.pi / 2) β§ cot (arccot t) = t)
: (β' n : β, arccot (n ^ 2 + n + 1) = putnam_1986_a3_solution) :=
sorry | import Mathlib
open Real
-- Real.pi / 2
/--
Evaluate $\sum_{n=0}^\infty \mathrm{Arccot}(n^2+n+1)$, where $\mathrm{Arccot}\,t$ for $t \geq 0$ denotes the number $\theta$ in the interval $0 < \theta \leq \pi/2$ with $\cot \theta = t$.
-/
theorem putnam_1986_a3
(cot : β β β)
(fcot : cot = fun ΞΈ β¦ cos ΞΈ / sin ΞΈ)
(arccot : β β β)
(harccot : β t : β, t β₯ 0 β arccot t β Set.Ioc 0 (Real.pi / 2) β§ cot (arccot t) = t)
: (β' n : β, arccot (n ^ 2 + n + 1) = putnam_1986_a3_solution) := by
| import Mathlib
open Real
noncomputable abbrev putnam_1986_a3_solution : β := sorry
-- Real.pi / 2
/--
Evaluate $\sum_{n=0}^\infty \mathrm{Arccot}(n^2+n+1)$, where $\mathrm{Arccot}\,t$ for $t \geq 0$ denotes the number $\theta$ in the interval $0 < \theta \leq \pi/2$ with $\cot \theta = t$.
-/
theorem putnam_1986_a3
(cot : β β β)
(fcot : cot = fun ΞΈ β¦ cos ΞΈ / sin ΞΈ)
(arccot : β β β)
(harccot : β t : β, t β₯ 0 β arccot t β Set.Ioc 0 (Real.pi / 2) β§ cot (arccot t) = t)
: (β' n : β, arccot (n ^ 2 + n + 1) = putnam_1986_a3_solution) :=
sorry
| Evaluate $\sum_{n=0}^\infty \mathrm{Arccot}(n^2+n+1)$, where $\mathrm{Arccot}\,t$ for $t \geq 0$ denotes the number $\theta$ in the interval $0 < \theta \leq \pi/2$ with $\cot \theta = t$. | Show that the sum equals $\pi/2$. | [
"analysis"
] | null | null |
|
putnam_1972_b5 | b8cf64b5-9ab2-5578-b9ff-9fae8a34266e | train | theorem putnam_1972_b5
(A B C D : EuclideanSpace β (Fin 3))
(hnonplanar : Β¬Coplanar β {A, B, C, D})
(hangles : β A B C = β C D A β§ β B C D = β D A B)
: dist A B = dist C D β§ dist B C = dist D A :=
sorry | import Mathlib
open EuclideanGeometry Set Metric
/--
Let $ABCD$ be a skew (non-planar) quadrilateral. Prove that if $\angle ABC = \angle CDA$ and $\angle BCD = \angle DAB$, then $AB = CD$ and $AD = BC$.
-/
theorem putnam_1972_b5
(A B C D : EuclideanSpace β (Fin 3))
(hnonplanar : Β¬Coplanar β {A, B, C, D})
(hangles : β A B C = β C D A β§ β B C D = β D A B)
: dist A B = dist C D β§ dist B C = dist D A := by
| import Mathlib
open EuclideanGeometry Set Metric
/--
Let $ABCD$ be a skew (non-planar) quadrilateral. Prove that if $\angle ABC = \angle CDA$ and $\angle BCD = \angle DAB$, then $AB = CD$ and $AD = BC$.
-/
theorem putnam_1972_b5
(A B C D : EuclideanSpace β (Fin 3))
(hnonplanar : Β¬Coplanar β {A, B, C, D})
(hangles : β A B C = β C D A β§ β B C D = β D A B)
: dist A B = dist C D β§ dist B C = dist D A :=
sorry
| Let $ABCD$ be a skew (non-planar) quadrilateral. Prove that if $\angle ABC = \angle CDA$ and $\angle BCD = \angle DAB$, then $AB = CD$ and $AD = BC$. | null | [
"geometry"
] | null | null |
|
putnam_1998_a2 | d564c555-f360-5a7e-8b46-9f7ac1cf6f27 | train | 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 | import Mathlib
/--
Let $s$ be any arc of the unit circle lying entirely in the first quadrant. Let $A$ be the area of the region lying below $s$ and above the $x$-axis and let $B$ be the area of the region lying to the right of the $y$-axis and to the left of $s$. Prove that $A+B$ depends only on the arc length, and not on the position, of $s$.
