name stringlengths 7 15 | uuid stringlengths 36 36 | split stringclasses 1
value | lean4_statement stringlengths 75 2.06k | full_text_no_abbrv stringlengths 223 2.05k | full_text stringlengths 226 2.1k | lean4_proof stringclasses 1
value | informal_statement stringlengths 4 898 | informal_solution stringlengths 4 282 | tags listlengths 0 3 | coq_statement stringclasses 1
value | isabelle_statement stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|---|
putnam_1987_a2 | 5bce53b3-b67e-5252-a4e8-df7427e65568 | train | abbrev putnam_1987_a2_solution : β := sorry
-- 1984
/--
The sequence of digits $123456789101112131415161718192021 \dots$ is obtained by writing the positive integers in order. If the $10^n$-th digit in this sequence occurs in the part of the sequence in which the $m$-digit numbers are placed, define $f(n)$ to be $m$. F... | import Mathlib
-- 1984
/--
The sequence of digits $123456789101112131415161718192021 \dots$ is obtained by writing the positive integers in order. If the $10^n$-th digit in this sequence occurs in the part of the sequence in which the $m$-digit numbers are placed, define $f(n)$ to be $m$. For example, $f(2)=2$ because... | import Mathlib
abbrev putnam_1987_a2_solution : β := sorry
-- 1984
/--
The sequence of digits $123456789101112131415161718192021 \dots$ is obtained by writing the positive integers in order. If the $10^n$-th digit in this sequence occurs in the part of the sequence in which the $m$-digit numbers are placed, define $f(... | The sequence of digits $123456789101112131415161718192021 \dots$ is obtained by writing the positive integers in order. If the $10^n$-th digit in this sequence occurs in the part of the sequence in which the $m$-digit numbers are placed, define $f(n)$ to be $m$. For example, $f(2)=2$ because the $100$th digit enters th... | Show that the value of $f(1987)$ is $1984$. | [
"algebra"
] | null | null | |
putnam_2002_b5 | 5e4c3ab8-2311-5d37-be85-ddb673ef2085 | train | theorem putnam_2002_b5
: β n : β, {b : β | b β₯ 1 β§ (Nat.digits b n).length = 3 β§ List.Palindrome (Nat.digits b n)}.ncard β₯ 2002 :=
sorry | import Mathlib
open Nat Set Topology Filter
/--
A palindrome in base $b$ is a positive integer whose base-$b$
digits read the same backwards and forwards; for example,
$2002$ is a 4-digit palindrome in base 10. Note that 200 is not
a palindrome in base 10, but it is the 3-digit palindrome
242 in base 9, and 404 in ba... | import Mathlib
open Nat Set Topology Filter
/--
A palindrome in base $b$ is a positive integer whose base-$b$
digits read the same backwards and forwards; for example,
$2002$ is a 4-digit palindrome in base 10. Note that 200 is not
a palindrome in base 10, but it is the 3-digit palindrome
242 in base 9, and 404 in ba... | A palindrome in base $b$ is a positive integer whose base-$b$
digits read the same backwards and forwards; for example,
$2002$ is a 4-digit palindrome in base 10. Note that 200 is not
a palindrome in base 10, but it is the 3-digit palindrome
242 in base 9, and 404 in base 7. Prove that there is an integer
which is a 3-... | null | [
"number_theory"
] | null | null | |
putnam_1981_b1 | cf7adaca-307b-5dda-aa5c-523634e57376 | train | abbrev putnam_1981_b1_solution : β := sorry
-- -1
/--
Find the value of $$\lim_{n \rightarrow \infty} \frac{1}{n^5}\sum_{h=1}^{n}\sum_{k=1}^{n}(5h^4 - 18h^2k^2 + 5k^4).$$
-/
theorem putnam_1981_b1
(f : β β β)
(hf : f = fun n : β => ((1 : β)/n^5) * β h in Finset.Icc 1 n, β k in Finset.Icc 1 n, (5*(h : β)^4 - 18*h^2*k^2 ... | import Mathlib
open Topology Filter Set Polynomial Function
-- -1
/--
Find the value of $$\lim_{n \rightarrow \infty} \frac{1}{n^5}\sum_{h=1}^{n}\sum_{k=1}^{n}(5h^4 - 18h^2k^2 + 5k^4).$$
-/
theorem putnam_1981_b1
(f : β β β)
(hf : f = fun n : β => ((1 : β)/n^5) * β h in Finset.Icc 1 n, β k in Finset.Icc 1 n, (5*(h : ... | import Mathlib
open Topology Filter Set Polynomial Function
abbrev putnam_1981_b1_solution : β := sorry
-- -1
/--
Find the value of $$\lim_{n \rightarrow \infty} \frac{1}{n^5}\sum_{h=1}^{n}\sum_{k=1}^{n}(5h^4 - 18h^2k^2 + 5k^4).$$
-/
theorem putnam_1981_b1
(f : β β β)
(hf : f = fun n : β => ((1 : β)/n^5) * β h in Fin... | Find the value of $$\lim_{n \rightarrow \infty} \frac{1}{n^5}\sum_{h=1}^{n}\sum_{k=1}^{n}(5h^4 - 18h^2k^2 + 5k^4).$$ | The limit equals $-1$. | [
"analysis"
] | null | null | |
putnam_2002_a3 | cbc068cd-3ee7-5a29-b478-c09312695265 | train | theorem putnam_2002_a3
(n Tn : β€)
(hn : n β₯ 2)
(hTn : Tn = Set.ncard {S : Set β€ | S β Set.Icc 1 n β§ Nonempty S β§ β k : β€, k = ((1 : β) / S.ncard) * (β' s : S, s.1)})
: Even (Tn - n) :=
sorry | import Mathlib
open Nat
/--
Let $n \geq 2$ be an integer and $T_n$ be the number of non-empty subsets $S$ of $\{1, 2, 3, \dots, n\}$ with the property that the average of the elements of $S$ is an integer. Prove that $T_n - n$ is always even.
-/
theorem putnam_2002_a3
(n Tn : β€)
(hn : n β₯ 2)
(hTn : Tn = Set.ncard {S ... | import Mathlib
open Nat
/--
Let $n \geq 2$ be an integer and $T_n$ be the number of non-empty subsets $S$ of $\{1, 2, 3, \dots, n\}$ with the property that the average of the elements of $S$ is an integer. Prove that $T_n - n$ is always even.
-/
theorem putnam_2002_a3
(n Tn : β€)
(hn : n β₯ 2)
(hTn : Tn = Set.ncard {S ... | Let $n \geq 2$ be an integer and $T_n$ be the number of non-empty subsets $S$ of $\{1, 2, 3, \dots, n\}$ with the property that the average of the elements of $S$ is an integer. Prove that $T_n - n$ is always even. | null | [
"algebra"
] | null | null | |
putnam_1981_a5 | 2e2e58f3-e2a6-5efa-886f-6af9cd957438 | train | abbrev putnam_1981_a5_solution : Prop := sorry
-- True
/--
Let $P(x)$ be a polynomial with real coefficients; let $$Q(x) = (x^2 + 1)P(x)P'(x) + x((P(x))^2 + (P'(x))^2).$$
Given that $P$ has $n$ distinct real roots all greater than $1$, prove or disprove that $Q$ must have at least $2n - 1$ distinct real roots.
-/
theor... | import Mathlib
open Topology Filter Set Polynomial Function
-- True
/--
Let $P(x)$ be a polynomial with real coefficients; let $$Q(x) = (x^2 + 1)P(x)P'(x) + x((P(x))^2 + (P'(x))^2).$$
Given that $P$ has $n$ distinct real roots all greater than $1$, prove or disprove that $Q$ must have at least $2n - 1$ distinct real ... | import Mathlib
open Topology Filter Set Polynomial Function
abbrev putnam_1981_a5_solution : Prop := sorry
-- True
/--
Let $P(x)$ be a polynomial with real coefficients; let $$Q(x) = (x^2 + 1)P(x)P'(x) + x((P(x))^2 + (P'(x))^2).$$
Given that $P$ has $n$ distinct real roots all greater than $1$, prove or disprove that... | Let $P(x)$ be a polynomial with real coefficients; let $$Q(x) = (x^2 + 1)P(x)P'(x) + x((P(x))^2 + (P'(x))^2).$$
Given that $P$ has $n$ distinct real roots all greater than $1$, prove or disprove that $Q$ must have at least $2n - 1$ distinct real roots. | $Q(x)$ must have at least $2n - 1$ distinct real roots. | [
"algebra"
] | null | null | |
putnam_1995_a3 | 39ef9e8e-8b6f-5c7e-b129-78b21ff62131 | train | theorem putnam_1995_a3
(relation : (Fin 9 β β€) β (Fin 9 β β€) β Prop)
(digits_to_num : (Fin 9 β β€) β β€)
(hdigits_to_num : digits_to_num = fun dig => β i : Fin 9, (dig i) * 10^i.1)
(hrelation : β d e : (Fin 9 β β€), relation d e β (β i : Fin 9, d i < 10 β§ d i β₯ 0 β§ e i < 10 β§ e i β₯ 0) β§ (β i : Fin 9, 7 β£ (digits_to_num (f... | import Mathlib
open Filter Topology Real
/--
The number $d_{1}d_{2}\dots d_{9}$ has nine (not necessarily distinct) decimal digits. The number $e_{1}e_{2}\dots e_{9}$ is such that each of the nine 9-digit numbers formed by replacing just one of the digits $d_{i}$ is $d_{1}d_{2}\dots d_{9}$ by the corresponding digit ... | import Mathlib
open Filter Topology Real
/--
The number $d_{1}d_{2}\dots d_{9}$ has nine (not necessarily distinct) decimal digits. The number $e_{1}e_{2}\dots e_{9}$ is such that each of the nine 9-digit numbers formed by replacing just one of the digits $d_{i}$ is $d_{1}d_{2}\dots d_{9}$ by the corresponding digit ... | The number $d_{1}d_{2}\dots d_{9}$ has nine (not necessarily distinct) decimal digits. The number $e_{1}e_{2}\dots e_{9}$ is such that each of the nine 9-digit numbers formed by replacing just one of the digits $d_{i}$ is $d_{1}d_{2}\dots d_{9}$ by the corresponding digit $e_{i}$ ($1 \leq i \leq 9$) is divisible by 7. ... | null | [
"number_theory"
] | null | null | |
putnam_2010_b5 | b0b878ef-5fad-57f0-a05d-790669ed41a0 | train | abbrev putnam_2010_b5_solution : Prop := sorry
-- False
/--
Is there a strictly increasing function $f: \mathbb{R} \to \mathbb{R}$ such that $f'(x) = f(f(x))$ for all $x$?
-/
theorem putnam_2010_b5 :
(β f : β β β, StrictMono f β§ Differentiable β f β§ (β x : β, deriv f x = f (f x))) β putnam_2010_b5_solution :=
sorry | import Mathlib
open Filter Topology Set
-- False
/--
Is there a strictly increasing function $f: \mathbb{R} \to \mathbb{R}$ such that $f'(x) = f(f(x))$ for all $x$?
-/
theorem putnam_2010_b5 :
(β f : β β β, StrictMono f β§ Differentiable β f β§ (β x : β, deriv f x = f (f x))) β putnam_2010_b5_solution := by
| import Mathlib
open Filter Topology Set
abbrev putnam_2010_b5_solution : Prop := sorry
-- False
/--
Is there a strictly increasing function $f: \mathbb{R} \to \mathbb{R}$ such that $f'(x) = f(f(x))$ for all $x$?
-/
theorem putnam_2010_b5 :
(β f : β β β, StrictMono f β§ Differentiable β f β§ (β x : β, deriv f x = f (f... | Is there a strictly increasing function $f: \mathbb{R} \to \mathbb{R}$ such that $f'(x) = f(f(x))$ for all $x$? | Show that the solution is no such function exists. | [
"analysis"
] | null | null | |
putnam_1984_b5 | 5e815945-72f3-54f1-a936-85c5995f5ec7 | train | abbrev putnam_1984_b5_solution : β€ Γ Polynomial β Γ Polynomial β := sorry
-- (2, (Polynomial.X * (Polynomial.X - 1)) / 2, Polynomial.X)
/--
For each nonnegative integer $k$, let $d(k)$ denote the number of $1$'s in the binary expansion of $k$ (for example, $d(0)=0$ and $d(5)=2$). Let $m$ be a positive integer. Express ... | import Mathlib
open Topology Filter Nat
-- (2, (Polynomial.X * (Polynomial.X - 1)) / 2, Polynomial.X)
/--
For each nonnegative integer $k$, let $d(k)$ denote the number of $1$'s in the binary expansion of $k$ (for example, $d(0)=0$ and $d(5)=2$). Let $m$ be a positive integer. Express $\sum_{k=0}^{2^m-1} (-1)^{d(k)}k... | import Mathlib
open Topology Filter Nat
noncomputable abbrev putnam_1984_b5_solution : β€ Γ Polynomial β Γ Polynomial β := sorry
-- (2, (Polynomial.X * (Polynomial.X - 1)) / 2, Polynomial.X)
/--
For each nonnegative integer $k$, let $d(k)$ denote the number of $1$'s in the binary expansion of $k$ (for example, $d(0)=0... | For each nonnegative integer $k$, let $d(k)$ denote the number of $1$'s in the binary expansion of $k$ (for example, $d(0)=0$ and $d(5)=2$). Let $m$ be a positive integer. Express $\sum_{k=0}^{2^m-1} (-1)^{d(k)}k^m$ in the form $(-1)^ma^{f(m)}(g(m))!$, where $a$ is an integer and $f$ and $g$ are polynomials. | Show that $\sum_{k=0}^{2^m-1} (-1)^{d(k)}k^m=(-1)^m2^{m(m-1)/2}m!$. | [
"algebra",
"analysis"
] | null | null | |
putnam_2017_b3 | 5ddca2c4-9951-5b34-906c-e1a77f325ee9 | train | theorem putnam_2017_b3
(f : β β β)
(c : β β β)
(hc : β n, c n = 0 β¨ c n = 1)
(hf : β x, f x = β' n : β, (c n) * x^n)
: f (2/3) = 3/2 β Irrational (f 1/2) :=
sorry | import Mathlib
open Topology Filter
/--
Suppose that $f(x) = \sum_{i=0}^\infty c_i x^i$ is a power series for which each coefficient $c_i$ is $0$ or $1$. Show that if $f(2/3) = 3/2$, then $f(1/2)$ must be irrational.
-/
theorem putnam_2017_b3
(f : β β β)
(c : β β β)
(hc : β n, c n = 0 β¨ c n = 1)
(hf : β x, f x = β' n... | import Mathlib
open Topology Filter
/--
Suppose that $f(x) = \sum_{i=0}^\infty c_i x^i$ is a power series for which each coefficient $c_i$ is $0$ or $1$. Show that if $f(2/3) = 3/2$, then $f(1/2)$ must be irrational.
-/
theorem putnam_2017_b3
(f : β β β)
(c : β β β)
(hc : β n, c n = 0 β¨ c n = 1)
(hf : β x, f x = β' n... | Suppose that $f(x) = \sum_{i=0}^\infty c_i x^i$ is a power series for which each coefficient $c_i$ is $0$ or $1$. Show that if $f(2/3) = 3/2$, then $f(1/2)$ must be irrational. | null | [
"number_theory"
] | null | null | |
putnam_1989_b6 | 2ad96435-dc80-5259-bc39-7cbcdcfc58bc | train | theorem putnam_1989_b6
(n : β)
(Sx : Set (Fin n β β))
(fprop : (β β β) β Prop)
(xext : (Fin n β β) β (β β β))
(fxsum : (β β β) β (Fin n β β) β β)
(fEV : (β β β) β β)
(hSx : Sx = {x : Fin n β β | 0 < x β§ StrictMono x β§ x < 1})
(hfprop : fprop = (fun f : β β β => ContinuousOn f (Set.Icc 0 1) β§ f 1 = 0))
(hfxsum : fxsum =... | import Mathlib
open Nat Filter Topology Set
-- Note: uses (β β β) instead of (Set.Icc 0 1 β β)
/--
Let $(x_1,x_2,\dots,x_n)$ be a point chosen at random from the $n$-dimensional region defined by $0<x_1<x_2<\dots<x_n<1$. Let $f$ be a continuous function on $[0,1]$ with $f(1)=0$. Set $x_0=0$ and $x_{n+1}=1$. Show that... | import Mathlib
open Nat Filter Topology Set
-- Note: uses (β β β) instead of (Set.Icc 0 1 β β)
/--
Let $(x_1,x_2,\dots,x_n)$ be a point chosen at random from the $n$-dimensional region defined by $0<x_1<x_2<\dots<x_n<1$. Let $f$ be a continuous function on $[0,1]$ with $f(1)=0$. Set $x_0=0$ and $x_{n+1}=1$. Show that... | Let $(x_1,x_2,\dots,x_n)$ be a point chosen at random from the $n$-dimensional region defined by $0<x_1<x_2<\dots<x_n<1$. Let $f$ be a continuous function on $[0,1]$ with $f(1)=0$. Set $x_0=0$ and $x_{n+1}=1$. Show that the expected value of the Riemann sum $\sum_{i=0}^n (x_{i+1}-x_i)f(x_{i+1})$ is $\int_0^1 f(t)P(t)\,... | null | [
"probability",
"analysis",
"algebra"
] | null | null | |
putnam_1971_b2 | 0f152c45-007e-5b3f-8024-dbaa1a1410b6 | train | abbrev putnam_1971_b2_solution : Set (β β β) := sorry
-- {fun x : β => (x^3 - x^2 - 1)/(2 * x * (x - 1))}
/--
Find all functions $F : \mathbb{R} \setminus \{0, 1\} \to \mathbb{R}$ that satisfy $F(x) + F\left(\frac{x - 1}{x}\right) = 1 + x$ for all $x \in \mathbb{R} \setminus \{0, 1\}$.
-/
theorem putnam_1971_b2
(S : Se... | import Mathlib
open Set MvPolynomial
-- {fun x : β => (x^3 - x^2 - 1)/(2 * x * (x - 1))}
/--
Find all functions $F : \mathbb{R} \setminus \{0, 1\} \to \mathbb{R}$ that satisfy $F(x) + F\left(\frac{x - 1}{x}\right) = 1 + x$ for all $x \in \mathbb{R} \setminus \{0, 1\}$.
-/
theorem putnam_1971_b2
(S : Set β)
(hS : S = ... | import Mathlib
open Set MvPolynomial
abbrev putnam_1971_b2_solution : Set (β β β) := sorry
-- {fun x : β => (x^3 - x^2 - 1)/(2 * x * (x - 1))}
/--
Find all functions $F : \mathbb{R} \setminus \{0, 1\} \to \mathbb{R}$ that satisfy $F(x) + F\left(\frac{x - 1}{x}\right) = 1 + x$ for all $x \in \mathbb{R} \setminus \{0, ... | Find all functions $F : \mathbb{R} \setminus \{0, 1\} \to \mathbb{R}$ that satisfy $F(x) + F\left(\frac{x - 1}{x}\right) = 1 + x$ for all $x \in \mathbb{R} \setminus \{0, 1\}$. | The only such function is $F(x) = \frac{x^3 - x^2 - 1}{2x(x - 1)}$. | [
"algebra"
] | null | null | |
putnam_2019_a1 | 809c958b-410b-5e57-b3d3-79f29094facd | train | abbrev putnam_2019_a1_solution : Set β€ := sorry
-- {n : β€ | n β₯ 0 β§ Β¬Int.ModEq 9 n 3 β§ Β¬Int.ModEq 9 n 6}
/--
Determine all possible values of the expression
\[
A^3+B^3+C^3-3ABC
\]
where $A, B$, and $C$ are nonnegative integers.