-/
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)) := by
| import Mathlib
/--
Let $s$ be any arc of the unit circle lying entirely in the first quadrant. Let $A$ be the area of the region lying below $s$ and above the $x$-axis and let $B$ be the area of the region lying to the right of the $y$-axis and to the left of $s$. Prove that $A+B$ depends only on the arc length, and not on the position, of $s$.
-/
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
| Let $s$ be any arc of the unit circle lying entirely in the first quadrant. Let $A$ be the area of the region lying below $s$ and above the $x$-axis and let $B$ be the area of the region lying to the right of the $y$-axis and to the left of $s$. Prove that $A+B$ depends only on the arc length, and not on the position, of $s$. | null | [
"geometry"
] | null | null |
|
putnam_1970_a2 | 0ddafeb3-2554-53ad-baed-46702ff99461 | train | theorem putnam_1970_a2
(A B C D E F G : β)
(hle : B^2 - 4*A*C < 0)
: β Ξ΄ > 0, Β¬β x y : β, x^2 + y^2 β Set.Ioo 0 (Ξ΄^2) β§
A*x^2 + B*x*y + C*y^2 + D*x^3 + E*x^2*y + F*x*y^2 + G*y^3 = 0 :=
sorry | import Mathlib
open Metric Set EuclideanGeometry
/--
Let $A$, $B$, $C$, $D$, $E$, $F$, and $G$ be real numbers satisfying $B^2 - 4AC < 0$. Prove that there exists some $\delta > 0$ such that no points $(x, y)$ in the punctured disk $0 < x^2 + y^2 < \delta$ satisfy $$Ax^2 + Bxy + Cy^2 + Dx^3 + Ex^2y + Fxy^2 + Gy^3 = 0.$$
-/
theorem putnam_1970_a2
(A B C D E F G : β)
(hle : B^2 - 4*A*C < 0)
: β Ξ΄ > 0, Β¬β x y : β, x^2 + y^2 β Set.Ioo 0 (Ξ΄^2) β§
A*x^2 + B*x*y + C*y^2 + D*x^3 + E*x^2*y + F*x*y^2 + G*y^3 = 0 := by
| import Mathlib
open Metric Set EuclideanGeometry
/--
Let $A$, $B$, $C$, $D$, $E$, $F$, and $G$ be real numbers satisfying $B^2 - 4AC < 0$. Prove that there exists some $\delta > 0$ such that no points $(x, y)$ in the punctured disk $0 < x^2 + y^2 < \delta$ satisfy $$Ax^2 + Bxy + Cy^2 + Dx^3 + Ex^2y + Fxy^2 + Gy^3 = 0.$$
-/
theorem putnam_1970_a2
(A B C D E F G : β)
(hle : B^2 - 4*A*C < 0)
: β Ξ΄ > 0, Β¬β x y : β, x^2 + y^2 β Set.Ioo 0 (Ξ΄^2) β§
A*x^2 + B*x*y + C*y^2 + D*x^3 + E*x^2*y + F*x*y^2 + G*y^3 = 0 :=
sorry
| Let $A$, $B$, $C$, $D$, $E$, $F$, and $G$ be real numbers satisfying $B^2 - 4AC < 0$. Prove that there exists some $\delta > 0$ such that no points $(x, y)$ in the punctured disk $0 < x^2 + y^2 < \delta$ satisfy $$Ax^2 + Bxy + Cy^2 + Dx^3 + Ex^2y + Fxy^2 + Gy^3 = 0.$$ | null | [
"analysis",
"algebra"
] | null | null |
|
putnam_1982_a2 | 00b73bd0-fcab-576b-98a7-b004695f369c | train | abbrev putnam_1982_a2_solution : Prop := sorry
-- True
/--
Let $B_n(x) = 1^x + 2^x + \dots + n^x$ and let $f(n) = \frac{B_n(\log_n 2)}{(n \log_2 n)^2}$. Does $f(2) + f(3) + f(4) + \dots$ converge?