-/
theorem putnam_2019_a1
: {n : β€ | β A B C : β€, A β₯ 0 β§ B β₯ 0 β§ C β₯ 0 β§ A^3 + B^3 + C^3 - ... | import Mathlib
open Topology Filter
-- {n : β€ | n β₯ 0 β§ Β¬Int.ModEq 9 n 3 β§ Β¬Int.ModEq 9 n 6}
/--
Determine all possible values of the expression
\[
A^3+B^3+C^3-3ABC
\]
where $A, B$, and $C$ are nonnegative integers.
-/
theorem putnam_2019_a1
: {n : β€ | β A B C : β€, A β₯ 0 β§ B β₯ 0 β§ C β₯ 0 β§ A^3 + B^3 + C^3 - 3*A*B*C = ... | import Mathlib
open Topology Filter
abbrev putnam_2019_a1_solution : Set β€ := sorry
-- {n : β€ | n β₯ 0 β§ Β¬Int.ModEq 9 n 3 β§ Β¬Int.ModEq 9 n 6}
/--
Determine all possible values of the expression
\[
A^3+B^3+C^3-3ABC
\]
where $A, B$, and $C$ are nonnegative integers.
-/
theorem putnam_2019_a1
: {n : β€ | β A B C : β€, A β₯ ... | Determine all possible values of the expression
\[
A^3+B^3+C^3-3ABC
\]
where $A, B$, and $C$ are nonnegative integers. | The answer is all nonnegative integers not congruent to $3$ or $6 \pmod{9}$. | [
"algebra"
] | null | null | |
putnam_1977_a2 | 3c04332a-fbcf-5760-bfbc-2a67a9db1c30 | train | abbrev putnam_1977_a2_solution : β β β β β β β β Prop := sorry
-- fun a b c d β¦ d = a β§ b = -c β¨ d = b β§ a = -c β¨ d = c β§ a = -b
/--
Find all real solutions $(a, b, c, d)$ to the equations $a + b + c = d$, $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{d}$.
-/
theorem putnam_1977_a2 :
β a b c d : β, putnam_197... | import Mathlib
-- fun a b c d β¦ d = a β§ b = -c β¨ d = b β§ a = -c β¨ d = c β§ a = -b
/--
Find all real solutions $(a, b, c, d)$ to the equations $a + b + c = d$, $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{d}$.
-/
theorem putnam_1977_a2 :
β a b c d : β, putnam_1977_a2_solution a b c d β
a β 0 β b β 0 β c... | import Mathlib
abbrev putnam_1977_a2_solution : β β β β β β β β Prop := sorry
-- fun a b c d β¦ d = a β§ b = -c β¨ d = b β§ a = -c β¨ d = c β§ a = -b
/--
Find all real solutions $(a, b, c, d)$ to the equations $a + b + c = d$, $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{d}$.
-/
theorem putnam_1977_a2 :
β a b c d... | Find all real solutions $(a, b, c, d)$ to the equations $a + b + c = d$, $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{d}$. | Prove that the solutions are $d = a$ and $b = -c$, $d = b$ and $a = -c$, or $d = c$ and $a = -b$, with $a, b, c, d$ nonzero. | [
"algebra"
] | null | null | |
putnam_1998_a5 | 17cb60f9-16e8-535a-877c-d7699d277d67 | train | theorem putnam_1998_a5
(k : β)
(c : Fin k β (EuclideanSpace β (Fin 2)))
(r : Fin k β β)
(hr : β i, r i > 0)
(E : Set (EuclideanSpace β (Fin 2)))
(hE : E β β i, ball (c i) (r i)) :
β (n : β) (t : Fin n β Fin k),
(β i j, i β j β (ball (c (t i)) (r (t i)) β© ball (c (t j)) (r (t j)) = β
)) β§
E β β i : ... | import Mathlib
open Set Function Metric
/--
Let $\mathcal F$ be a finite collection of open discs in $\mathbb R^2$ whose union contains a set $E\subseteq \mathbb R^2$. Show that there is a pairwise disjoint subcollection $D_1,\ldots, D_n$ in $\mathcal F$ such that \[E\subseteq \cup_{j=1}^n 3D_j.\] Here, if $D$ is th... | import Mathlib
open Set Function Metric
/--
Let $\mathcal F$ be a finite collection of open discs in $\mathbb R^2$ whose union contains a set $E\subseteq \mathbb R^2$. Show that there is a pairwise disjoint subcollection $D_1,\ldots, D_n$ in $\mathcal F$ such that \[E\subseteq \cup_{j=1}^n 3D_j.\] Here, if $D$ is th... | Let $\mathcal F$ be a finite collection of open discs in $\mathbb R^2$ whose union contains a set $E\subseteq \mathbb R^2$. Show that there is a pairwise disjoint subcollection $D_1,\ldots, D_n$ in $\mathcal F$ such that \[E\subseteq \cup_{j=1}^n 3D_j.\] Here, if $D$ is the disc of radius $r$ and center $P$, then $3D$... | null | [
"analysis"
] | null | null | |
putnam_1995_a6 | 2fcd9fcb-46ea-5256-b7a8-59fa62c551bf | train | theorem putnam_1995_a6
(SM : (n : β) β Set (Matrix (Fin 3) (Fin n) β€))
(Srowsums : (n : β) β Matrix (Fin 3) (Fin n) β€ β Set β€)
(sumsprog : (Set β€) β Prop)
(hSM : β n, SM n = {M : Matrix (Fin 3) (Fin n) β€ | β j : Fin n, {M i j | i : Fin 3} = {1, 2, 3}})
(hSrowsums : β n M, Srowsums n M = {β j : Fin n, M i j | i : Fin 3}... | import Mathlib
open Filter Topology Real
/--
Suppose that each of $n$ people writes down the numbers $1,2,3$ in random order in one column of a $3 \times n$ matrix, with all orders equally likely and with the orders for different columns independent of each other. Let the row sums $a,b,c$ of the resulting matrix be r... | import Mathlib
open Filter Topology Real
/--
Suppose that each of $n$ people writes down the numbers $1,2,3$ in random order in one column of a $3 \times n$ matrix, with all orders equally likely and with the orders for different columns independent of each other. Let the row sums $a,b,c$ of the resulting matrix be r... | Suppose that each of $n$ people writes down the numbers $1,2,3$ in random order in one column of a $3 \times n$ matrix, with all orders equally likely and with the orders for different columns independent of each other. Let the row sums $a,b,c$ of the resulting matrix be rearranged (if necessary) so that $a \leq b \leq... | null | [
"algebra"
] | null | null | |
putnam_2021_a6 | 39ac01e0-6ab1-57c5-82b2-bab484e8ef04 | train | abbrev putnam_2021_a6_solution : Prop := sorry
-- True
/--
Let $P(x)$ be a polynomial whose coefficients are all either $0$ or $1$. Suppose that $P(x)$ can be written as a product of two nonconstant polynomials with integer coefficients. Does it follow that $P(2)$ is a composite integer?
-/
theorem putnam_2021_a6 :
(... | import Mathlib
open Filter Topology
-- True
/--
Let $P(x)$ be a polynomial whose coefficients are all either $0$ or $1$. Suppose that $P(x)$ can be written as a product of two nonconstant polynomials with integer coefficients. Does it follow that $P(2)$ is a composite integer?
-/
theorem putnam_2021_a6 :
(β P, (β n... | import Mathlib
open Filter Topology
abbrev putnam_2021_a6_solution : Prop := sorry
-- True
/--
Let $P(x)$ be a polynomial whose coefficients are all either $0$ or $1$. Suppose that $P(x)$ can be written as a product of two nonconstant polynomials with integer coefficients. Does it follow that $P(2)$ is a composite in... | Let $P(x)$ be a polynomial whose coefficients are all either $0$ or $1$. Suppose that $P(x)$ can be written as a product of two nonconstant polynomials with integer coefficients. Does it follow that $P(2)$ is a composite integer? | Show that it does follow that $P(2)$ is a composite integer. | [
"number_theory",
"algebra"
] | null | null | |
putnam_1972_b6 | ce7cf111-3ea0-5ea4-bc90-c17ed42f145a | train | theorem putnam_1972_b6
(k : β)
(hk : k β₯ 1)
(n : Fin k β β€)
(hn : β i : Fin k, n i > 0)
(hn' : β i j : Fin k, i < j β n i < n j)
(zpoly : β β β)
(hzpoly : zpoly = fun z => 1 + β i : Fin k, z^(n i))
: β z : β, z β ball 0 ((Real.sqrt 5 - 1)/2) β zpoly z β 0 :=
sorry | import Mathlib
open EuclideanGeometry Filter Topology Set MeasureTheory Metric
/--
Let $n_1 < n_2 < \dots < n_k$ be a set of positive integers. Prove that the polynomail $1 + z^{n_1} + z^{n_2} + \dots + z^{n_k}$ has not roots inside the circle $|z| < (\frac{\sqrt{5}-1}{2}$.
-/
theorem putnam_1972_b6
(k : β)
(hk : k β₯... | import Mathlib
open EuclideanGeometry Filter Topology Set MeasureTheory Metric
/--
Let $n_1 < n_2 < \dots < n_k$ be a set of positive integers. Prove that the polynomail $1 + z^{n_1} + z^{n_2} + \dots + z^{n_k}$ has not roots inside the circle $|z| < (\frac{\sqrt{5}-1}{2}$.
-/
theorem putnam_1972_b6
(k : β)
(hk : k β₯... | Let $n_1 < n_2 < \dots < n_k$ be a set of positive integers. Prove that the polynomail $1 + z^{n_1} + z^{n_2} + \dots + z^{n_k}$ has not roots inside the circle $|z| < (\frac{\sqrt{5}-1}{2}$. | null | [
"analysis"
] | null | null | |
putnam_1988_b4 | f95e2134-7b38-5ee5-8868-503fc0946b22 | train | theorem putnam_1988_b4
(a : β β β)
(IsPosConv : (β β β) β Prop)
(IsPosConv_def : β a' : β β β, IsPosConv a' β
(β n β₯ 1, a' n > 0) β§
(β s : β, Tendsto (fun N : β => β n : Set.Icc 1 N, a' n) atTop (π s))) :
(IsPosConv a) β IsPosConv (fun n : β => (a n) ^ ((n : β) / (n + 1))) :=
sorry | import Mathlib
open Set Filter Topology
/--
Prove that if $\sum_{n=1}^\infty a_n$ is a convergent series of positive real numbers, then so is $\sum_{n=1}^\infty (a_n)^{n/(n+1)}$.
-/
theorem putnam_1988_b4
(a : β β β)
(IsPosConv : (β β β) β Prop)
(IsPosConv_def : β a' : β β β, IsPosConv a' β
(β n β₯ 1... | import Mathlib
open Set Filter Topology
/--
Prove that if $\sum_{n=1}^\infty a_n$ is a convergent series of positive real numbers, then so is $\sum_{n=1}^\infty (a_n)^{n/(n+1)}$.
-/
theorem putnam_1988_b4
(a : β β β)
(IsPosConv : (β β β) β Prop)
(IsPosConv_def : β a' : β β β, IsPosConv a' β
(β n β₯ 1... | Prove that if $\sum_{n=1}^\infty a_n$ is a convergent series of positive real numbers, then so is $\sum_{n=1}^\infty (a_n)^{n/(n+1)}$. | null | [
"analysis"
] | null | null | |
putnam_1996_a6 | 0b28e551-871f-53f9-bcf9-ed2c53fac06e | train | abbrev putnam_1996_a6_solution : β β Set (β β β) := sorry
-- (fun c : β => if c β€ 1 / 4 then {f : β β β | β d : β, β x : β, f x = d} else {f : β β β | ContinuousOn f (Set.Icc 0 c) β§ f 0 = f c β§ (β x > 0, f x = f (x ^ 2 + c)) β§ (β x < 0, f x = f (-x))})
/--
Let $c>0$ be a constant. Give a complete description, with proo... | import Mathlib
open Function
-- (fun c : β => if c β€ 1 / 4 then {f : β β β | β d : β, β x : β, f x = d} else {f : β β β | ContinuousOn f (Set.Icc 0 c) β§ f 0 = f c β§ (β x > 0, f x = f (x ^ 2 + c)) β§ (β x < 0, f x = f (-x))})
/--
Let $c>0$ be a constant. Give a complete description, with proof, of the set of all contin... | import Mathlib
open Function
abbrev putnam_1996_a6_solution : β β Set (β β β) := sorry
-- (fun c : β => if c β€ 1 / 4 then {f : β β β | β d : β, β x : β, f x = d} else {f : β β β | ContinuousOn f (Set.Icc 0 c) β§ f 0 = f c β§ (β x > 0, f x = f (x ^ 2 + c)) β§ (β x < 0, f x = f (-x))})
/--
Let $c>0$ be a constant. Give a ... | Let $c>0$ be a constant. Give a complete description, with proof, of the set of all continuous functions $f:\mathbb{R} \to \mathbb{R}$ such that $f(x)=f(x^2+c)$ for all $x \in \mathbb{R}$. | Show that if $c \leq 1/4$ then $f$ must be constant, and if $c>1/4$ then $f$ can be defined on $[0,c]$ as any continuous function with equal values on the endpoints, then extended to $x>c$ by the relation $f(x)=f(x^2+c)$, then extended further to $x<0$ by the relation $f(x)=f(-x)$. | [
"analysis",
"algebra"
] | null | null | |
putnam_2023_b5 | 1a5195a1-b018-5133-9c1e-3fcc1f16aa3c | train | abbrev putnam_2023_b5_solution : Set β := sorry
-- {n : β | n = 1 β¨ n β‘ 2 [MOD 4]}
/--
Determine which positive integers $n$ have the following property: For all integers $m$ that are relatively prime to $n$, there exists a permutation $\pi:\{1,2,\dots,n\} \to \{1,2,\dots,n\}$ such that $\pi(\pi(k)) \equiv mk \pmod{n}$... | import Mathlib
open Nat Topology Filter
-- {n : β | n = 1 β¨ n β‘ 2 [MOD 4]}
/--
Determine which positive integers $n$ have the following property: For all integers $m$ that are relatively prime to $n$, there exists a permutation $\pi:\{1,2,\dots,n\} \to \{1,2,\dots,n\}$ such that $\pi(\pi(k)) \equiv mk \pmod{n}$ for a... | import Mathlib
open Nat Topology Filter
abbrev putnam_2023_b5_solution : Set β := sorry
-- {n : β | n = 1 β¨ n β‘ 2 [MOD 4]}
/--
Determine which positive integers $n$ have the following property: For all integers $m$ that are relatively prime to $n$, there exists a permutation $\pi:\{1,2,\dots,n\} \to \{1,2,\dots,n\}$ ... | Determine which positive integers $n$ have the following property: For all integers $m$ that are relatively prime to $n$, there exists a permutation $\pi:\{1,2,\dots,n\} \to \{1,2,\dots,n\}$ such that $\pi(\pi(k)) \equiv mk \pmod{n}$ for all $k \in \{1,2,\dots,n\}$. | Show that the desired property holds if and only if $n=1$ or $n \equiv 2 \pmod{4}$. | [
"number_theory"
] | null | null | |
putnam_2006_b2 | 9325a8ef-9b4b-5248-8ac1-eff8d3178da5 | train | theorem putnam_2006_b2
(n : β)
(npos : n > 0)
(X : Finset β)
(hXcard : X.card = n)
: (β S β X, S β β
β§ β m : β€, |m + β s in S, s| β€ 1 / (n + 1)) :=
sorry | import Mathlib
/--
Prove that, for every set $X = \{x_1, x_2, \dots, x_n\}$ of $n$ real numbers, there exists a non-empty subset $S$ of $X$ and an integer $m$ such that
\[
\left| m + \sum_{s \in S} s \right| \leq \frac{1}{n+1}.
\]
-/
theorem putnam_2006_b2
(n : β)
(npos : n > 0)
(X : Finset β)
(hXcard : X.card = n)
: ... | import Mathlib
/--
Prove that, for every set $X = \{x_1, x_2, \dots, x_n\}$ of $n$ real numbers, there exists a non-empty subset $S$ of $X$ and an integer $m$ such that
\[
\left| m + \sum_{s \in S} s \right| \leq \frac{1}{n+1}.
\]
-/
theorem putnam_2006_b2
(n : β)
(npos : n > 0)
(X : Finset β)
(hXcard : X.card = n)
: ... | Prove that, for every set $X = \{x_1, x_2, \dots, x_n\}$ of $n$ real numbers, there exists a non-empty subset $S$ of $X$ and an integer $m$ such that
\[
\left| m + \sum_{s \in S} s \right| \leq \frac{1}{n+1}.
\] | null | [
"analysis"
] | null | null | |
putnam_2018_b3 | a672c119-54e2-523e-b731-cb8e538615f0 | train | abbrev putnam_2018_b3_solution : Set β := sorry
-- {2^2, 2^4, 2^16, 2^256}
/--
Find all positive integers $n < 10^{100}$ for which simultaneously $n$ divides $2^n$, $n-1$ divides $2^n-1$, and $n-2$ divides $2^n - 2$.
-/
theorem putnam_2018_b3
(n : β) (hn : n < 0) :
(n < 10^100 β§ ((n : β€) β£ (2^n : β€) β§ (n - 1 : β€) β£... | import Mathlib
-- {2^2, 2^4, 2^16, 2^256}
/--
Find all positive integers $n < 10^{100}$ for which simultaneously $n$ divides $2^n$, $n-1$ divides $2^n-1$, and $n-2$ divides $2^n - 2$.
-/
theorem putnam_2018_b3
(n : β) (hn : n < 0) :
(n < 10^100 β§ ((n : β€) β£ (2^n : β€) β§ (n - 1 : β€) β£ (2^n - 1 : β€) β§ (n - 2 : β€) β£ (... | import Mathlib
abbrev putnam_2018_b3_solution : Set β := sorry
-- {2^2, 2^4, 2^16, 2^256}
/--
Find all positive integers $n < 10^{100}$ for which simultaneously $n$ divides $2^n$, $n-1$ divides $2^n-1$, and $n-2$ divides $2^n - 2$.
-/
theorem putnam_2018_b3
(n : β) (hn : n < 0) :
(n < 10^100 β§ ((n : β€) β£ (2^n : β€)... | Find all positive integers $n < 10^{100}$ for which simultaneously $n$ divides $2^n$, $n-1$ divides $2^n-1$, and $n-2$ divides $2^n - 2$. | Show that the solution is the set $\{2^2, 2^4, 2^8, 2^16\}$. | [
"number_theory"
] | null | null | |
putnam_2008_b6 | 84945d81-b313-525f-a8c8-8960c12ac7bc | train | theorem putnam_2008_b6
(n k : β)
(hnk : n > 0 β§ k > 0)
: Odd (Set.ncard {s : Equiv.Perm (Fin n) | klimited k n s}) β (n β‘ 0 [MOD 2*k+1] β¨ n β‘ 1 [MOD 2*k+1]) :=
sorry | import Mathlib
open Filter Topology Set Nat
def klimited (k n : β) (s : Equiv.Perm (Fin n)) := β i, |((s i) : β€) - i| β€ k
/--
Let $n$ and $k$ be positive integers. Say that a permutation $\sigma$ of $\{1,2,\dots,n\} is $k-limited$ if \|\sigma(i) - i\| \leq k$ for all $i$. Prove that the number of $k-limited$ permutat... | import Mathlib
open Filter Topology Set Nat
def klimited (k n : β) (s : Equiv.Perm (Fin n)) := β i, |((s i) : β€) - i| β€ k
/--
Let $n$ and $k$ be positive integers. Say that a permutation $\sigma$ of $\{1,2,\dots,n\} is $k-limited$ if \|\sigma(i) - i\| \leq k$ for all $i$. Prove that the number of $k-limited$ permutat... | Let $n$ and $k$ be positive integers. Say that a permutation $\sigma$ of $\{1,2,\dots,n\} is $k-limited$ if \|\sigma(i) - i\| \leq k$ for all $i$. Prove that the number of $k-limited$ permutations $\{1,2,\dots,n\}$ is odd if and only if $n \equiv 0$ or $1 (mod 2k+1)$. | null | [
"number_theory"
] | null | null | |
putnam_1989_a6 | 3c93059f-1f14-5c56-89fa-8dde3b383ef9 | train | theorem putnam_1989_a6
(F : Type*) [Field F] [Fintype F]
(hF : Fintype.card F = 2)
(Ξ± : PowerSeries F)
(hΞ± : β n : β, let bin := [1] ++ (digits 2 n) ++ [1]; PowerSeries.coeff F n Ξ± = ite (β i j : Fin bin.length, i < j β bin.get i = 1 β bin.get j = 1 β (β k, i < k β k < j β bin.get k = 0) β Even ((j : β) - (i : β) - 1))... | import Mathlib
open Nat
/--
Let $\alpha=1+a_1x+a_2x^2+\cdots$ be a formal power series with coefficients in the field of two elements. Let
\[
a_n =
\begin{cases}
1 & \parbox{2in}{if every block of zeros in the binary expansion of $n$ has an even number of zeros in the block} \\[.3in]
0 & \text{otherwise.}
\end{cases}... | import Mathlib
open Nat
/--
Let $\alpha=1+a_1x+a_2x^2+\cdots$ be a formal power series with coefficients in the field of two elements. Let
\[
a_n =
\begin{cases}
1 & \parbox{2in}{if every block of zeros in the binary expansion of $n$ has an even number of zeros in the block} \\[.3in]
0 & \text{otherwise.}
\end{cases}... | Let $\alpha=1+a_1x+a_2x^2+\cdots$ be a formal power series with coefficients in the field of two elements. Let
\[
a_n =
\begin{cases}
1 & \parbox{2in}{if every block of zeros in the binary expansion of $n$ has an even number of zeros in the block} \\[.3in]
0 & \text{otherwise.}
\end{cases}
\]
(For example, $a_{36}=1$ b... | null | [
"algebra",
"abstract_algebra"
] | null | null | |
putnam_1973_a4 | 0bba6054-4a50-5ba7-9bcd-cf26b7a96936 | train | abbrev putnam_1973_a4_solution : β := sorry
-- 3
/--
How many zeros does the function $f(x) = 2^x - 1 - x^2$ have on the real line?
-/
theorem putnam_1973_a4
(f : β β β)
(hf : f = fun x => 2^x - 1 - x^2)
: putnam_1973_a4_solution = {x : β | f x = 0}.ncard :=
sorry | import Mathlib
open Nat Set MeasureTheory Topology Filter
-- 3
/--
How many zeros does the function $f(x) = 2^x - 1 - x^2$ have on the real line?
-/
theorem putnam_1973_a4
(f : β β β)
(hf : f = fun x => 2^x - 1 - x^2)
: putnam_1973_a4_solution = {x : β | f x = 0}.ncard := by
| import Mathlib
open Nat Set MeasureTheory Topology Filter
abbrev putnam_1973_a4_solution : β := sorry
-- 3
/--
How many zeros does the function $f(x) = 2^x - 1 - x^2$ have on the real line?
-/
theorem putnam_1973_a4
(f : β β β)
(hf : f = fun x => 2^x - 1 - x^2)
: putnam_1973_a4_solution = {x : β | f x = 0}.ncard :=
s... | How many zeros does the function $f(x) = 2^x - 1 - x^2$ have on the real line? | Show that the solution is 3. | [
"analysis"
] | null | null | |
putnam_2014_a4 | 98772546-3ebb-5b60-959a-568073c9d2b6 | train | abbrev putnam_2014_a4_solution : β := sorry
-- 1 / 3
/--
Suppose $X$ is a random variable that takes on only nonnegative integer values, with $E\left[X\right]=1$, $E\left[X^2\right]=2$, and $E\left[X^3\right]=5$. (Here $E\left[Y\right]$ denotes the expectation of the random variable $Y$.) Determine the smallest possibl... | import Mathlib
open Topology Filter Nat
-- 1 / 3
/--
Suppose $X$ is a random variable that takes on only nonnegative integer values, with $E\left[X\right]=1$, $E\left[X^2\right]=2$, and $E\left[X^3\right]=5$. (Here $E\left[Y\right]$ denotes the expectation of the random variable $Y$.) Determine the smallest possible ... | import Mathlib
open Topology Filter Nat
noncomputable abbrev putnam_2014_a4_solution : β := sorry
-- 1 / 3
/--
Suppose $X$ is a random variable that takes on only nonnegative integer values, with $E\left[X\right]=1$, $E\left[X^2\right]=2$, and $E\left[X^3\right]=5$. (Here $E\left[Y\right]$ denotes the expectation of ... | Suppose $X$ is a random variable that takes on only nonnegative integer values, with $E\left[X\right]=1$, $E\left[X^2\right]=2$, and $E\left[X^3\right]=5$. (Here $E\left[Y\right]$ denotes the expectation of the random variable $Y$.) Determine the smallest possible value of the probability of the event $X=0$. | Show that the answer is $\frac{1}{3}$. | [
"probability",
"analysis"
] | null | null | |
putnam_2002_b6 | decbcbc1-0884-52f7-b570-0108fe9b2d8d | train | theorem putnam_2002_b6
(p : β)
(hp : Nat.Prime p)
(M : Matrix (Fin 3) (Fin 3) (MvPolynomial (Fin 3) β€))
(hM : M = fun (r c : Fin 3) => ((X c)^(p^(r : β)) : MvPolynomial (Fin 3) β€))
(cong : β β MvPolynomial (Fin 3) β€ Γ MvPolynomial (Fin 3) β€ β Prop)
(hcong : cong = fun p : β => fun (f, g) => β n : Fin 3 ββ β, Int.ModEq ... | import Mathlib
open Nat Set Topology Filter Matrix MvPolynomial
/--
Let $p$ be a prime number. Prove that the determinant of the matrix
\[
\begin{pmatrix}
x & y & z \\
x^p & y^p & z^p \\
x^{p^2} & y^{p^2} & z^{p^2}
\end{pmatrix}
\]
is congruent modulo $p$ to a product of polynomials of the form
$ax+by+cz$, where $a,b... | import Mathlib
open Nat Set Topology Filter Matrix MvPolynomial
/--
Let $p$ be a prime number. Prove that the determinant of the matrix
\[
\begin{pmatrix}
x & y & z \\
x^p & y^p & z^p \\
x^{p^2} & y^{p^2} & z^{p^2}
\end{pmatrix}
\]
is congruent modulo $p$ to a product of polynomials of the form
$ax+by+cz$, where $a,b... | Let $p$ be a prime number. Prove that the determinant of the matrix
\[
\begin{pmatrix}
x & y & z \\
x^p & y^p & z^p \\
x^{p^2} & y^{p^2} & z^{p^2}
\end{pmatrix}
\]
is congruent modulo $p$ to a product of polynomials of the form
$ax+by+cz$, where $a,b,c$ are integers. (We say two integer
polynomials are congruent modulo... | null | [
"linear_algebra",
"number_theory",
"algebra"
] | null | null | |
putnam_1976_b5 | 9d59352d-d938-56b3-9aa6-946fad040835 | train | abbrev putnam_1976_b5_solution : β β Polynomial β€ := sorry
-- fun n => C (Nat.factorial n)
/--
Find $$\sum_{k=0}^{n} (-1)^k {n \choose k} (x - k)^n.$$
-/
theorem putnam_1976_b5
: β n : β, β k in Finset.range (n + 1), C ((-(1 : β€))^k * Nat.choose n k) * (X - (C (k : β€)))^n = putnam_1976_b5_solution n :=
sorry | import Mathlib
open Polynomial Filter Topology ProbabilityTheory MeasureTheory
-- fun n => C (Nat.factorial n)
/--
Find $$\sum_{k=0}^{n} (-1)^k {n \choose k} (x - k)^n.$$
-/
theorem putnam_1976_b5
: β n : β, β k in Finset.range (n + 1), C ((-(1 : β€))^k * Nat.choose n k) * (X - (C (k : β€)))^n = putnam_1976_b5_solution... | import Mathlib
open Polynomial Filter Topology ProbabilityTheory MeasureTheory
noncomputable abbrev putnam_1976_b5_solution : β β Polynomial β€ := sorry
-- fun n => C (Nat.factorial n)
/--
Find $$\sum_{k=0}^{n} (-1)^k {n \choose k} (x - k)^n.$$
-/
theorem putnam_1976_b5
: β n : β, β k in Finset.range (n + 1), C ((-(1 ... | Find $$\sum_{k=0}^{n} (-1)^k {n \choose k} (x - k)^n.$$ | The sum equals $n!$. | [
"algebra"
] | null | null | |
putnam_1974_b2 | 2bc1da42-a678-5228-b03b-df097d2a43ba | train | theorem putnam_1974_b2
(y : β β β)
(ycontdiff : ContDiff β 1 y)
(limy : Tendsto (fun x : β => (deriv y x)^2 + (y x)^3) atTop (π 0))
: Tendsto y atTop (π 0) β§ Tendsto (deriv y) atTop (π 0) :=
sorry | import Mathlib
open Set Nat Polynomial Filter Topology
/--
Let $y(x)$ be a continuously differentiable real-valued function of a real vairable $x$. Show that if $(y')^2 + y^3 \to 0$ as $x \to +\infty$, then $y(x)$ and $y'(x) \to 0$ as $x \to +\infty$.
-/
theorem putnam_1974_b2
(y : β β β)
(ycontdiff : ContDiff β 1 y)... | import Mathlib
open Set Nat Polynomial Filter Topology
/--
Let $y(x)$ be a continuously differentiable real-valued function of a real vairable $x$. Show that if $(y')^2 + y^3 \to 0$ as $x \to +\infty$, then $y(x)$ and $y'(x) \to 0$ as $x \to +\infty$.
-/
theorem putnam_1974_b2
(y : β β β)
(ycontdiff : ContDiff β 1 y)... | Let $y(x)$ be a continuously differentiable real-valued function of a real vairable $x$. Show that if $(y')^2 + y^3 \to 0$ as $x \to +\infty$, then $y(x)$ and $y'(x) \to 0$ as $x \to +\infty$. | null | [
"analysis"
] | null | null | |
putnam_2001_a5 | e335529b-36bd-54b2-b2ee-4a534369da4a | train | theorem putnam_2001_a5
: β! an : β€ Γ β, let (a, n) := an; a > 0 β§ n > 0 β§ a^(n+1) - (a+1)^n = 2001 :=
sorry | import Mathlib
open Topology Filter Polynomial Set
/--
Prove that there are unique positive integers $a$, $n$ such that $a^{n+1}-(a+1)^n=2001$.
-/
theorem putnam_2001_a5
: β! an : β€ Γ β, let (a, n) := an; a > 0 β§ n > 0 β§ a^(n+1) - (a+1)^n = 2001 := by
| import Mathlib
open Topology Filter Polynomial Set
/--
Prove that there are unique positive integers $a$, $n$ such that $a^{n+1}-(a+1)^n=2001$.
-/
theorem putnam_2001_a5
: β! an : β€ Γ β, let (a, n) := an; a > 0 β§ n > 0 β§ a^(n+1) - (a+1)^n = 2001 :=
sorry
| Prove that there are unique positive integers $a$, $n$ such that $a^{n+1}-(a+1)^n=2001$. | null | [
"number_theory"
] | null | null | |
putnam_2000_a6 | 02414c8b-71b9-50b9-b668-7c315c2b1b52 | train | theorem putnam_2000_a6
(f : Polynomial β€)
(a : β β β€)
(ha0 : a 0 = 0)
(ha : β n : β, a (n + 1) = f.eval (a n))
: ((β m > 0, a m = 0) β (a 1 = 0 β¨ a 2 = 0)) :=
sorry | import Mathlib
open Topology Filter
/--
Let $f(x)$ be a polynomial with integer coefficients. Define a sequence $a_0,a_1,\ldots$ of integers such that $a_0=0$ and $a_{n+1}=f(a_n)$ for all $n\geq 0$. Prove that if there exists a positive integer $m$ for which $a_m=0$ then either $a_1=0$ or $a_2=0$.
-/
theorem putnam_2... | import Mathlib
open Topology Filter
/--
Let $f(x)$ be a polynomial with integer coefficients. Define a sequence $a_0,a_1,\ldots$ of integers such that $a_0=0$ and $a_{n+1}=f(a_n)$ for all $n\geq 0$. Prove that if there exists a positive integer $m$ for which $a_m=0$ then either $a_1=0$ or $a_2=0$.
-/
theorem putnam_2... | Let $f(x)$ be a polynomial with integer coefficients. Define a sequence $a_0,a_1,\ldots$ of integers such that $a_0=0$ and $a_{n+1}=f(a_n)$ for all $n\geq 0$. Prove that if there exists a positive integer $m$ for which $a_m=0$ then either $a_1=0$ or $a_2=0$. | null | [
"algebra"
] | null | null | |
putnam_1997_b2 | 5853bc8c-4356-5e10-b751-0910fa17c787 | train | theorem putnam_1997_b2
(f g : β β β)
(hg : β x : β, g x β₯ 0)
(hfderiv1 : ContDiff β 1 f)
(hfderiv2 : Differentiable β (deriv f))
(hfg : β x : β, f x + iteratedDeriv 2 f x = -x * g x * deriv f x)
: IsBounded (range (fun x => |f x|)) :=
sorry | import Mathlib
open Filter Topology Bornology Set
/--
Let $f$ be a twice-differentiable real-valued function satisfying \[f(x)+f''(x)=-xg(x)f'(x),\] where $g(x)\geq 0$ for all real $x$. Prove that $|f(x)|$ is bounded.
-/
theorem putnam_1997_b2
(f g : β β β)
(hg : β x : β, g x β₯ 0)
(hfderiv1 : ContDiff β 1 f)
(hfderi... | import Mathlib
open Filter Topology Bornology Set
/--
Let $f$ be a twice-differentiable real-valued function satisfying \[f(x)+f''(x)=-xg(x)f'(x),\] where $g(x)\geq 0$ for all real $x$. Prove that $|f(x)|$ is bounded.
-/
theorem putnam_1997_b2
(f g : β β β)
(hg : β x : β, g x β₯ 0)
(hfderiv1 : ContDiff β 1 f)
(hfderi... | Let $f$ be a twice-differentiable real-valued function satisfying \[f(x)+f''(x)=-xg(x)f'(x),\] where $g(x)\geq 0$ for all real $x$. Prove that $|f(x)|$ is bounded. | null | [
"analysis"
] | null | null | |
putnam_1992_b1 | 0a2d5787-4803-58b5-8d02-02248df7c00d | train | abbrev putnam_1992_b1_solution : β β β€ := sorry
-- fun n β¦ 2 * n - 3
/--
Let $S$ be a set of $n$ distinct real numbers. Let $A_S$ be the set of numbers that occur as averages of two distinct elements of $S$. For a given $n \geq 2$, what is the smallest possible number of elements in $A_S$?
-/
theorem putnam_1992_b1
(... | import Mathlib
open Topology Filter Nat Function
-- fun n β¦ 2 * n - 3
/--
Let $S$ be a set of $n$ distinct real numbers. Let $A_S$ be the set of numbers that occur as averages of two distinct elements of $S$. For a given $n \geq 2$, what is the smallest possible number of elements in $A_S$?
-/
theorem putnam_1992_b1
... | import Mathlib
open Topology Filter Nat Function
abbrev putnam_1992_b1_solution : β β β€ := sorry
-- fun n β¦ 2 * n - 3
/--
Let $S$ be a set of $n$ distinct real numbers. Let $A_S$ be the set of numbers that occur as averages of two distinct elements of $S$. For a given $n \geq 2$, what is the smallest possible number ... | Let $S$ be a set of $n$ distinct real numbers. Let $A_S$ be the set of numbers that occur as averages of two distinct elements of $S$. For a given $n \geq 2$, what is the smallest possible number of elements in $A_S$? | Show that the answer is $2n - 3$. | [
"algebra"
] | null | null | |
putnam_1978_a3 | b0e4432a-e03d-5188-a4ed-2a9ba72fb074 | train | abbrev putnam_1978_a3_solution : β := sorry
-- 2
/--
Let $p(x) = 2(x^6 + 1) + 4(x^5 + x) + 3(x^4 + x^2) + 5x^3$. For $k$ with $0 < k < 5$, let
\[
I_k = \int_0^{\infty} \frac{x^k}{p(x)} \, dx.
\]
For which $k$ is $I_k$ smallest?
-/
theorem putnam_1978_a3
(p : Polynomial β)
(hp : p = 2 * (X ^ 6 + 1) + 4 * (X ^ 5 ... | import Mathlib
open Set Polynomial
-- 2
/--
Let $p(x) = 2(x^6 + 1) + 4(x^5 + x) + 3(x^4 + x^2) + 5x^3$. For $k$ with $0 < k < 5$, let
\[
I_k = \int_0^{\infty} \frac{x^k}{p(x)} \, dx.
\]
For which $k$ is $I_k$ smallest?
-/
theorem putnam_1978_a3
(p : Polynomial β)
(hp : p = 2 * (X ^ 6 + 1) + 4 * (X ^ 5 + X) + ... | import Mathlib
open Set Polynomial
abbrev putnam_1978_a3_solution : β := sorry
-- 2
/--
Let $p(x) = 2(x^6 + 1) + 4(x^5 + x) + 3(x^4 + x^2) + 5x^3$. For $k$ with $0 < k < 5$, let
\[
I_k = \int_0^{\infty} \frac{x^k}{p(x)} \, dx.
\]
For which $k$ is $I_k$ smallest?
-/
theorem putnam_1978_a3
(p : Polynomial β)
(h... | Let $p(x) = 2(x^6 + 1) + 4(x^5 + x) + 3(x^4 + x^2) + 5x^3$. For $k$ with $0 < k < 5$, let
\[
I_k = \int_0^{\infty} \frac{x^k}{p(x)} \, dx.