-/
theorem putnam_1982_a2
(B : β β β β β)
(hB : B = fun (n : β) (x : β) β¦ β k in Finset.Icc 1 n, (k : β) ^ x)
(f : β β β)
(hf : f = fun n β¦ B n (logb n 2) / (n * logb 2 n) ^ 2)
: (β L : β, Tendsto (fun N β¦ β j in Finset.Icc 2 N, f j) atTop (π L)) β putnam_1982_a2_solution :=
sorry | import Mathlib
open Set Function Filter Topology Polynomial Real
-- True
/--
Let $B_n(x) = 1^x + 2^x + \dots + n^x$ and let $f(n) = \frac{B_n(\log_n 2)}{(n \log_2 n)^2}$. Does $f(2) + f(3) + f(4) + \dots$ converge?
-/
theorem putnam_1982_a2
(B : β β β β β)
(hB : B = fun (n : β) (x : β) β¦ β k in Finset.Icc 1 n, (k : β) ^ x)
(f : β β β)
(hf : f = fun n β¦ B n (logb n 2) / (n * logb 2 n) ^ 2)
: (β L : β, Tendsto (fun N β¦ β j in Finset.Icc 2 N, f j) atTop (π L)) β putnam_1982_a2_solution := by
| import Mathlib
open Set Function Filter Topology Polynomial Real
abbrev putnam_1982_a2_solution : Prop := sorry
-- True
/--
Let $B_n(x) = 1^x + 2^x + \dots + n^x$ and let $f(n) = \frac{B_n(\log_n 2)}{(n \log_2 n)^2}$. Does $f(2) + f(3) + f(4) + \dots$ converge?
-/
theorem putnam_1982_a2
(B : β β β β β)
(hB : B = fun (n : β) (x : β) β¦ β k in Finset.Icc 1 n, (k : β) ^ x)
(f : β β β)
(hf : f = fun n β¦ B n (logb n 2) / (n * logb 2 n) ^ 2)
: (β L : β, Tendsto (fun N β¦ β j in Finset.Icc 2 N, f j) atTop (π L)) β putnam_1982_a2_solution :=
sorry
| Let $B_n(x) = 1^x + 2^x + \dots + n^x$ and let $f(n) = \frac{B_n(\log_n 2)}{(n \log_2 n)^2}$. Does $f(2) + f(3) + f(4) + \dots$ converge? | Prove that the series converges. | [
"algebra"
] | null | null |
|
putnam_1993_a5 | b2503ca3-5680-595c-ab88-539f4750df04 | train | 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 | import Mathlib
/--
Show that $\int_{-100}^{-10} (\frac{x^2-x}{x^3-3x+1})^2\,dx+\int_{\frac{1}{101}}^{\frac{1}{11}} (\frac{x^2-x}{x^3-3x+1})^2\,dx+\int_{\frac{101}{100}}^{\frac{11}{10}} (\frac{x^2-x}{x^3-3x+1})^2\,dx$ is a rational number.
-/
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))) := by
| import Mathlib
/--
Show that $\int_{-100}^{-10} (\frac{x^2-x}{x^3-3x+1})^2\,dx+\int_{\frac{1}{101}}^{\frac{1}{11}} (\frac{x^2-x}{x^3-3x+1})^2\,dx+\int_{\frac{101}{100}}^{\frac{11}{10}} (\frac{x^2-x}{x^3-3x+1})^2\,dx$ is a rational number.
-/
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
| Show that $\int_{-100}^{-10} (\frac{x^2-x}{x^3-3x+1})^2\,dx+\int_{\frac{1}{101}}^{\frac{1}{11}} (\frac{x^2-x}{x^3-3x+1})^2\,dx+\int_{\frac{101}{100}}^{\frac{11}{10}} (\frac{x^2-x}{x^3-3x+1})^2\,dx$ is a rational number. | null | [
"analysis"
] | null | null |
|
putnam_1987_b6 | 268391ba-acb3-536d-b97e-f1bfc1ce45bc | train | theorem putnam_1987_b6
(p : β)
(F : Type*) [Field F] [Fintype F]
(S : Set F)
(hp : Odd p β§ Nat.Prime p)
(Fcard : Fintype.card F = p ^ 2)
(Snz : β x β S, x β 0)
(Scard : S.ncard = ((p : β€) ^ 2 - 1) / 2)
(hS : β a : F, a β 0 β Xor' (a β S) (-a β S)) :
(Even ((S β© {x | β a β S, x = 2 * a}).ncard)) :=
sorry | import Mathlib
open MvPolynomial Real Nat Filter Topology
/--
Let $F$ be the field of $p^2$ elements, where $p$ is an odd prime. Suppose $S$ is a set of $(p^2-1)/2$ distinct nonzero elements of $F$ with the property that for each $a\neq 0$ in $F$, exactly one of $a$ and $-a$ is in $S$. Let $N$ be the number of elements in the intersection $S \cap \{2a: a \in S\}$. Prove that $N$ is even.