\]
For which $k$ is $I_k$ smallest? | Show that $I_k$ is smallest for $k = 2$. | [
"analysis",
"algebra"
] | null | null | |
putnam_1967_b2 | 17cd65b4-6ac9-5243-9aa2-95c2fb3a17d6 | train | theorem putnam_1967_b2
(p r A B C Ξ± Ξ² Ξ³ : β)
(prbound : 0 β€ p β§ p β€ 1 β§ 0 β€ r β§ r β€ 1)
(id1 : β x y : β, (p * x + (1 - p) * y) ^ 2 = A * x ^ 2 + B * x * y + C * y ^ 2)
(id2 : β x y : β, (p * x + (1 - p) * y) * (r * x + (1 - r) * y) = Ξ± * x ^ 2 + Ξ² * x * y + Ξ³ * y ^ 2)
: max (max A B) C β₯ 4 / 9 β§ max (max Ξ± Ξ²) Ξ³ β₯ 4 / 9... | import Mathlib
open Nat Topology Filter
/--
Let $0 \leq p \leq 1$ and $0 \leq r \leq 1$ and consider the identities
\begin{enumerate}
\item[(a)] $(px+(1-p)y)^2=Ax^2+Bxy+Cy^2$,
\item[(b)] $(px+(1-p)y)(rx+(1-r)y)=\alpha x^2+\beta xy+\gamma y^2$.
\end{enumerate}
Show that (with respect to $p$ and $r$)
\begin{enumerate}
... | import Mathlib
open Nat Topology Filter
/--
Let $0 \leq p \leq 1$ and $0 \leq r \leq 1$ and consider the identities
\begin{enumerate}
\item[(a)] $(px+(1-p)y)^2=Ax^2+Bxy+Cy^2$,
\item[(b)] $(px+(1-p)y)(rx+(1-r)y)=\alpha x^2+\beta xy+\gamma y^2$.
\end{enumerate}
Show that (with respect to $p$ and $r$)
\begin{enumerate}
... | Let $0 \leq p \leq 1$ and $0 \leq r \leq 1$ and consider the identities
\begin{enumerate}
\item[(a)] $(px+(1-p)y)^2=Ax^2+Bxy+Cy^2$,
\item[(b)] $(px+(1-p)y)(rx+(1-r)y)=\alpha x^2+\beta xy+\gamma y^2$.
\end{enumerate}
Show that (with respect to $p$ and $r$)
\begin{enumerate}
\item[(a)] $\max\{A,B,C\} \geq 4/9$,
\item[(b)... | null | [
"algebra"
] | null | null | |
putnam_1995_b4 | ff6da42a-7c4e-5d08-a542-6a1d101621e7 | train | abbrev putnam_1995_b4_solution : β€ Γ β€ Γ β€ Γ β€ := sorry
-- β¨3,1,5,2β©
/--
Evaluate \[ \sqrt[8]{2207 - \frac{1}{2207-\frac{1}{2207-\dots}}}. \] Express your answer in the form $\frac{a+b\sqrt{c}}{d}$, where $a,b,c,d$ are integers.
-/
theorem putnam_1995_b4
(contfrac : β)
(hcontfrac : contfrac = 2207 - 1 / contfra... | import Mathlib
open Filter Topology Real Nat
-- β¨3,1,5,2β©
/--
Evaluate \[ \sqrt[8]{2207 - \frac{1}{2207-\frac{1}{2207-\dots}}}. \] Express your answer in the form $\frac{a+b\sqrt{c}}{d}$, where $a,b,c,d$ are integers.
-/
theorem putnam_1995_b4
(contfrac : β)
(hcontfrac : contfrac = 2207 - 1 / contfrac)
(h... | import Mathlib
open Filter Topology Real Nat
abbrev putnam_1995_b4_solution : β€ Γ β€ Γ β€ Γ β€ := sorry
-- β¨3,1,5,2β©
/--
Evaluate \[ \sqrt[8]{2207 - \frac{1}{2207-\frac{1}{2207-\dots}}}. \] Express your answer in the form $\frac{a+b\sqrt{c}}{d}$, where $a,b,c,d$ are integers.
-/
theorem putnam_1995_b4
(contfrac : β)... | Evaluate \[ \sqrt[8]{2207 - \frac{1}{2207-\frac{1}{2207-\dots}}}. \] Express your answer in the form $\frac{a+b\sqrt{c}}{d}$, where $a,b,c,d$ are integers. | Show that the solution is $(3 + 1*\sqrt{5})/2. | [
"algebra"
] | null | null | |
putnam_2008_a6 | fef4518c-5305-524a-adf9-1cfed1923d57 | train | theorem putnam_2008_a6
: β c : β, c > 0 β§ β G [Group G] (fing : Fintype G), fing.card > 1 β β s : List G, s.length β€ c * Real.log (fing.card : β) β§ β g : G, β t : List G, t.Sublist s β§ t.prod = g :=
sorry | import Mathlib
open Filter Topology Set
/--
Prove that there exists a constant $c>0$ such that in every nontrivial finite group $G$ there exists a sequence of length at most $c \log |G|$ with the property that each element of $G$ equals the product of some subsequence. (The elements of $G$ in the sequence are not req... | import Mathlib
open Filter Topology Set
/--
Prove that there exists a constant $c>0$ such that in every nontrivial finite group $G$ there exists a sequence of length at most $c \log |G|$ with the property that each element of $G$ equals the product of some subsequence. (The elements of $G$ in the sequence are not req... | Prove that there exists a constant $c>0$ such that in every nontrivial finite group $G$ there exists a sequence of length at most $c \log |G|$ with the property that each element of $G$ equals the product of some subsequence. (The elements of $G$ in the sequence are not required to be distinct. A \emph{subsequence} of ... | null | [
"abstract_algebra"
] | null | null | |
putnam_2009_b5 | 4e0ec67b-bff6-5a5d-9f02-fe9cd9bd4f63 | train | theorem putnam_2009_b5
(f : β β β)
(hfdiff : DifferentiableOn β f (Ioi 1))
(hf : β x > 1, deriv f x = (x ^ 2 - (f x) ^ 2) / ((x ^ 2) * ((f x) ^ 2 + 1)))
: (Tendsto f atTop atTop) :=
sorry | import Mathlib
open Topology MvPolynomial Filter Set Metric
/--
Let $f: (1, \infty) \to \mathbb{R}$ be a differentiable function such that
\[
f'(x) = \frac{x^2 - f(x)^2}{x^2 (f(x)^2 + 1)}
\qquad \mbox{for all $x>1$.}
\]
Prove that $\lim_{x \to \infty} f(x) = \infty$.
-/
theorem putnam_2009_b5
(f : β β β)
(hfdiff : D... | import Mathlib
open Topology MvPolynomial Filter Set Metric
/--
Let $f: (1, \infty) \to \mathbb{R}$ be a differentiable function such that
\[
f'(x) = \frac{x^2 - f(x)^2}{x^2 (f(x)^2 + 1)}
\qquad \mbox{for all $x>1$.}
\]
Prove that $\lim_{x \to \infty} f(x) = \infty$.
-/
theorem putnam_2009_b5
(f : β β β)
(hfdiff : D... | Let $f: (1, \infty) \to \mathbb{R}$ be a differentiable function such that
\[
f'(x) = \frac{x^2 - f(x)^2}{x^2 (f(x)^2 + 1)}
\qquad \mbox{for all $x>1$.}
\]
Prove that $\lim_{x \to \infty} f(x) = \infty$. | null | [
"analysis"
] | null | null | |
putnam_2020_b6 | d45de3d5-9f86-5a0e-bef4-90f27cf0bb73 | train | theorem putnam_2020_b6
(n : β)
(npos : n > 0)
: β k in Finset.Icc 1 n, ((-1) ^ Int.floor (k * (Real.sqrt 2 - 1)) : β) β₯ 0 :=
sorry | import Mathlib
open Filter Topology Set
/--
Let $n$ be a positive integer. Prove that $\sum_{k=1}^n(-1)^{\lfloor k(\sqrt{2}-1) \rfloor} \geq 0$.
-/
theorem putnam_2020_b6
(n : β)
(npos : n > 0)
: β k in Finset.Icc 1 n, ((-1) ^ Int.floor (k * (Real.sqrt 2 - 1)) : β) β₯ 0 := by
| import Mathlib
open Filter Topology Set
/--
Let $n$ be a positive integer. Prove that $\sum_{k=1}^n(-1)^{\lfloor k(\sqrt{2}-1) \rfloor} \geq 0$.
-/
theorem putnam_2020_b6
(n : β)
(npos : n > 0)
: β k in Finset.Icc 1 n, ((-1) ^ Int.floor (k * (Real.sqrt 2 - 1)) : β) β₯ 0 :=
sorry
| Let $n$ be a positive integer. Prove that $\sum_{k=1}^n(-1)^{\lfloor k(\sqrt{2}-1) \rfloor} \geq 0$. | null | [
"algebra"
] | null | null | |
putnam_2018_a2 | f635cde6-8968-5cfc-8f5d-c457bf1308a7 | train | abbrev putnam_2018_a2_solution : β β β := sorry
-- (fun n : β => if n = 1 then 1 else -1)
/--
Let \( S_1, S_2, \ldots, S_{2^n-1} \) be the nonempty subsets of \( \{1, 2, \ldots, n\} \) in some order, and let \( M \) be the \( (2^n - 1) \times (2^n - 1) \) matrix whose \((i, j)\) entry is $m_{ij} = \begin{cases} 0 & \te... | import Mathlib
-- (fun n : β => if n = 1 then 1 else -1)
/--
Let \( S_1, S_2, \ldots, S_{2^n-1} \) be the nonempty subsets of \( \{1, 2, \ldots, n\} \) in some order, and let \( M \) be the \( (2^n - 1) \times (2^n - 1) \) matrix whose \((i, j)\) entry is $m_{ij} = \begin{cases} 0 & \text{if } S_i \cap S_j = \emptyset... | import Mathlib
abbrev putnam_2018_a2_solution : β β β := sorry
-- (fun n : β => if n = 1 then 1 else -1)
/--
Let \( S_1, S_2, \ldots, S_{2^n-1} \) be the nonempty subsets of \( \{1, 2, \ldots, n\} \) in some order, and let \( M \) be the \( (2^n - 1) \times (2^n - 1) \) matrix whose \((i, j)\) entry is $m_{ij} = \begi... | Let \( S_1, S_2, \ldots, S_{2^n-1} \) be the nonempty subsets of \( \{1, 2, \ldots, n\} \) in some order, and let \( M \) be the \( (2^n - 1) \times (2^n - 1) \) matrix whose \((i, j)\) entry is $m_{ij} = \begin{cases} 0 & \text{if } S_i \cap S_j = \emptyset; \\ 1 & \text{otherwise}. \end{cases} $ Calculate the determi... | Show that the solution is 1 if n = 1, and otherwise -1. | [
"linear_algebra"
] | null | null | |
putnam_2014_b2 | eb1eff11-97dc-504a-98ff-bc32dd98e280 | train | abbrev putnam_2014_b2_solution : β := sorry
-- Real.log (4 / 3)
/--
Suppose that \( f \) is a function on the interval \([1,3]\) such that \(-1 \leq f(x) \leq 1\) for all \( x \) and \( \int_{1}^{3} f(x) \, dx = 0 \). How large can \(\int_{1}^{3} \frac{f(x)}{x} \, dx \) be?
-/
theorem putnam_2014_b2 :
IsGreatest {t |... | import Mathlib
open Topology Filter Nat
-- Note: uses (β β β) instead of (Set.Icc (1 : β) 3 β β)
-- Real.log (4 / 3)
/--
Suppose that \( f \) is a function on the interval \([1,3]\) such that \(-1 \leq f(x) \leq 1\) for all \( x \) and \( \int_{1}^{3} f(x) \, dx = 0 \). How large can \(\int_{1}^{3} \frac{f(x)}{x} \, ... | import Mathlib
open Topology Filter Nat
-- Note: uses (β β β) instead of (Set.Icc (1 : β) 3 β β)
noncomputable abbrev putnam_2014_b2_solution : β := sorry
-- Real.log (4 / 3)
/--
Suppose that \( f \) is a function on the interval \([1,3]\) such that \(-1 \leq f(x) \leq 1\) for all \( x \) and \( \int_{1}^{3} f(x) \, ... | Suppose that \( f \) is a function on the interval \([1,3]\) such that \(-1 \leq f(x) \leq 1\) for all \( x \) and \( \int_{1}^{3} f(x) \, dx = 0 \). How large can \(\int_{1}^{3} \frac{f(x)}{x} \, dx \) be? | Show that the solution is log (4 / 3). | [
"analysis"
] | null | null | |
putnam_1987_a1 | ec9f9a30-687f-5913-9bb9-9ace43f412c8 | train | theorem putnam_1987_a1
(A B C D : Set (β Γ β))
(hA : A = {(x, y) : β Γ β | x ^ 2 + y ^ 2 β 0 β§ x ^ 2 - y ^ 2 = x / (x ^ 2 + y ^ 2)})
(hB : B = {(x, y) : β Γ β | x ^ 2 + y ^ 2 β 0 β§ 2 * x * y + y / (x ^ 2 + y ^ 2) = 3})
(hC : C = {(x, y) : β Γ β | x ^ 3 - 3 * x * y ^ 2 + 3 * y = 1})
(hD : D = {(x, y) : β Γ β | 3 * x ^ 2... | import Mathlib
/--
Curves $A$, $B$, $C$, and $D$ are defined in the plane as follows:
\begin{align*}
A&=\left\{ (x,y):x^2-y^2=\frac{x}{x^2+y^2} \right\}, \\
B&=\left\{ (x,y):2xy+\frac{y}{x^2+y^2}=3 \right\}, \\
C&=\left\{ (x,y):x^3-3xy^2+3y=1 \right\}, \\
D&=\left\{ (x,y):3x^2y-3x-y^3=0 \right\}.
\end{align*}
Prove th... | import Mathlib
/--
Curves $A$, $B$, $C$, and $D$ are defined in the plane as follows:
\begin{align*}
A&=\left\{ (x,y):x^2-y^2=\frac{x}{x^2+y^2} \right\}, \\
B&=\left\{ (x,y):2xy+\frac{y}{x^2+y^2}=3 \right\}, \\
C&=\left\{ (x,y):x^3-3xy^2+3y=1 \right\}, \\
D&=\left\{ (x,y):3x^2y-3x-y^3=0 \right\}.
\end{align*}
Prove th... | Curves $A$, $B$, $C$, and $D$ are defined in the plane as follows:
\begin{align*}
A&=\left\{ (x,y):x^2-y^2=\frac{x}{x^2+y^2} \right\}, \\
B&=\left\{ (x,y):2xy+\frac{y}{x^2+y^2}=3 \right\}, \\
C&=\left\{ (x,y):x^3-3xy^2+3y=1 \right\}, \\
D&=\left\{ (x,y):3x^2y-3x-y^3=0 \right\}.
\end{align*}
Prove that $A \cap B=C \cap ... | null | [
"algebra"
] | null | null | |
putnam_1996_b3 | 3e3bc971-6258-57f9-9978-2613ff827f51 | train | abbrev putnam_1996_b3_solution : β β β := sorry
-- (fun n : β => (2 * n ^ 3 + 3 * n ^ 2 - 11 * n + 18) / 6)
/--
Given that $\{x_1,x_2,\ldots,x_n\}=\{1,2,\ldots,n\}$, find, with proof, the largest possible value, as a function of $n$ (with $n \geq 2$), of $x_1x_2+x_2x_3+\cdots+x_{n-1}x_n+x_nx_1$.
-/
theorem putnam_1996_... | import Mathlib
open Function
-- Note: uses (β β β) instead of (Fin n β β)
-- (fun n : β => (2 * n ^ 3 + 3 * n ^ 2 - 11 * n + 18) / 6)
/--
Given that $\{x_1,x_2,\ldots,x_n\}=\{1,2,\ldots,n\}$, find, with proof, the largest possible value, as a function of $n$ (with $n \geq 2$), of $x_1x_2+x_2x_3+\cdots+x_{n-1}x_n+x_nx... | import Mathlib
open Function
-- Note: uses (β β β) instead of (Fin n β β)
abbrev putnam_1996_b3_solution : β β β := sorry
-- (fun n : β => (2 * n ^ 3 + 3 * n ^ 2 - 11 * n + 18) / 6)
/--
Given that $\{x_1,x_2,\ldots,x_n\}=\{1,2,\ldots,n\}$, find, with proof, the largest possible value, as a function of $n$ (with $n \g... | Given that $\{x_1,x_2,\ldots,x_n\}=\{1,2,\ldots,n\}$, find, with proof, the largest possible value, as a function of $n$ (with $n \geq 2$), of $x_1x_2+x_2x_3+\cdots+x_{n-1}x_n+x_nx_1$. | Show that the maximum is $(2n^3+3n^2-11n+18)/6$. | [
"algebra"
] | null | null | |
putnam_1994_b4 | 5f231a3b-9df6-566f-a4f9-33ffdb53c327 | train | theorem putnam_1994_b4
(matgcd : Matrix (Fin 2) (Fin 2) β€ β β€)
(A : Matrix (Fin 2) (Fin 2) β€)
(d : β β β€)
(hmatgcd : β M, matgcd M = Int.gcd (Int.gcd (Int.gcd (M 0 0) (M 0 1)) (M 1 0)) (M 1 1))
(hA : A 0 0 = 3 β§ A 0 1 = 2 β§ A 1 0 = 4 β§ A 1 1 = 3)
(hd : β n β₯ 1, d n = matgcd (A ^ n - 1))
: Tendsto d atTop atTop :=
sorry | import Mathlib
open Filter Topology
/--
For $n \geq 1$, let $d_n$ be the greatest common divisor of the entries of $A^n-I$, where $A=\begin{pmatrix} 3 & 2 \\ 4 & 3 \end{pmatrix}$ and $I=\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$. Show that $\lim_{n \to \infty} d_n=\infty$.
-/
theorem putnam_1994_b4
(matgcd : Matri... | import Mathlib
open Filter Topology
/--
For $n \geq 1$, let $d_n$ be the greatest common divisor of the entries of $A^n-I$, where $A=\begin{pmatrix} 3 & 2 \\ 4 & 3 \end{pmatrix}$ and $I=\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$. Show that $\lim_{n \to \infty} d_n=\infty$.
-/
theorem putnam_1994_b4
(matgcd : Matri... | For $n \geq 1$, let $d_n$ be the greatest common divisor of the entries of $A^n-I$, where $A=\begin{pmatrix} 3 & 2 \\ 4 & 3 \end{pmatrix}$ and $I=\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$. Show that $\lim_{n \to \infty} d_n=\infty$. | null | [
"linear_algebra",
"number_theory",
"analysis"
] | null | null | |
putnam_2004_a4 | f293844d-e392-5bb6-a90b-7da7ee04be79 | train | theorem putnam_2004_a4
(n : β) (npos : n > 0)
(x : Fin n β β)
(avals : β β (β β Fin n β β) β Prop)
(havals : β N a, avals N a β β (i : Fin N) (j : Fin n), (a i j = -1 β¨ a i j = 0 β¨ a i j = 1)) :
β (N : β) (c : Fin N β β) (a : β β Fin n β β),
avals N a β§
(β i : Fin n, x i) = β i : Fin N, c i * (β j : F... | import Mathlib
open Nat Topology Filter
-- Note: uses (β β Fin n β β) instead of (Fin N β Fin n β β)
/--
Show that for any positive integer $n$ there is an integer $N$ such that the product $x_1x_2 \cdots x_n$ can be expressed identically in the form $x_1x_2 \cdots x_n=\sum_{i=1}^Nc_i(a_{i1}x_1+a_{i2}x_2+\cdots+a_{in... | import Mathlib
open Nat Topology Filter
-- Note: uses (β β Fin n β β) instead of (Fin N β Fin n β β)
/--
Show that for any positive integer $n$ there is an integer $N$ such that the product $x_1x_2 \cdots x_n$ can be expressed identically in the form $x_1x_2 \cdots x_n=\sum_{i=1}^Nc_i(a_{i1}x_1+a_{i2}x_2+\cdots+a_{in... | Show that for any positive integer $n$ there is an integer $N$ such that the product $x_1x_2 \cdots x_n$ can be expressed identically in the form $x_1x_2 \cdots x_n=\sum_{i=1}^Nc_i(a_{i1}x_1+a_{i2}x_2+\cdots+a_{in}x_n)^n$ where the $c_i$ are rational numbers and each $a_{ij}$ is one of the numbers $-1,0,1$. | null | [
"algebra"
] | null | null | |
putnam_2012_a2 | 4b49224b-73e8-584a-adb4-78e7d8776c9b | train | theorem putnam_2012_a2
(S : Type*) [CommSemigroup S]
(a b c : S)
(hS : β x y : S, β z : S, x * z = y)
(habc : a * c = b * c)
: a = b :=
sorry | import Mathlib
open Matrix
/--
Let $*$ be a commutative and associative binary operation on a set $S$. Assume that for every $x$ and $y$ in $S$, there exists $z$ in $S$ such that $x*z=y$. (This $z$ may depend on $x$ and $y$.) Show that if $a,b,c$ are in $S$ and $a*c=b*c$, then $a=b$.
-/
theorem putnam_2012_a2
(S : Ty... | import Mathlib
open Matrix
/--
Let $*$ be a commutative and associative binary operation on a set $S$. Assume that for every $x$ and $y$ in $S$, there exists $z$ in $S$ such that $x*z=y$. (This $z$ may depend on $x$ and $y$.) Show that if $a,b,c$ are in $S$ and $a*c=b*c$, then $a=b$.
-/
theorem putnam_2012_a2
(S : Ty... | Let $*$ be a commutative and associative binary operation on a set $S$. Assume that for every $x$ and $y$ in $S$, there exists $z$ in $S$ such that $x*z=y$. (This $z$ may depend on $x$ and $y$.) Show that if $a,b,c$ are in $S$ and $a*c=b*c$, then $a=b$. | null | [
"abstract_algebra"
] | null | null | |
putnam_1990_b3 | 121f7794-1ff4-5198-a15e-1c4ab9fe92fb | train | theorem putnam_1990_b3
(S : Set (Matrix (Fin 2) (Fin 2) β))
(hS : β A β S, β i j : Fin 2, (β x : β€, A i j = x ^ 2) β§ A i j β€ 200)
: (S.encard > 50387) β (β A β S, β B β S, A β B β§ A * B = B * A) :=
sorry | import Mathlib
open Filter Topology Nat
/--
Let $S$ be a set of $2 \times 2$ integer matrices whose entries $a_{ij}$ (1) are all squares of integers, and, (2) satisfy $a_{ij} \leq 200$. Show that if $S$ has more than $50387$ ($=15^4-15^2-15+2$) elements, then it has two elements that commute.
-/
theorem putnam_1990_b... | import Mathlib
open Filter Topology Nat
/--
Let $S$ be a set of $2 \times 2$ integer matrices whose entries $a_{ij}$ (1) are all squares of integers, and, (2) satisfy $a_{ij} \leq 200$. Show that if $S$ has more than $50387$ ($=15^4-15^2-15+2$) elements, then it has two elements that commute.
-/
theorem putnam_1990_b... | Let $S$ be a set of $2 \times 2$ integer matrices whose entries $a_{ij}$ (1) are all squares of integers, and, (2) satisfy $a_{ij} \leq 200$. Show that if $S$ has more than $50387$ ($=15^4-15^2-15+2$) elements, then it has two elements that commute. | null | [
"linear_algebra"
] | null | null | |
putnam_2019_a3 | ea19e0cf-e487-52fe-8964-7badda686f1f | train | abbrev putnam_2019_a3_solution : β := sorry
-- 2019^(-(1:β)/2019)
/--
Given real numbers $b_0, b_1, \dots, b_{2019}$ with $b_{2019} \neq 0$, let $z_1,z_2,\dots,z_{2019}$ be
the roots in the complex plane of the polynomial
\[
P(z) = \sum_{k=0}^{2019} b_k z^k.
\]
Let $\mu = (|z_1| + \cdots + |z_{2019}|)/2019$ be the aver... | import Mathlib
open Topology Filter
-- 2019^(-(1:β)/2019)
/--
Given real numbers $b_0, b_1, \dots, b_{2019}$ with $b_{2019} \neq 0$, let $z_1,z_2,\dots,z_{2019}$ be
the roots in the complex plane of the polynomial
\[
P(z) = \sum_{k=0}^{2019} b_k z^k.
\]
Let $\mu = (|z_1| + \cdots + |z_{2019}|)/2019$ be the average of... | import Mathlib
open Topology Filter
noncomputable abbrev putnam_2019_a3_solution : β := sorry
-- 2019^(-(1:β)/2019)
/--
Given real numbers $b_0, b_1, \dots, b_{2019}$ with $b_{2019} \neq 0$, let $z_1,z_2,\dots,z_{2019}$ be
the roots in the complex plane of the polynomial
\[
P(z) = \sum_{k=0}^{2019} b_k z^k.
\]
Let $\... | Given real numbers $b_0, b_1, \dots, b_{2019}$ with $b_{2019} \neq 0$, let $z_1,z_2,\dots,z_{2019}$ be
the roots in the complex plane of the polynomial
\[
P(z) = \sum_{k=0}^{2019} b_k z^k.
\]
Let $\mu = (|z_1| + \cdots + |z_{2019}|)/2019$ be the average of the distances from $z_1,z_2,\dots,z_{2019}$ to the origin. Dete... | The answer is $M = 2019^{-1/2019}$. | [
"algebra"
] | null | null | |
putnam_1988_b3 | f7f87ddf-92fc-51c1-bebe-d611c193f397 | train | abbrev putnam_1988_b3_solution : β := sorry
-- (1 + Real.sqrt 3) / 2
/--
For every $n$ in the set $N=\{1,2,\dots\}$ of positive integers, let $r_n$ be the minimum value of $|c-d \sqrt{3}|$ for all nonnegative integers $c$ and $d$ with $c+d=n$. Find, with proof, the smallest positive real number $g$ with $r_n \leq g$ fo... | import Mathlib
open Set Filter Topology
-- (1 + Real.sqrt 3) / 2
/--
For every $n$ in the set $N=\{1,2,\dots\}$ of positive integers, let $r_n$ be the minimum value of $|c-d \sqrt{3}|$ for all nonnegative integers $c$ and $d$ with $c+d=n$. Find, with proof, the smallest positive real number $g$ with $r_n \leq g$ for ... | import Mathlib
open Set Filter Topology
noncomputable abbrev putnam_1988_b3_solution : β := sorry
-- (1 + Real.sqrt 3) / 2
/--
For every $n$ in the set $N=\{1,2,\dots\}$ of positive integers, let $r_n$ be the minimum value of $|c-d \sqrt{3}|$ for all nonnegative integers $c$ and $d$ with $c+d=n$. Find, with proof, th... | For every $n$ in the set $N=\{1,2,\dots\}$ of positive integers, let $r_n$ be the minimum value of $|c-d \sqrt{3}|$ for all nonnegative integers $c$ and $d$ with $c+d=n$. Find, with proof, the smallest positive real number $g$ with $r_n \leq g$ for all $n \in N$. | Show that the smallest such $g$ is $(1+\sqrt{3})/2$. | [
"algebra"
] | null | null | |
putnam_1998_b6 | 59a6c8ff-b1d2-518b-90f7-8cc2887d3acf | train | theorem putnam_1998_b6
: β a b c : β€, β n : β€, n > 0 β§ Β¬(β k : β€, k = Real.sqrt (n^3 + a * n^2 + b * n + c)) :=
sorry | import Mathlib
open Set Function Metric
/--
Prove that, for any integers $a, b, c$, there exists a positive integer $n$ such that $\sqrt{n^3+an^2+bn+c}$ is not an integer.
-/
theorem putnam_1998_b6
: β a b c : β€, β n : β€, n > 0 β§ Β¬(β k : β€, k = Real.sqrt (n^3 + a * n^2 + b * n + c)) := by
| import Mathlib
open Set Function Metric
/--
Prove that, for any integers $a, b, c$, there exists a positive integer $n$ such that $\sqrt{n^3+an^2+bn+c}$ is not an integer.
-/
theorem putnam_1998_b6
: β a b c : β€, β n : β€, n > 0 β§ Β¬(β k : β€, k = Real.sqrt (n^3 + a * n^2 + b * n + c)) :=
sorry
| Prove that, for any integers $a, b, c$, there exists a positive integer $n$ such that $\sqrt{n^3+an^2+bn+c}$ is not an integer. | null | [
"number_theory"
] | null | null | |
putnam_2013_b3 | 79328774-9835-59c0-8354-c438fce75ad9 | train | abbrev putnam_2013_b3_solution : Prop := sorry
-- True
/--
Let $\mathcal{P}$ be a nonempty collection of subsets of $\{1,\dots, n\}$ such that: \begin{enumerate} \item[(i)] if $S, S' \in \mathcal{P}$, then $S \cup S' \in \mathcal{P}$ and $S \cap S' \in \mathcal{P}$, and \item[(ii)] if $S \in \mathcal{P}$ and $S \neq \e... | import Mathlib
open Function Set
-- True
/--
Let $\mathcal{P}$ be a nonempty collection of subsets of $\{1,\dots, n\}$ such that: \begin{enumerate} \item[(i)] if $S, S' \in \mathcal{P}$, then $S \cup S' \in \mathcal{P}$ and $S \cap S' \in \mathcal{P}$, and \item[(ii)] if $S \in \mathcal{P}$ and $S \neq \emptyset$, th... | import Mathlib
open Function Set
abbrev putnam_2013_b3_solution : Prop := sorry
-- True
/--
Let $\mathcal{P}$ be a nonempty collection of subsets of $\{1,\dots, n\}$ such that: \begin{enumerate} \item[(i)] if $S, S' \in \mathcal{P}$, then $S \cup S' \in \mathcal{P}$ and $S \cap S' \in \mathcal{P}$, and \item[(ii)] if... | Let $\mathcal{P}$ be a nonempty collection of subsets of $\{1,\dots, n\}$ such that: \begin{enumerate} \item[(i)] if $S, S' \in \mathcal{P}$, then $S \cup S' \in \mathcal{P}$ and $S \cap S' \in \mathcal{P}$, and \item[(ii)] if $S \in \mathcal{P}$ and $S \neq \emptyset$, then there is a subset $T \subset S$ such that $T... | null | [
"set_theory"
] | null | null | |
putnam_1987_b5 | dc6dbfc1-9825-596f-9cf4-2d29e9073921 | train | theorem putnam_1987_b5
(n : β)
(npos : n > 0)
(M : Matrix (Fin (2 * n)) (Fin n) β)
(hM : β z : Matrix (Fin 1) (Fin (2 * n)) β, z * M = 0 β (Β¬β i : Fin (2 * n), z 0 i = 0) β β i : Fin (2 * n), (z 0 i).im β 0)
: (β r : Matrix (Fin (2 * n)) (Fin 1) β, β w : Matrix (Fin n) (Fin 1) β, β i : (Fin (2 * n)), ((M * w) i 0).re =... | import Mathlib
open MvPolynomial Real Nat Filter Topology
/--
Let $O_n$ be the $n$-dimensional vector $(0,0,\cdots, 0)$. Let $M$ be a $2n \times n$ matrix of complex numbers such that whenever $(z_1, z_2, \dots, z_{2n})M = O_n$, with complex $z_i$, not all zero, then at least one of the $z_i$ is not real. Prove that ... | import Mathlib
open MvPolynomial Real Nat Filter Topology
/--
Let $O_n$ be the $n$-dimensional vector $(0,0,\cdots, 0)$. Let $M$ be a $2n \times n$ matrix of complex numbers such that whenever $(z_1, z_2, \dots, z_{2n})M = O_n$, with complex $z_i$, not all zero, then at least one of the $z_i$ is not real. Prove that ... | Let $O_n$ be the $n$-dimensional vector $(0,0,\cdots, 0)$. Let $M$ be a $2n \times n$ matrix of complex numbers such that whenever $(z_1, z_2, \dots, z_{2n})M = O_n$, with complex $z_i$, not all zero, then at least one of the $z_i$ is not real. Prove that for arbitrary real numbers $r_1, r_2, \dots, r_{2n}$, there are ... | null | [
"linear_algebra"
] | null | null | |
putnam_2010_a2 | a2e969dc-ca3c-508c-a385-55d08bf3087e | train | abbrev putnam_2010_a2_solution : Set (β β β) := sorry
-- {f : β β β | β c d : β, β x : β, f x = c*x + d}
/--
Find all differentiable functions $f:\mathbb{R} \to \mathbb{R}$ such that
\[
f'(x) = \frac{f(x+n)-f(x)}{n}
\]
for all real numbers $x$ and all positive integers $n$.
-/
theorem putnam_2010_a2
: {f : β β β | Diff... | import Mathlib
-- {f : β β β | β c d : β, β x : β, f x = c*x + d}
/--
Find all differentiable functions $f:\mathbb{R} \to \mathbb{R}$ such that
\[
f'(x) = \frac{f(x+n)-f(x)}{n}
\]
for all real numbers $x$ and all positive integers $n$.
-/
theorem putnam_2010_a2
: {f : β β β | Differentiable β f β§
β x : β, β n : β€, n >... | import Mathlib
abbrev putnam_2010_a2_solution : Set (β β β) := sorry
-- {f : β β β | β c d : β, β x : β, f x = c*x + d}
/--
Find all differentiable functions $f:\mathbb{R} \to \mathbb{R}$ such that
\[
f'(x) = \frac{f(x+n)-f(x)}{n}
\]
for all real numbers $x$ and all positive integers $n$.
-/
theorem putnam_2010_a2
: {... | Find all differentiable functions $f:\mathbb{R} \to \mathbb{R}$ such that
\[
f'(x) = \frac{f(x+n)-f(x)}{n}
\]
for all real numbers $x$ and all positive integers $n$. | The solution consists of all functions of the form $f(x) = cx+d$ for some real numbers $c,d$. | [
"analysis"
] | null | null | |
putnam_2023_b3 | 96031ff9-84d6-5729-8e44-f48b95374c5b | train | abbrev putnam_2023_b3_solution : β β β := sorry
-- (fun n : β β¦ (2 * (n : β) + 2) / 3)
/--
A sequence $y_1, y_2, \ldots, y_k$ of real numbers is called zigzag if $k = 1$, or if $y_2 - y_1, y_3 - y_2, \ldots, y_k - y_{k-1}$ are nonzero and alternate in sign. Let $X_1, X_2,\ldots, X_n$ be chosen independently from the un... | import Mathlib
open ProbabilityTheory Set
-- (fun n : β β¦ (2 * (n : β) + 2) / 3)
/--
A sequence $y_1, y_2, \ldots, y_k$ of real numbers is called zigzag if $k = 1$, or if $y_2 - y_1, y_3 - y_2, \ldots, y_k - y_{k-1}$ are nonzero and alternate in sign. Let $X_1, X_2,\ldots, X_n$ be chosen independently from the unifor... | import Mathlib
open ProbabilityTheory Set
noncomputable abbrev putnam_2023_b3_solution : β β β := sorry
-- (fun n : β β¦ (2 * (n : β) + 2) / 3)
/--
A sequence $y_1, y_2, \ldots, y_k$ of real numbers is called zigzag if $k = 1$, or if $y_2 - y_1, y_3 - y_2, \ldots, y_k - y_{k-1}$ are nonzero and alternate in sign. Let ... | A sequence $y_1, y_2, \ldots, y_k$ of real numbers is called zigzag if $k = 1$, or if $y_2 - y_1, y_3 - y_2, \ldots, y_k - y_{k-1}$ are nonzero and alternate in sign. Let $X_1, X_2,\ldots, X_n$ be chosen independently from the uniform distribution on $[0,1]$. Let $a(X_1, X_2, \ldots, X_n)$ be the largest value of $k$ f... | Show that the expected value is \frac{2n + 2}{3}. | [
"probability",
"combinatorics"
] | null | null | |
putnam_1993_b4 | 400ee065-c782-5576-aece-75893ca56ba0 | train | theorem putnam_1993_b4
(K : β Γ β β β)
(f g : β β β)
(Kpos : β x y : Set.Icc (0 : β) 1, K (x, y) > 0)
(Kcont : ContinuousOn K {(x, y) : β Γ β | x β Set.Icc 0 1 β§ y β Set.Icc 0 1})
(fgpos : β x : Set.Icc (0 : β) 1, f x > 0 β§ g x > 0)
(fgcont : ContinuousOn f (Set.Icc 0 1) β§ ContinuousOn g (Set.Icc 0 1))
(fgint : β x : S... | import Mathlib
/--
The function $K(x,y)$ is positive and continuous for $0 \leq x \leq 1,0 \leq y \leq 1$, and the functions $f(x)$ and $g(x)$ are positive and continuous for $0 \leq x \leq 1$. Suppose that for all $x$, $0 \leq x \leq 1$, $\int_0^1 f(y)K(x,y)\,dy=g(x)$ and $\int_0^1 g(y)K(x,y)\,dy=f(x)$. Show that $f(... | import Mathlib
/--
The function $K(x,y)$ is positive and continuous for $0 \leq x \leq 1,0 \leq y \leq 1$, and the functions $f(x)$ and $g(x)$ are positive and continuous for $0 \leq x \leq 1$. Suppose that for all $x$, $0 \leq x \leq 1$, $\int_0^1 f(y)K(x,y)\,dy=g(x)$ and $\int_0^1 g(y)K(x,y)\,dy=f(x)$. Show that $f(... | The function $K(x,y)$ is positive and continuous for $0 \leq x \leq 1,0 \leq y \leq 1$, and the functions $f(x)$ and $g(x)$ are positive and continuous for $0 \leq x \leq 1$. Suppose that for all $x$, $0 \leq x \leq 1$, $\int_0^1 f(y)K(x,y)\,dy=g(x)$ and $\int_0^1 g(y)K(x,y)\,dy=f(x)$. Show that $f(x)=g(x)$ for $0 \leq... | null | [
"analysis"
] | null | null | |
putnam_1963_b3 | 9c91f28c-bfef-5481-8758-b0f50e30eecf | train | abbrev putnam_1963_b3_solution : Set (β β β) := sorry
-- {(fun u : β => A * Real.sinh (k * u)) | (A : β) (k : β)} βͺ {(fun u : β => A * u) | A : β} βͺ {(fun u : β => A * Real.sin (k * u)) | (A : β) (k : β)}
/--
Find every twice-differentiable real-valued function $f$ with domain the set of all real numbers and satisfying... | import Mathlib
open Topology Filter Polynomial
-- {(fun u : β => A * Real.sinh (k * u)) | (A : β) (k : β)} βͺ {(fun u : β => A * u) | A : β} βͺ {(fun u : β => A * Real.sin (k * u)) | (A : β) (k : β)}
/--
Find every twice-differentiable real-valued function $f$ with domain the set of all real numbers and satisfying the ... | import Mathlib
open Topology Filter Polynomial
abbrev putnam_1963_b3_solution : Set (β β β) := sorry
-- {(fun u : β => A * Real.sinh (k * u)) | (A : β) (k : β)} βͺ {(fun u : β => A * u) | A : β} βͺ {(fun u : β => A * Real.sin (k * u)) | (A : β) (k : β)}
/--
Find every twice-differentiable real-valued function $f$ with ... | Find every twice-differentiable real-valued function $f$ with domain the set of all real numbers and satisfying the functional equation $(f(x))^2-(f(y))^2=f(x+y)f(x-y)$ for all real numbers $x$ and $y$. | Show that the solution is the sets of functions $f(u)=A\sinh ku$, $f(u)=Au$, and $f(u)=A\sin ku$ with $A,k \in \mathbb{R}$. | [
"analysis"
] | null | null | |
putnam_2009_a4 | 1f52368c-4ab9-5a48-8906-e619618d50b1 | train | abbrev putnam_2009_a4_solution : Prop := sorry
-- False
/--
Let $S$ be a set of rational numbers such that
\begin{enumerate}
\item[(a)] $0 \in S$;
\item[(b)] If $x \in S$ then $x+1\in S$ and $x-1\in S$; and
\item[(c)] If $x\in S$ and $x\not\in\{0,1\}$, then $\frac{1}{x(x-1)}\in S$.