-/
theorem putnam_1987_b6
(p : β)
(F : Type*) [Field F] [Fintype F]
(S : Set F)
(hp : Odd p β§ Nat.Prime p)
(Fcard : Fintype.card F = p ^ 2)
(Snz : β x β S, x β 0)
(Scard : S.ncard = ((p : β€) ^ 2 - 1) / 2)
(hS : β a : F, a β 0 β Xor' (a β S) (-a β S)) :
(Even ((S β© {x | β a β S, x = 2 * a}).ncard)) := by
| import Mathlib
open MvPolynomial Real Nat Filter Topology
/--
Let $F$ be the field of $p^2$ elements, where $p$ is an odd prime. Suppose $S$ is a set of $(p^2-1)/2$ distinct nonzero elements of $F$ with the property that for each $a\neq 0$ in $F$, exactly one of $a$ and $-a$ is in $S$. Let $N$ be the number of elements in the intersection $S \cap \{2a: a \in S\}$. Prove that $N$ is even.
-/
theorem putnam_1987_b6
(p : β)
(F : Type*) [Field F] [Fintype F]
(S : Set F)
(hp : Odd p β§ Nat.Prime p)
(Fcard : Fintype.card F = p ^ 2)
(Snz : β x β S, x β 0)
(Scard : S.ncard = ((p : β€) ^ 2 - 1) / 2)
(hS : β a : F, a β 0 β Xor' (a β S) (-a β S)) :
(Even ((S β© {x | β a β S, x = 2 * a}).ncard)) :=
sorry
| Let $F$ be the field of $p^2$ elements, where $p$ is an odd prime. Suppose $S$ is a set of $(p^2-1)/2$ distinct nonzero elements of $F$ with the property that for each $a\neq 0$ in $F$, exactly one of $a$ and $-a$ is in $S$. Let $N$ be the number of elements in the intersection $S \cap \{2a: a \in S\}$. Prove that $N$ is even. | null | [
"abstract_algebra"
] | null | null |
|
putnam_2022_b2 | 87d17693-cac3-5990-aff5-daf99a0869c3 | train | abbrev putnam_2022_b2_solution : Set β := sorry
-- {1, 7}
/--
Let $\times$ represent the cross product in $\mathbb{R}^3$. For what positive integers $n$ does there exist a set $S \subset \mathbb{R}^3$ with exactly $n$ elements such that $S=\{v \times w:v,w \in S\}$?
-/
theorem putnam_2022_b2
(n : β)
(P : Finset (Fin 3 β β) β Prop)
(P_def : β S : Finset (Fin 3 β β), P S β (S = {u : Fin 3 β β | β v w : S, u = crossProduct v w})) :
(0 < n β§ β S : Finset (Fin 3 β β), S.card = n β§ P S) β n β putnam_2022_b2_solution :=
sorry | import Mathlib
open Polynomial
-- {1, 7}
/--
Let $\times$ represent the cross product in $\mathbb{R}^3$. For what positive integers $n$ does there exist a set $S \subset \mathbb{R}^3$ with exactly $n$ elements such that $S=\{v \times w:v,w \in S\}$?
-/
theorem putnam_2022_b2
(n : β)
(P : Finset (Fin 3 β β) β Prop)
(P_def : β S : Finset (Fin 3 β β), P S β (S = {u : Fin 3 β β | β v w : S, u = crossProduct v w})) :
(0 < n β§ β S : Finset (Fin 3 β β), S.card = n β§ P S) β n β putnam_2022_b2_solution := by
| import Mathlib
open Polynomial
abbrev putnam_2022_b2_solution : Set β := sorry
-- {1, 7}
/--
Let $\times$ represent the cross product in $\mathbb{R}^3$. For what positive integers $n$ does there exist a set $S \subset \mathbb{R}^3$ with exactly $n$ elements such that $S=\{v \times w:v,w \in S\}$?