\end{enumerate}
Must $S$ contain all ... | import Mathlib
open Topology MvPolynomial Filter Set
-- False
/--
Let $S$ be a set of rational numbers such that
\begin{enumerate}
\item[(a)] $0 \in S$;
\item[(b)] If $x \in S$ then $x+1\in S$ and $x-1\in S$; and
\item[(c)] If $x\in S$ and $x\not\in\{0,1\}$, then $\frac{1}{x(x-1)}\in S$.
\end{enumerate}
Must $S$ cont... | import Mathlib
open Topology MvPolynomial Filter Set
abbrev putnam_2009_a4_solution : Prop := sorry
-- False
/--
Let $S$ be a set of rational numbers such that
\begin{enumerate}
\item[(a)] $0 \in S$;
\item[(b)] If $x \in S$ then $x+1\in S$ and $x-1\in S$; and
\item[(c)] If $x\in S$ and $x\not\in\{0,1\}$, then $\frac{... | Let $S$ be a set of rational numbers such that
\begin{enumerate}
\item[(a)] $0 \in S$;
\item[(b)] If $x \in S$ then $x+1\in S$ and $x-1\in S$; and
\item[(c)] If $x\in S$ and $x\not\in\{0,1\}$, then $\frac{1}{x(x-1)}\in S$.
\end{enumerate}
Must $S$ contain all rational numbers? | Prove that $S$ need not contain all rationals. | [
"number_theory"
] | null | null | |
putnam_1978_a2 | 7327b79e-c562-5cc9-b082-e019e44c509c | train | theorem putnam_1978_a2
(n : β)
(npos : n > 0)
(a b : β)
(hab : a β b)
(c : Fin n β β)
(A : Matrix (Fin n) (Fin n) β)
(hA : A = fun i j β¦ ite (i < j) a (ite (i > j) b (c i)))
(p : β β β)
(hp : p = fun x β¦ β i : Fin n, (c i - x))
: (A.det = ((b * p a - a * p b) / (b - a))) :=
sorry | import Mathlib
/--
Let $A$ be the real $n \times n$ matrix $(a_{ij})$ where $a_{ij} = a$ for $i < j$, $b \; (\neq a)$ for $i > j$, and $c_i$ for $i = j$. Show that $\det A = \frac{b p(a) - a p(b)}{b - a}$, where $p(x) = \prod_{i=1}^n (c_i - x)$.
-/
theorem putnam_1978_a2
(n : β)
(npos : n > 0)
(a b : β)
(hab : a β b)
... | import Mathlib
/--
Let $A$ be the real $n \times n$ matrix $(a_{ij})$ where $a_{ij} = a$ for $i < j$, $b \; (\neq a)$ for $i > j$, and $c_i$ for $i = j$. Show that $\det A = \frac{b p(a) - a p(b)}{b - a}$, where $p(x) = \prod_{i=1}^n (c_i - x)$.
-/
theorem putnam_1978_a2
(n : β)
(npos : n > 0)
(a b : β)
(hab : a β b)
... | Let $A$ be the real $n \times n$ matrix $(a_{ij})$ where $a_{ij} = a$ for $i < j$, $b \; (\neq a)$ for $i > j$, and $c_i$ for $i = j$. Show that $\det A = \frac{b p(a) - a p(b)}{b - a}$, where $p(x) = \prod_{i=1}^n (c_i - x)$. | null | [
"linear_algebra"
] | null | null | |
putnam_1980_a2 | dbcac05e-f56d-5ec0-b6d5-ff3207141412 | train | abbrev putnam_1980_a2_solution : β β β β β := sorry
-- (fun r s : β => (1 + 4 * r + 6 * r ^ 2) * (1 + 4 * s + 6 * s ^ 2))
/--
Let $r$ and $s$ be positive integers. Derive a formula for the number of ordered quadruples $(a,b,c,d)$ of positive integers such that $3^r \cdot 7^s=\text{lcm}[a,b,c]=\text{lcm}[a,b,d]=\text{lc... | import Mathlib
-- (fun r s : β => (1 + 4 * r + 6 * r ^ 2) * (1 + 4 * s + 6 * s ^ 2))
/--
Let $r$ and $s$ be positive integers. Derive a formula for the number of ordered quadruples $(a,b,c,d)$ of positive integers such that $3^r \cdot 7^s=\text{lcm}[a,b,c]=\text{lcm}[a,b,d]=\text{lcm}[a,c,d]=\text{lcm}[b,c,d]$. The an... | import Mathlib
abbrev putnam_1980_a2_solution : β β β β β := sorry
-- (fun r s : β => (1 + 4 * r + 6 * r ^ 2) * (1 + 4 * s + 6 * s ^ 2))
/--
Let $r$ and $s$ be positive integers. Derive a formula for the number of ordered quadruples $(a,b,c,d)$ of positive integers such that $3^r \cdot 7^s=\text{lcm}[a,b,c]=\text{lcm}... | Let $r$ and $s$ be positive integers. Derive a formula for the number of ordered quadruples $(a,b,c,d)$ of positive integers such that $3^r \cdot 7^s=\text{lcm}[a,b,c]=\text{lcm}[a,b,d]=\text{lcm}[a,c,d]=\text{lcm}[b,c,d]$. The answer should be a function of $r$ and $s$. (Note that $\text{lcm}[x,y,z]$ denotes the least... | Show that the number is $(1+4r+6r^2)(1+4s+6s^2)$. | [
"number_theory"
] | null | null | |
putnam_2015_a5 | ca6fa0b3-64e9-57cf-8c9c-cf2996f4e5bb | train | theorem putnam_2015_a5
(q : β)
(Nq : β)
(qodd : Odd q)
(qpos : q > 0)
(hNq : Nq = {a : β | 0 < a β§ a < (q : β) / 4 β§ Nat.gcd a q = 1}.encard)
: Odd Nq β β p k : β, q = p ^ k β§ k > 0 β§ p.Prime β§ (p % 8 = 5 β¨ p % 8 = 7) :=
sorry | import Mathlib
/--
Let $q$ be an odd positive integer, and let $N_q$ denote the number of integers $a$ such that $0<a<q/4$ and $\gcd(a,q)=1$. Show that $N_q$ is odd if and only if $q$ is of the form $p^k$ with $k$ a positive integer and $p$ a prime congruent to $5$ or $7$ modulo $8$.
-/
theorem putnam_2015_a5
(q : β)
... | import Mathlib
/--
Let $q$ be an odd positive integer, and let $N_q$ denote the number of integers $a$ such that $0<a<q/4$ and $\gcd(a,q)=1$. Show that $N_q$ is odd if and only if $q$ is of the form $p^k$ with $k$ a positive integer and $p$ a prime congruent to $5$ or $7$ modulo $8$.
-/
theorem putnam_2015_a5
(q : β)
... | Let $q$ be an odd positive integer, and let $N_q$ denote the number of integers $a$ such that $0<a<q/4$ and $\gcd(a,q)=1$. Show that $N_q$ is odd if and only if $q$ is of the form $p^k$ with $k$ a positive integer and $p$ a prime congruent to $5$ or $7$ modulo $8$. | null | [
"number_theory"
] | null | null | |
putnam_2011_a5 | c03f3f10-59d2-57b7-b28f-066387bf32e7 | train | theorem putnam_2011_a5
(F : (Fin 2 β β) β β)
(g : β β β)
(vec : β β β β (Fin 2 β β))
(Fgrad : (Fin 2 β β) β (Fin 2 β β))
(parallel : (Fin 2 β β) β (Fin 2 β β) β Prop)
(hparallel : parallel = (fun u v : Fin 2 β β => β c : β, u = c β’ v))
(Fgdiff : ContDiff β 2 F β§ ContDiff β 2 g)
(prop1 : β uu : Fin 2 β β, uu 0 = uu 1 β ... | import Mathlib
open Topology Filter Matrix
/--
Let $F:\mathbb{R}^2 \to \mathbb{R}$ and $g:\mathbb{R} \to \mathbb{R}$ be twice continuously differentiable functions with the following properties:
\begin{itemize}
\item $F(u,u)=0$ for every $u \in \mathbb{R}$;
\item for every $x \in \mathbb{R}$, $g(x)>0$ and $x^2g(x) \l... | import Mathlib
open Topology Filter Matrix
/--
Let $F:\mathbb{R}^2 \to \mathbb{R}$ and $g:\mathbb{R} \to \mathbb{R}$ be twice continuously differentiable functions with the following properties:
\begin{itemize}
\item $F(u,u)=0$ for every $u \in \mathbb{R}$;
\item for every $x \in \mathbb{R}$, $g(x)>0$ and $x^2g(x) \l... | Let $F:\mathbb{R}^2 \to \mathbb{R}$ and $g:\mathbb{R} \to \mathbb{R}$ be twice continuously differentiable functions with the following properties:
\begin{itemize}
\item $F(u,u)=0$ for every $u \in \mathbb{R}$;
\item for every $x \in \mathbb{R}$, $g(x)>0$ and $x^2g(x) \leq 1$;
\item for every $(u,v) \in \mathbb{R}^2$, ... | null | [
"analysis"
] | null | null | |
putnam_1995_b6 | 54fa8ca2-9eba-58bc-bd96-2d846016507d | train | theorem putnam_1995_b6
(S : β β Set β)
(hS : S = fun (Ξ± : β) => {x : β | β n : β, n β₯ 1 β§ x = floor (n * Ξ±)}) :
Β¬ β Ξ± Ξ² Ξ³,
Ξ± > 0 β§ Ξ² > 0 β§ Ξ³ > 0 β§
(S Ξ±) β© (S Ξ²) = β
β§ (S Ξ²) β© (S Ξ³) = β
β§ (S Ξ±) β© (S Ξ³) = β
β§
Set.Ici 1 = (S Ξ±) βͺ (S Ξ²) βͺ (S Ξ³) :=
sorry | import Mathlib
open Filter Topology Real Nat
/--
For a positive real number $\alpha$, define \[ S(\alpha) = \{ \lfloor n\alpha \rfloor : n = 1,2,3,\dots \}. \] Prove that $\{1,2,3,\dots\}$ cannot be expressed as the disjoint union of three sets $S(\alpha), S(\beta)$ and $S(\gamma)$. [As usual, $\lfloor x \rfloor$ is ... | import Mathlib
open Filter Topology Real Nat
/--
For a positive real number $\alpha$, define \[ S(\alpha) = \{ \lfloor n\alpha \rfloor : n = 1,2,3,\dots \}. \] Prove that $\{1,2,3,\dots\}$ cannot be expressed as the disjoint union of three sets $S(\alpha), S(\beta)$ and $S(\gamma)$. [As usual, $\lfloor x \rfloor$ is ... | For a positive real number $\alpha$, define \[ S(\alpha) = \{ \lfloor n\alpha \rfloor : n = 1,2,3,\dots \}. \] Prove that $\{1,2,3,\dots\}$ cannot be expressed as the disjoint union of three sets $S(\alpha), S(\beta)$ and $S(\gamma)$. [As usual, $\lfloor x \rfloor$ is the greatest integer $\leq x$.] | null | [
"algebra",
"number_theory"
] | null | null | |
putnam_1979_a6 | 01fc52aa-7118-561e-8672-d6109fdf5adf | train | theorem putnam_1979_a6
(n : β)
(p : β β β)
(hp : β i β Finset.range n, p i β Icc 0 1)
: β x β Icc 0 1, (β i β Finset.range n, x β p i) β§ β i in Finset.range n, 1/|x - p i| β€ 8*n*β i in Finset.range n, (1 : β)/(2*i + 1) :=
sorry | import Mathlib
open Set
/--
For all $i \in \{0, 1, \dots, n - 1\}$, let $p_i \in [0, 1]$. Prove that there exists some $x \in [0, 1]$ such that $$\sum_{i = 0}^{n - 1} \frac{1}{|x - p_i|} \le 8n\left(\sum_{i = 0}^{n-1} \frac{1}{2i + 1}\right).$$
-/
theorem putnam_1979_a6
(n : β)
(p : β β β)
(hp : β i β Finset.range n,... | import Mathlib
open Set
/--
For all $i \in \{0, 1, \dots, n - 1\}$, let $p_i \in [0, 1]$. Prove that there exists some $x \in [0, 1]$ such that $$\sum_{i = 0}^{n - 1} \frac{1}{|x - p_i|} \le 8n\left(\sum_{i = 0}^{n-1} \frac{1}{2i + 1}\right).$$
-/
theorem putnam_1979_a6
(n : β)
(p : β β β)
(hp : β i β Finset.range n,... | For all $i \in \{0, 1, \dots, n - 1\}$, let $p_i \in [0, 1]$. Prove that there exists some $x \in [0, 1]$ such that $$\sum_{i = 0}^{n - 1} \frac{1}{|x - p_i|} \le 8n\left(\sum_{i = 0}^{n-1} \frac{1}{2i + 1}\right).$$ | null | [
"algebra"
] | null | null | |
putnam_1997_b5 | b5afb8da-96d6-586b-947b-dec418d74657 | train | theorem putnam_1997_b5
(n : β)
(hn : n β₯ 2)
: tetration 2 n β‘ tetration 2 (n-1) [MOD n] :=
sorry | import Mathlib
def tetration : β β β β β
| _, 0 => 1
| b, (m + 1) => b^(tetration b m)
/--
Prove that for $n\geq 2$, \[\overbrace{2^{2^{\cdots^{2}}}}^{\mbox{$n$ terms}} \equiv \overbrace{2^{2^{\cdots^{2}}}}^{\mbox{$n-1$ terms}} \quad \pmod{n}.\]
-/
theorem putnam_1997_b5
(n : β)
(hn : n β₯ 2)
: tetration 2 n β‘ tetr... | import Mathlib
def tetration : β β β β β
| _, 0 => 1
| b, (m + 1) => b^(tetration b m)
/--
Prove that for $n\geq 2$, \[\overbrace{2^{2^{\cdots^{2}}}}^{\mbox{$n$ terms}} \equiv \overbrace{2^{2^{\cdots^{2}}}}^{\mbox{$n-1$ terms}} \quad \pmod{n}.\]
-/
theorem putnam_1997_b5
(n : β)
(hn : n β₯ 2)
: tetration 2 n β‘ tetr... | Prove that for $n\geq 2$, \[\overbrace{2^{2^{\cdots^{2}}}}^{\mbox{$n$ terms}} \equiv \overbrace{2^{2^{\cdots^{2}}}}^{\mbox{$n-1$ terms}} \quad \pmod{n}.\] | null | [
"number_theory"
] | null | null | |
putnam_1983_b5 | d15768c8-2b6a-5a31-bc35-216c261e04ee | train | abbrev putnam_1983_b5_solution : β := sorry
-- log (4 / Real.pi)
/--
Define $\left\lVert x \right\rVert$ as the distance from $x$ to the nearest integer. Find $\lim_{n \to \infty} \frac{1}{n} \int_{1}^{n} \left\lVert \frac{n}{x} \right\rVert \, dx$. You may assume that $\prod_{n=1}^{\infty} \frac{2n}{(2n-1)} \cdot \fra... | import Mathlib
open Nat Filter Topology Real
-- log (4 / Real.pi)
/--
Define $\left\lVert x \right\rVert$ as the distance from $x$ to the nearest integer. Find $\lim_{n \to \infty} \frac{1}{n} \int_{1}^{n} \left\lVert \frac{n}{x} \right\rVert \, dx$. You may assume that $\prod_{n=1}^{\infty} \frac{2n}{(2n-1)} \cdot \... | import Mathlib
open Nat Filter Topology Real
noncomputable abbrev putnam_1983_b5_solution : β := sorry
-- log (4 / Real.pi)
/--
Define $\left\lVert x \right\rVert$ as the distance from $x$ to the nearest integer. Find $\lim_{n \to \infty} \frac{1}{n} \int_{1}^{n} \left\lVert \frac{n}{x} \right\rVert \, dx$. You may a... | Define $\left\lVert x \right\rVert$ as the distance from $x$ to the nearest integer. Find $\lim_{n \to \infty} \frac{1}{n} \int_{1}^{n} \left\lVert \frac{n}{x} \right\rVert \, dx$. You may assume that $\prod_{n=1}^{\infty} \frac{2n}{(2n-1)} \cdot \frac{2n}{(2n+1)} = \frac{\pi}{2}$. | Show that the limit equals $\ln \left( \frac{4}{\pi} \right)$. | [
"analysis"
] | null | null | |
putnam_1975_b5 | eef33668-1bab-501d-8704-ec9e2c4f7897 | train | theorem putnam_1975_b5
(e : β)
(he : e = Real.exp 1)
(f : β β β β β)
(h0 : β x : β, f 0 x = e^x)
(hf : β n : β, β x : β, f (n + 1) x = x * (deriv (f n) x))
: β' n : β, (f n 1)/(Nat.factorial n) = e^e :=
sorry | import Mathlib
open Polynomial Real Complex Matrix Filter Topology Multiset
/--
Let $f_0(x) = e^x$ and $f_{n+1}(x) = xf'_n(x)$ for all $n \ge 0$. Prove that $$\sum_{n=0}^{\infty} \frac{f_n(1)}{n!} = e^e.$$
-/
theorem putnam_1975_b5
(e : β)
(he : e = Real.exp 1)
(f : β β β β β)
(h0 : β x : β, f 0 x = e^x)
(hf : β n : ... | import Mathlib
open Polynomial Real Complex Matrix Filter Topology Multiset
/--
Let $f_0(x) = e^x$ and $f_{n+1}(x) = xf'_n(x)$ for all $n \ge 0$. Prove that $$\sum_{n=0}^{\infty} \frac{f_n(1)}{n!} = e^e.$$
-/
theorem putnam_1975_b5
(e : β)
(he : e = Real.exp 1)
(f : β β β β β)
(h0 : β x : β, f 0 x = e^x)
(hf : β n : ... | Let $f_0(x) = e^x$ and $f_{n+1}(x) = xf'_n(x)$ for all $n \ge 0$. Prove that $$\sum_{n=0}^{\infty} \frac{f_n(1)}{n!} = e^e.$$ | null | [
"analysis",
"algebra"
] | null | null | |
putnam_2008_b1 | 92596519-5bdc-577d-952a-0d75264935ea | train | abbrev putnam_2008_b1_solution : β := sorry
-- 2
def is_rational_point (p : Fin 2 β β) : Prop := β (a b : β), a = p 0 β§ b = p 1
def real_circle (c : EuclideanSpace β (Fin 2)) (r : β) : Set (EuclideanSpace β (Fin 2)) := {p | dist p c = r}
/--
What is the maximum number of rational points that can lie on a circle in $\ma... | import Mathlib
open Filter Topology Set
-- 2
def is_rational_point (p : Fin 2 β β) : Prop := β (a b : β), a = p 0 β§ b = p 1
def real_circle (c : EuclideanSpace β (Fin 2)) (r : β) : Set (EuclideanSpace β (Fin 2)) := {p | dist p c = r}
/--
What is the maximum number of rational points that can lie on a circle in $\math... | import Mathlib
open Filter Topology Set
abbrev putnam_2008_b1_solution : β := sorry
-- 2
def is_rational_point (p : Fin 2 β β) : Prop := β (a b : β), a = p 0 β§ b = p 1
def real_circle (c : EuclideanSpace β (Fin 2)) (r : β) : Set (EuclideanSpace β (Fin 2)) := {p | dist p c = r}
/--
What is the maximum number of ration... | What is the maximum number of rational points that can lie on a circle in $\mathbb{R}^2$ whose center is not a rational point? (A \emph{rational point} is a point both of whose coordinates are rational numbers.) | Show that the maximum number is $2$. | [
"geometry",
"number_theory"
] | null | null | |
putnam_2008_a5 | 4c40e1cf-79d1-5b36-aa45-33db461e7c7a | train | theorem putnam_2008_a5
(n : β)
(nge3 : n β₯ 3)
(f g : Polynomial β)
(hfg : β O z : β, z β 0 β§ β k : β, k β Icc 1 n β (f.eval (k : β)) + Complex.I * (g.eval (k : β)) = O + z * Complex.exp (Complex.I * 2 * Real.pi * k / n))
: (f.natDegree β₯ n - 1 β¨ g.natDegree β₯ n - 1) :=
sorry | import Mathlib
open Filter Topology Set
/--
Let $n \geq 3$ be an integer. Let $f(x)$ and $g(x)$ be polynomials with real coefficients such that the points $(f(1), g(1)), (f(2), g(2)), \dots, (f(n), g(n))$ in $\mathbb{R}^2$ are the vertices of a regular $n$-gon in counterclockwise order. Prove that at least one of $f(... | import Mathlib
open Filter Topology Set
/--
Let $n \geq 3$ be an integer. Let $f(x)$ and $g(x)$ be polynomials with real coefficients such that the points $(f(1), g(1)), (f(2), g(2)), \dots, (f(n), g(n))$ in $\mathbb{R}^2$ are the vertices of a regular $n$-gon in counterclockwise order. Prove that at least one of $f(... | Let $n \geq 3$ be an integer. Let $f(x)$ and $g(x)$ be polynomials with real coefficients such that the points $(f(1), g(1)), (f(2), g(2)), \dots, (f(n), g(n))$ in $\mathbb{R}^2$ are the vertices of a regular $n$-gon in counterclockwise order. Prove that at least one of $f(x)$ and $g(x)$ has degree greater than or equa... | null | [
"algebra",
"geometry"
] | null | null | |
putnam_1969_b2 | 1e680b8e-4938-51f3-af09-dc9d5dd0dbad | train | abbrev putnam_1969_b2_solution : Prop := sorry
-- False
/--
Show that a finite group can not be the union of two of its proper subgroups. Does the statement remain true if 'two' is replaced by 'three'?
-/
theorem putnam_1969_b2
(P : β β Prop)
(P_def : β n, P n β β (G : Type) [Group G] [Finite G],
β H : Fi... | import Mathlib
open Matrix Filter Topology Set Nat
-- False
/--
Show that a finite group can not be the union of two of its proper subgroups. Does the statement remain true if 'two' is replaced by 'three'?
-/
theorem putnam_1969_b2
(P : β β Prop)
(P_def : β n, P n β β (G : Type) [Group G] [Finite G],
β ... | import Mathlib
open Matrix Filter Topology Set Nat
abbrev putnam_1969_b2_solution : Prop := sorry
-- False
/--
Show that a finite group can not be the union of two of its proper subgroups. Does the statement remain true if 'two' is replaced by 'three'?
-/
theorem putnam_1969_b2
(P : β β Prop)
(P_def : β n, P ... | Show that a finite group can not be the union of two of its proper subgroups. Does the statement remain true if 'two' is replaced by 'three'? | Show that the statement is no longer true if 'two' is replaced by 'three'. | [
"abstract_algebra"
] | null | null | |
putnam_2023_b6 | 12195118-b6f7-5dfa-b81a-3b4ce677e02c | train | abbrev putnam_2023_b6_solution : β β β€ := sorry
-- (fun n : β => (-1) ^ (β(n / 2 : β)ββ + 1) * 2 * β(n / 2 : β)ββ)
/--
Let $n$ be a positive integer. For $i$ and $j$ in $\{1,2,\dots,n\}$, let $s(i,j)$ be the number of pairs $(a,b)$ of nonnegative integers satisfying $ai+bj=n$. Let $S$ be the $n$-by-$n$ matrix whose $(i... | import Mathlib
open Nat Topology Filter
-- (fun n : β => (-1) ^ (β(n / 2 : β)ββ + 1) * 2 * β(n / 2 : β)ββ)
/--
Let $n$ be a positive integer. For $i$ and $j$ in $\{1,2,\dots,n\}$, let $s(i,j)$ be the number of pairs $(a,b)$ of nonnegative integers satisfying $ai+bj=n$. Let $S$ be the $n$-by-$n$ matrix whose $(i,j)$ e... | import Mathlib
open Nat Topology Filter
abbrev putnam_2023_b6_solution : β β β€ := sorry
-- (fun n : β => (-1) ^ (β(n / 2 : β)ββ + 1) * 2 * β(n / 2 : β)ββ)
/--
Let $n$ be a positive integer. For $i$ and $j$ in $\{1,2,\dots,n\}$, let $s(i,j)$ be the number of pairs $(a,b)$ of nonnegative integers satisfying $ai+bj=n$. ... | Let $n$ be a positive integer. For $i$ and $j$ in $\{1,2,\dots,n\}$, let $s(i,j)$ be the number of pairs $(a,b)$ of nonnegative integers satisfying $ai+bj=n$. Let $S$ be the $n$-by-$n$ matrix whose $(i,j)$ entry is $s(i,j)$. For example, when $n=5$, we have $S = \begin{bmatrix}
6 & 3 & 2 & 2 & 2 \\
3 & 0 & 1 & 0 & 1 \\... | Show that the determinant equals $(-1)^{\lceil n/2 \rceil-1}2\lceil\frac{n}{2}\rceil$. | [
"linear_algebra"
] | null | null | |
putnam_1982_b5 | 83898fab-503f-5278-914f-37d217acbab1 | train | theorem putnam_1982_b5
(T : Set β)
(hT : T = Ioi (Real.exp (Real.exp 1)))
(S : β β β β β)
(hS : β x β T, S x 0 = (Real.exp 1) β§ β n : β, S x (n + 1) = Real.logb (S x n) x)
(g : β β β)
: β x β T, (β L : β, Tendsto (S x) atTop (π L)) β§
(β x β T, Tendsto (S x) atTop (π (g x))) β ContinuousOn g T :=
sorry | import Mathlib
open Set Function Filter Topology Polynomial Real
/--
For all $x > e^e$, let $S = u_0, u_1, \dots$ be a recursively defined sequence with $u_0 = e$ and $u_{n+1} = \log_{u_n} x$ for all $n \ge 0$. Prove that $S_x$ converges to some real number $g(x)$ and that this function $g$ is continuous for $x > e^e... | import Mathlib
open Set Function Filter Topology Polynomial Real
/--
For all $x > e^e$, let $S = u_0, u_1, \dots$ be a recursively defined sequence with $u_0 = e$ and $u_{n+1} = \log_{u_n} x$ for all $n \ge 0$. Prove that $S_x$ converges to some real number $g(x)$ and that this function $g$ is continuous for $x > e^e... | For all $x > e^e$, let $S = u_0, u_1, \dots$ be a recursively defined sequence with $u_0 = e$ and $u_{n+1} = \log_{u_n} x$ for all $n \ge 0$. Prove that $S_x$ converges to some real number $g(x)$ and that this function $g$ is continuous for $x > e^e$. | null | [
"analysis"
] | null | null | |
putnam_2003_a3 | 10e1dec8-2b3c-5535-962f-c20c7ca275f0 | train | abbrev putnam_2003_a3_solution : β := sorry
-- 2 * Real.sqrt 2 - 1
/--
Find the minimum value of $|\sin x+\cos x+\tan x+\cot x+\sec x+\csc x|$ for real numbers $x$.
-/
theorem putnam_2003_a3
(f : β β β)
(hf : β x : β, f x = |Real.sin x + Real.cos x + Real.tan x + 1 / Real.tan x + 1 / Real.cos x + 1 / Real.sin x... | import Mathlib
open Set
-- 2 * Real.sqrt 2 - 1
/--
Find the minimum value of $|\sin x+\cos x+\tan x+\cot x+\sec x+\csc x|$ for real numbers $x$.
-/
theorem putnam_2003_a3
(f : β β β)
(hf : β x : β, f x = |Real.sin x + Real.cos x + Real.tan x + 1 / Real.tan x + 1 / Real.cos x + 1 / Real.sin x|) :
IsLeast (... | import Mathlib
open Set
noncomputable abbrev putnam_2003_a3_solution : β := sorry
-- 2 * Real.sqrt 2 - 1
/--
Find the minimum value of $|\sin x+\cos x+\tan x+\cot x+\sec x+\csc x|$ for real numbers $x$.
-/
theorem putnam_2003_a3
(f : β β β)
(hf : β x : β, f x = |Real.sin x + Real.cos x + Real.tan x + 1 / Real... | Find the minimum value of $|\sin x+\cos x+\tan x+\cot x+\sec x+\csc x|$ for real numbers $x$. | Show that the minimum is $2\sqrt{2}-1$. | [
"analysis"
] | null | null | |
putnam_1981_b3 | 373df209-5160-5dc3-b61b-828c7fd6c6ee | train | theorem putnam_1981_b3
(P : β β Prop)
(hP : β n, P n β
β p : β, (Nat.Prime p β§ p β£ n^2 + 3) β
β k : β, (p : β€) β£ (k : β€)^2 + 3 β§ k^2 < n) :
β n : β, β m : β, (m : β€) > n β§ P m :=
sorry | import Mathlib
open Topology Filter Set Polynomial Function
/--
Prove that, for infinitely many positive integers $n$, all primes $p$ that divide $n^2 + 3$ also divide $k^2 + 3$ for some integer $k$ such that $k^2 < n$.
-/
theorem putnam_1981_b3
(P : β β Prop)
(hP : β n, P n β
β p : β, (Nat.Prime p β§ p ... | import Mathlib
open Topology Filter Set Polynomial Function
/--
Prove that, for infinitely many positive integers $n$, all primes $p$ that divide $n^2 + 3$ also divide $k^2 + 3$ for some integer $k$ such that $k^2 < n$.
-/
theorem putnam_1981_b3
(P : β β Prop)
(hP : β n, P n β
β p : β, (Nat.Prime p β§ p ... | Prove that, for infinitely many positive integers $n$, all primes $p$ that divide $n^2 + 3$ also divide $k^2 + 3$ for some integer $k$ such that $k^2 < n$. | null | [
"number_theory"
] | null | null | |
putnam_1993_a2 | ba08f679-e7b0-52e2-8036-e28357c76aba | train | theorem putnam_1993_a2
(x : β β β)
(xnonzero : β n : β, x n β 0)
(hx : β n β₯ 1, (x n) ^ 2 - x (n - 1) * x (n + 1) = 1)
: β a : β, β n β₯ 1, x (n + 1) = a * x n - x (n - 1) :=
sorry | import Mathlib
/--
Let $(x_n)_{n \geq 0}$ be a sequence of nonzero real numbers such that $x_n^2-x_{n-1}x_{n+1}=1$ for $n=1,2,3,\dots$. Prove there exists a real number $a$ such that $x_{n+1}=ax_n-x_{n-1}$ for all $n \geq 1$.
-/
theorem putnam_1993_a2
(x : β β β)
(xnonzero : β n : β, x n β 0)
(hx : β n β₯ 1, (x n) ^ 2 ... | import Mathlib
/--
Let $(x_n)_{n \geq 0}$ be a sequence of nonzero real numbers such that $x_n^2-x_{n-1}x_{n+1}=1$ for $n=1,2,3,\dots$. Prove there exists a real number $a$ such that $x_{n+1}=ax_n-x_{n-1}$ for all $n \geq 1$.
-/
theorem putnam_1993_a2
(x : β β β)
(xnonzero : β n : β, x n β 0)
(hx : β n β₯ 1, (x n) ^ 2 ... | Let $(x_n)_{n \geq 0}$ be a sequence of nonzero real numbers such that $x_n^2-x_{n-1}x_{n+1}=1$ for $n=1,2,3,\dots$. Prove there exists a real number $a$ such that $x_{n+1}=ax_n-x_{n-1}$ for all $n \geq 1$. | null | [
"algebra"
] | null | null | |
putnam_1991_a4 | 081fb2f1-5e09-5bc7-a974-c2b49b787196 | train | abbrev putnam_1991_a4_solution : Prop := sorry
-- True
/--
Does there exist an infinite sequence of closed discs $D_1,D_2,D_3,\dots$ in the plane, with centers $c_1,c_2,c_3,\dots$, respectively, such that
\begin{enumerate}
\item the $c_i$ have no limit point in the finite plane,
\item the sum of the areas of the $D_i$ ... | import Mathlib
open Filter FiniteDimensional Metric Topology
-- True
/--
Does there exist an infinite sequence of closed discs $D_1,D_2,D_3,\dots$ in the plane, with centers $c_1,c_2,c_3,\dots$, respectively, such that
\begin{enumerate}
\item the $c_i$ have no limit point in the finite plane,
\item the sum of the are... | import Mathlib
open Filter FiniteDimensional Metric Topology
abbrev putnam_1991_a4_solution : Prop := sorry
-- True
/--
Does there exist an infinite sequence of closed discs $D_1,D_2,D_3,\dots$ in the plane, with centers $c_1,c_2,c_3,\dots$, respectively, such that
\begin{enumerate}
\item the $c_i$ have no limit poin... | Does there exist an infinite sequence of closed discs $D_1,D_2,D_3,\dots$ in the plane, with centers $c_1,c_2,c_3,\dots$, respectively, such that
\begin{enumerate}
\item the $c_i$ have no limit point in the finite plane,
\item the sum of the areas of the $D_i$ is finite, and
\item every line in the plane intersects at ... | Show that the answer is yes, such a sequence of closed discs exists. | [
"geometry",
"analysis"
] | null | null | |
putnam_1987_a6 | 330228f0-9ab5-52ea-9a21-5c5d6495ebab | train | abbrev putnam_1987_a6_solution : Set β := sorry
-- {x : β | x > 0 β§ x < 25}
/--
For each positive integer $n$, let $a(n)$ be the number of zeroes in the base $3$ representation of $n$. For which positive real numbers $x$ does the series
\[
\sum_{n=1}^\infty \frac{x^{a(n)}}{n^3}
\]
converge?
-/
theorem putnam_1987_a6
(a... | import Mathlib
open MvPolynomial Real Nat
-- {x : β | x > 0 β§ x < 25}
/--
For each positive integer $n$, let $a(n)$ be the number of zeroes in the base $3$ representation of $n$. For which positive real numbers $x$ does the series
\[
\sum_{n=1}^\infty \frac{x^{a(n)}}{n^3}
\]
converge?
-/
theorem putnam_1987_a6
(a : β... | import Mathlib
open MvPolynomial Real Nat
abbrev putnam_1987_a6_solution : Set β := sorry
-- {x : β | x > 0 β§ x < 25}
/--
For each positive integer $n$, let $a(n)$ be the number of zeroes in the base $3$ representation of $n$. For which positive real numbers $x$ does the series
\[
\sum_{n=1}^\infty \frac{x^{a(n)}}{n^... | For each positive integer $n$, let $a(n)$ be the number of zeroes in the base $3$ representation of $n$. For which positive real numbers $x$ does the series
\[
\sum_{n=1}^\infty \frac{x^{a(n)}}{n^3}
\]
converge? | Show that for positive $x$, the series converges if and only if $x < 25$. | [
"algebra",
"analysis"
] | null | null | |
putnam_1965_a6 | 23c83676-ac0d-5b49-abd8-1155814b3350 | train | theorem putnam_1965_a6
(u v m : β)
(hu : 0 < u)
(hv : 0 < v)
(hm : 1 < m) :
(βα΅ (x > 0) (y > 0),
u * x + v * y = 1 β§
x ^ m + y ^ m = 1 β§
u = x ^ (m - 1) β§
v = y ^ (m - 1)) β
β n, u ^ n + v ^ n = 1 β§ mβ»ΒΉ + nβ»ΒΉ = 1 :=
sorry | import Mathlib
open EuclideanGeometry Topology Filter Complex
/--
Prove that the line $ux + vy = 1$ (where $u \ge 0$ and $v \ge 0$) will lie tangent to the curve $x^m + y^m = 1$ (where $m > 1$) if and only if $u^n + v^n = 1$ for some $n$ such that $m^{-1} + n^{-1} = 1$.
-/
theorem putnam_1965_a6
(u v m : β)
(... | import Mathlib
open EuclideanGeometry Topology Filter Complex
/--
Prove that the line $ux + vy = 1$ (where $u \ge 0$ and $v \ge 0$) will lie tangent to the curve $x^m + y^m = 1$ (where $m > 1$) if and only if $u^n + v^n = 1$ for some $n$ such that $m^{-1} + n^{-1} = 1$.
-/
theorem putnam_1965_a6
(u v m : β)
(... | Prove that the line $ux + vy = 1$ (where $u \ge 0$ and $v \ge 0$) will lie tangent to the curve $x^m + y^m = 1$ (where $m > 1$) if and only if $u^n + v^n = 1$ for some $n$ such that $m^{-1} + n^{-1} = 1$. | null | [
"geometry"
] | null | null | |
putnam_1972_a2 | 762440ae-b383-559d-85b4-b2b6596a776a | train | theorem putnam_1972_a2
: (β (S : Type*) (_ : Mul S), (β x y : S, x * (x * y) = y β§ ((y * x) * x) = y) β (β x y : S, x * y = y * x)) β§ β (S : Type*) (_ : Mul S), (β x y : S, x * (x * y) = y β§ ((y * x) * x) = y) β§ Β¬(β x y z : S, x * (y * z) = (x * y) * z) :=
sorry | import Mathlib
open EuclideanGeometry Filter Topology Set
/--
Let $S$ be a set and $\cdot$ be a binary operation on $S$ satisfying: (1) for all $x,y$ in $S$, $x \cdot (x \cdot y) = y$ (2) for all $x,y$ in $S$, $(y \cdot x) \cdot x = y$. Show that $\cdot$ is commutative but not necessarily associative.