-/
theorem putnam_2022_b2
(n : β)
(P : Finset (Fin 3 β β) β Prop)
(P_def : β S : Finset (Fin 3 β β), P S β (S = {u : Fin 3 β β | β v w : S, u = crossProduct v w})) :
(0 < n β§ β S : Finset (Fin 3 β β), S.card = n β§ P S) β n β putnam_2022_b2_solution :=
sorry
| Let $\times$ represent the cross product in $\mathbb{R}^3$. For what positive integers $n$ does there exist a set $S \subset \mathbb{R}^3$ with exactly $n$ elements such that $S=\{v \times w:v,w \in S\}$? | Show that the possible values of $n$ are $1$ and $7$. | [
"algebra"
] | null | null |
|
putnam_1968_a4 | c49c3f88-80fc-5cee-85a8-f6e221b4af65 | train | theorem putnam_1968_a4
(n : β)
(S : Fin n β (EuclideanSpace β (Fin 3)))
(hS : β i : Fin n, dist 0 (S i) = 1)
: β i : Fin n, β j : Fin n, (if i < j then (dist (S i) (S j))^2 else (0 : β)) β€ n^2 :=
sorry | import Mathlib
open Finset
/--
Prove that the sum of the squares of the distances between any $n$ points on the unit sphere $\{(x, y, z) \mid x^2 + y^2 + z^2 = 1\}$ is at most $n^2$.
-/
theorem putnam_1968_a4
(n : β)
(S : Fin n β (EuclideanSpace β (Fin 3)))
(hS : β i : Fin n, dist 0 (S i) = 1)
: β i : Fin n, β j : Fin n, (if i < j then (dist (S i) (S j))^2 else (0 : β)) β€ n^2 := by
| import Mathlib
open Finset
/--
Prove that the sum of the squares of the distances between any $n$ points on the unit sphere $\{(x, y, z) \mid x^2 + y^2 + z^2 = 1\}$ is at most $n^2$.
-/
theorem putnam_1968_a4
(n : β)
(S : Fin n β (EuclideanSpace β (Fin 3)))
(hS : β i : Fin n, dist 0 (S i) = 1)
: β i : Fin n, β j : Fin n, (if i < j then (dist (S i) (S j))^2 else (0 : β)) β€ n^2 :=
sorry
| Prove that the sum of the squares of the distances between any $n$ points on the unit sphere $\{(x, y, z) \mid x^2 + y^2 + z^2 = 1\}$ is at most $n^2$. | null | [
"geometry",
"algebra"
] | null | null |
|
putnam_1986_a2 | ccfc0deb-3741-5142-8537-552e38712b97 | train | 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 | import Mathlib
-- 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) := by
| import Mathlib
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
| What is the units (i.e., rightmost) digit of
\[
\left\lfloor \frac{10^{20000}}{10^{100}+3}\right\rfloor ?
\] | Show that the answer is $3$. | [
"algebra"
] | null | null |
|
putnam_1978_b6 | 9cce7deb-47fb-525e-9d59-b0e6bc92ef7d | train | 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 | import Mathlib
open Set Real Filter Topology Polynomial
/--
Let $a_{ij}$ be real numbers in $[0, 1]$. Show that
\[
\left( \sum_{i=1}^n \sum_{j=1}^{mi} \frac{a_{ij}}{i} \right) ^2 \leq 2m \sum_{i=1}^n \sum_{j=1}^{mi} a_{ij}.
\]
-/
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) := by
| import Mathlib
open Set Real Filter Topology Polynomial
/--
Let $a_{ij}$ be real numbers in $[0, 1]$. Show that
\[
\left( \sum_{i=1}^n \sum_{j=1}^{mi} \frac{a_{ij}}{i} \right) ^2 \leq 2m \sum_{i=1}^n \sum_{j=1}^{mi} a_{ij}.
\]
-/
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
| Let $a_{ij}$ be real numbers in $[0, 1]$. Show that
\[
\left( \sum_{i=1}^n \sum_{j=1}^{mi} \frac{a_{ij}}{i} \right) ^2 \leq 2m \sum_{i=1}^n \sum_{j=1}^{mi} a_{ij}.
\] | null | [
"algebra"
] | null | null |