-/
theorem putn... | import Mathlib
open EuclideanGeometry Filter Topology Set
/--
Let $S$ be a set and $\cdot$ be a binary operation on $S$ satisfying: (1) for all $x,y$ in $S$, $x \cdot (x \cdot y) = y$ (2) for all $x,y$ in $S$, $(y \cdot x) \cdot x = y$. Show that $\cdot$ is commutative but not necessarily associative.
-/
theorem putn... | Let $S$ be a set and $\cdot$ be a binary operation on $S$ satisfying: (1) for all $x,y$ in $S$, $x \cdot (x \cdot y) = y$ (2) for all $x,y$ in $S$, $(y \cdot x) \cdot x = y$. Show that $\cdot$ is commutative but not necessarily associative. | null | [
"abstract_algebra"
] | null | null | |
putnam_1977_a6 | 1e6e09d0-9de2-5cdd-b73a-9d253d509438 | train | abbrev putnam_1977_a6_solution : Prop := sorry
-- True
/--
Let $X$ be the square $[0, 1] \times [0, 1]$, and let $f : X \to \mathbb{R}$ be continuous. If $\int_Y f(x, y) \, dx \, dy = 0$ for all squares $Y$ such that
\begin{itemize}
\item[(1)] $Y \subseteq X$,
\item[(2)] $Y$ has sides parallel to those of $X$,
\item[(3... | import Mathlib
open RingHom Set Nat
-- True
/--
Let $X$ be the square $[0, 1] \times [0, 1]$, and let $f : X \to \mathbb{R}$ be continuous. If $\int_Y f(x, y) \, dx \, dy = 0$ for all squares $Y$ such that
\begin{itemize}
\item[(1)] $Y \subseteq X$,
\item[(2)] $Y$ has sides parallel to those of $X$,
\item[(3)] at lea... | import Mathlib
open RingHom Set Nat
abbrev putnam_1977_a6_solution : Prop := sorry
-- True
/--
Let $X$ be the square $[0, 1] \times [0, 1]$, and let $f : X \to \mathbb{R}$ be continuous. If $\int_Y f(x, y) \, dx \, dy = 0$ for all squares $Y$ such that
\begin{itemize}
\item[(1)] $Y \subseteq X$,
\item[(2)] $Y$ has si... | Let $X$ be the square $[0, 1] \times [0, 1]$, and let $f : X \to \mathbb{R}$ be continuous. If $\int_Y f(x, y) \, dx \, dy = 0$ for all squares $Y$ such that
\begin{itemize}
\item[(1)] $Y \subseteq X$,
\item[(2)] $Y$ has sides parallel to those of $X$,
\item[(3)] at least one of $Y$'s sides is contained in the boundary... | Prove that $f(x,y)$ must be identically zero. | [
"analysis"
] | null | null | |
putnam_1963_b1 | f0a91301-efc9-56eb-9fb2-1c2902d9c7f5 | train | abbrev putnam_1963_b1_solution : β€ := sorry
-- 2
/--
For what integer $a$ does $x^2-x+a$ divide $x^{13}+x+90$?
-/
theorem putnam_1963_b1
: β a : β€, (X^2 - X + (C a)) β£ (X ^ 13 + X + (C 90)) β a = putnam_1963_b1_solution :=
sorry | import Mathlib
open Topology Filter Polynomial
-- 2
/--
For what integer $a$ does $x^2-x+a$ divide $x^{13}+x+90$?
-/
theorem putnam_1963_b1
: β a : β€, (X^2 - X + (C a)) β£ (X ^ 13 + X + (C 90)) β a = putnam_1963_b1_solution := by
| import Mathlib
open Topology Filter Polynomial
abbrev putnam_1963_b1_solution : β€ := sorry
-- 2
/--
For what integer $a$ does $x^2-x+a$ divide $x^{13}+x+90$?
-/
theorem putnam_1963_b1
: β a : β€, (X^2 - X + (C a)) β£ (X ^ 13 + X + (C 90)) β a = putnam_1963_b1_solution :=
sorry
| For what integer $a$ does $x^2-x+a$ divide $x^{13}+x+90$? | Show that $a=2$. | [
"algebra"
] | null | null | |
putnam_2013_a2 | b3186283-cef8-501a-ad56-6641dd7c739f | train | theorem putnam_2013_a2
(S : Set β€)
(hS : S = {n : β€ | n > 0 β§ Β¬β m : β€, m ^ 2 = n})
(P : β€ β List β€ β Prop)
(hP : β n a, P n a β
a.length > 0 β§ n < a[0]! β§
(β m : β€, m ^ 2 = n * a.prod) β§
(β i : Fin (a.length - 1), a[i] < a[i+(1:β)]))
(T : β€ β Set β€)
(hT : T = fun n : β€ => {m : β€ | β a : List β€,... | import Mathlib
open Function Set
/--
Let $S$ be the set of all positive integers that are \emph{not} perfect squares. For $n$ in $S$, consider choices of integers
$a_1, a_2, \dots, a_r$ such that $n < a_1< a_2 < \cdots < a_r$
and $n \cdot a_1 \cdot a_2 \cdots a_r$ is a perfect square, and
let $f(n)$ be the minumum o... | import Mathlib
open Function Set
/--
Let $S$ be the set of all positive integers that are \emph{not} perfect squares. For $n$ in $S$, consider choices of integers
$a_1, a_2, \dots, a_r$ such that $n < a_1< a_2 < \cdots < a_r$
and $n \cdot a_1 \cdot a_2 \cdots a_r$ is a perfect square, and
let $f(n)$ be the minumum o... | Let $S$ be the set of all positive integers that are \emph{not} perfect squares. For $n$ in $S$, consider choices of integers
$a_1, a_2, \dots, a_r$ such that $n < a_1< a_2 < \cdots < a_r$
and $n \cdot a_1 \cdot a_2 \cdots a_r$ is a perfect square, and
let $f(n)$ be the minumum of $a_r$ over all such choices. For exam... | null | [
"number_theory",
"algebra"
] | null | null | |
putnam_1991_b2 | 2198c242-28a4-5a47-bfe6-c4b62369f350 | train | theorem putnam_1991_b2
(f g : β β β)
(fgnconst : Β¬β c, f = Function.const β c β¨ g = Function.const β c)
(fgdiff : Differentiable β f β§ Differentiable β g)
(fadd : β x y, f (x + y) = f x * f y - g x * g y)
(gadd : β x y, g (x + y) = f x * g y + g x * f y) :
(deriv f 0 = 0) β (β x, (f x) ^ 2 + (g x) ^ 2 = 1) ... | import Mathlib
open Filter Topology
/--
Suppose $f$ and $g$ are non-constant, differentiable, real-valued functions defined on $(-\infty,\infty)$. Furthermore, suppose that for each pair of real numbers $x$ and $y$,
\begin{align*}
f(x+y)&=f(x)f(y)-g(x)g(y), \\
g(x+y)&=f(x)g(y)+g(x)f(y).
\end{align*}
If $f'(0)=0$, pro... | import Mathlib
open Filter Topology
/--
Suppose $f$ and $g$ are non-constant, differentiable, real-valued functions defined on $(-\infty,\infty)$. Furthermore, suppose that for each pair of real numbers $x$ and $y$,
\begin{align*}
f(x+y)&=f(x)f(y)-g(x)g(y), \\
g(x+y)&=f(x)g(y)+g(x)f(y).
\end{align*}
If $f'(0)=0$, pro... | Suppose $f$ and $g$ are non-constant, differentiable, real-valued functions defined on $(-\infty,\infty)$. Furthermore, suppose that for each pair of real numbers $x$ and $y$,
\begin{align*}
f(x+y)&=f(x)f(y)-g(x)g(y), \\
g(x+y)&=f(x)g(y)+g(x)f(y).
\end{align*}
If $f'(0)=0$, prove that $(f(x))^2+(g(x))^2=1$ for all $x$. | null | [
"analysis"
] | null | null | |
putnam_1982_b3 | 0685f413-dc72-57e4-937e-b82b12f0ccbe | train | abbrev putnam_1982_b3_solution : β := sorry
-- 4/3 * (Real.sqrt 2 - 1)
/--
Let $p_n$ denote the probability that $c + d$ will be a perfect square if $c$ and $d$ are selected independently and uniformly at random from $\{1, 2, 3, \dots, n\}$. Express $\lim_{n \rightarrow \infty} p_n \sqrt{n}$ in the form $r(\sqrt{s} - t... | import Mathlib
open Set Function Filter Topology Polynomial Real
-- 4/3 * (Real.sqrt 2 - 1)
/--
Let $p_n$ denote the probability that $c + d$ will be a perfect square if $c$ and $d$ are selected independently and uniformly at random from $\{1, 2, 3, \dots, n\}$. Express $\lim_{n \rightarrow \infty} p_n \sqrt{n}$ in t... | import Mathlib
open Set Function Filter Topology Polynomial Real
noncomputable abbrev putnam_1982_b3_solution : β := sorry
-- 4/3 * (Real.sqrt 2 - 1)
/--
Let $p_n$ denote the probability that $c + d$ will be a perfect square if $c$ and $d$ are selected independently and uniformly at random from $\{1, 2, 3, \dots, n\}... | Let $p_n$ denote the probability that $c + d$ will be a perfect square if $c$ and $d$ are selected independently and uniformly at random from $\{1, 2, 3, \dots, n\}$. Express $\lim_{n \rightarrow \infty} p_n \sqrt{n}$ in the form $r(\sqrt{s} - t)$ for integers $s$ and $t$ and rational $r$. | The limit equals $\frac{4}{3}(\sqrt{2} - 1)$. | [
"analysis",
"number_theory",
"probability"
] | null | null | |
putnam_1983_a6 | b5f22a24-a779-5cd3-b041-24c5edc75792 | train | abbrev putnam_1983_a6_solution : β := sorry
-- 2 / 9
/--
Let $T$ be the triangle with vertices $(0, 0)$, $(a, 0)$, and $(0, a)$. Find $\lim_{a \to \infty} a^4 \exp(-a^3) \int_T \exp(x^3+y^3) \, dx \, dy$.
-/
theorem putnam_1983_a6
(F : β β β)
(hF : F = fun a β¦ (a ^ 4 / exp (a ^ 3)) * β« x in (0)..a, β« y in (0)..(a - x),... | import Mathlib
open Nat Filter Topology Real
-- 2 / 9
/--
Let $T$ be the triangle with vertices $(0, 0)$, $(a, 0)$, and $(0, a)$. Find $\lim_{a \to \infty} a^4 \exp(-a^3) \int_T \exp(x^3+y^3) \, dx \, dy$.
-/
theorem putnam_1983_a6
(F : β β β)
(hF : F = fun a β¦ (a ^ 4 / exp (a ^ 3)) * β« x in (0)..a, β« y in (0)..(a - ... | import Mathlib
open Nat Filter Topology Real
noncomputable abbrev putnam_1983_a6_solution : β := sorry
-- 2 / 9
/--
Let $T$ be the triangle with vertices $(0, 0)$, $(a, 0)$, and $(0, a)$. Find $\lim_{a \to \infty} a^4 \exp(-a^3) \int_T \exp(x^3+y^3) \, dx \, dy$.
-/
theorem putnam_1983_a6
(F : β β β)
(hF : F = fun a ... | Let $T$ be the triangle with vertices $(0, 0)$, $(a, 0)$, and $(0, a)$. Find $\lim_{a \to \infty} a^4 \exp(-a^3) \int_T \exp(x^3+y^3) \, dx \, dy$. | Show that the integral evaluates to $\frac{2}{9}$. | [
"analysis"
] | null | null | |
putnam_2016_b5 | 24d67365-ed8d-505c-9c01-43ed8a697845 | train | abbrev putnam_2016_b5_solution : Set (Set.Ioi (1 : β) β Set.Ioi (1 : β)) := sorry
-- {f : Set.Ioi (1 : β) β Set.Ioi (1 : β) | β c : β, c > 0 β§ β x : Set.Ioi (1 : β), (f x : β) = x ^ c}
/--
Find all functions $f$ from the interval $(1,\infty)$ to $(1,\infty)$ with the following property: if $x,y \in (1,\infty)$ and $x^2... | import Mathlib
open Polynomial Filter Topology Real Set Nat List
-- {f : Set.Ioi (1 : β) β Set.Ioi (1 : β) | β c : β, c > 0 β§ β x : Set.Ioi (1 : β), (f x : β) = x ^ c}
/--
Find all functions $f$ from the interval $(1,\infty)$ to $(1,\infty)$ with the following property: if $x,y \in (1,\infty)$ and $x^2 \leq y \leq x^... | import Mathlib
open Polynomial Filter Topology Real Set Nat List
abbrev putnam_2016_b5_solution : Set (Set.Ioi (1 : β) β Set.Ioi (1 : β)) := sorry
-- {f : Set.Ioi (1 : β) β Set.Ioi (1 : β) | β c : β, c > 0 β§ β x : Set.Ioi (1 : β), (f x : β) = x ^ c}
/--
Find all functions $f$ from the interval $(1,\infty)$ to $(1,\in... | Find all functions $f$ from the interval $(1,\infty)$ to $(1,\infty)$ with the following property: if $x,y \in (1,\infty)$ and $x^2 \leq y \leq x^3$, then $(f(x))^2 \leq f(y) \leq (f(x))^3$. | Show that the only such functions are the functions $f(x)=x^c$ for some $c>0$. | [
"algebra"
] | null | null | |
putnam_2021_a1 | 981d8408-cce4-5db8-9764-a3148386302a | train | abbrev putnam_2021_a1_solution : β := sorry
-- 578
/--
A grasshopper starts at the origin in the coordinate plane and makes a sequence of hops.
Each hop has length $5$, and after each hop the grasshopper is at a point whose coordinates are both integers; thus, there are $12$ possible locations for the grasshopper after... | import Mathlib
open Filter Topology
-- 578
/--
A grasshopper starts at the origin in the coordinate plane and makes a sequence of hops.
Each hop has length $5$, and after each hop the grasshopper is at a point whose coordinates are both integers; thus, there are $12$ possible locations for the grasshopper after the f... | import Mathlib
open Filter Topology
abbrev putnam_2021_a1_solution : β := sorry
-- 578
/--
A grasshopper starts at the origin in the coordinate plane and makes a sequence of hops.
Each hop has length $5$, and after each hop the grasshopper is at a point whose coordinates are both integers; thus, there are $12$ possib... | A grasshopper starts at the origin in the coordinate plane and makes a sequence of hops.
Each hop has length $5$, and after each hop the grasshopper is at a point whose coordinates are both integers; thus, there are $12$ possible locations for the grasshopper after the first hop.
What is the smallest number of hops nee... | The answer is $578$. | [
"geometry"
] | null | null | |
putnam_2007_a2 | c6577d28-fce8-5bbd-8541-0c30f1d3d79a | train | abbrev putnam_2007_a2_solution : ENNReal := sorry
-- 4
/--
Find the least possible area of a convex set in the plane that intersects both branches of the hyperbola $xy=1$ and both branches of the hyperbola $xy=-1$. (A set $S$ in the plane is called \emph{convex} if for any two points in $S$ the line segment connecting ... | import Mathlib
open MeasureTheory
-- 4
/--
Find the least possible area of a convex set in the plane that intersects both branches of the hyperbola $xy=1$ and both branches of the hyperbola $xy=-1$. (A set $S$ in the plane is called \emph{convex} if for any two points in $S$ the line segment connecting them is contai... | import Mathlib
open MeasureTheory
noncomputable abbrev putnam_2007_a2_solution : ENNReal := sorry
-- 4
/--
Find the least possible area of a convex set in the plane that intersects both branches of the hyperbola $xy=1$ and both branches of the hyperbola $xy=-1$. (A set $S$ in the plane is called \emph{convex} if for ... | Find the least possible area of a convex set in the plane that intersects both branches of the hyperbola $xy=1$ and both branches of the hyperbola $xy=-1$. (A set $S$ in the plane is called \emph{convex} if for any two points in $S$ the line segment connecting them is contained in $S$.) | Show that the minimum is $4$. | [
"geometry"
] | null | null | |
putnam_1967_b4 | 4ae87954-620e-528d-8690-f62ba63ce060 | train | theorem putnam_1967_b4
(n : β)
(lockers : β β Set.Icc 1 n β Bool)
(npos : n β₯ 1)
(hlockers0 : β i : Set.Icc 1 n, lockers 0 i = false)
(hlockersk : β k β Set.Icc 1 n, β i : Set.Icc 1 n, lockers k i = if k β£ i then !(lockers (k - 1) i) else (lockers (k - 1) i))
: β i : Set.Icc 1 n, lockers n i β (β j : β€, j ^ 2 = i) :=
s... | import Mathlib
open Nat Topology Filter
-- Note: uses (β β Set.Icc 1 n β Bool) instead of (Set.Icc 0 n β Set.Icc 1 n β Bool)
/--
A certain locker room contains $n$ lockers numbered $1,2,3,\cdots,n$ and all are originally locked. An attendant performs a sequence of operations $T_1,T_2,\cdots,T_n$ whereby with the oper... | import Mathlib
open Nat Topology Filter
-- Note: uses (β β Set.Icc 1 n β Bool) instead of (Set.Icc 0 n β Set.Icc 1 n β Bool)
/--
A certain locker room contains $n$ lockers numbered $1,2,3,\cdots,n$ and all are originally locked. An attendant performs a sequence of operations $T_1,T_2,\cdots,T_n$ whereby with the oper... | A certain locker room contains $n$ lockers numbered $1,2,3,\cdots,n$ and all are originally locked. An attendant performs a sequence of operations $T_1,T_2,\cdots,T_n$ whereby with the operation $T_k$, $1 \leq k \leq n$, the condition of being locked or unlocked is changed for all those lockers and only those lockers w... | null | [
"number_theory"
] | null | null | |
putnam_2004_a5 | f946682e-4ad5-59db-9896-54771d198134 | train | theorem putnam_2004_a5
(m n : β)
(mnpos : Inhabited (Fin m Γ Fin n))
(adj : (Fin m Γ Fin n) β (Fin m Γ Fin n) β Prop)
(connected : (Fin m Γ Fin n β Prop) β (Fin m Γ Fin n) β (Fin m Γ Fin n) β Prop)
(cmr : (Fin m Γ Fin n β Prop) β β)
(hadj : adj = fun (β¨a, _β©, β¨b, _β©) (β¨c, _β©, β¨d, _β©) β¦ a = c β§ Nat.dist b d = 1 β¨ b = d ... | import Mathlib
open Nat Topology Filter
/--
An $m \times n$ checkerboard is colored randomly: each square is independently assigned red or black with probability $1/2$. We say that two squares, $p$ and $q$, are in the same connected monochromatic region if there is a sequence of squares, all of the same color, starti... | import Mathlib
open Nat Topology Filter
/--
An $m \times n$ checkerboard is colored randomly: each square is independently assigned red or black with probability $1/2$. We say that two squares, $p$ and $q$, are in the same connected monochromatic region if there is a sequence of squares, all of the same color, starti... | An $m \times n$ checkerboard is colored randomly: each square is independently assigned red or black with probability $1/2$. We say that two squares, $p$ and $q$, are in the same connected monochromatic region if there is a sequence of squares, all of the same color, starting at $p$ and ending at $q$, in which successi... | null | [
"combinatorics"
] | null | null | |
putnam_2021_a5 | 6d642e7f-042f-5610-867e-33924f338118 | train | abbrev putnam_2021_a5_solution : Set β := sorry
-- {j : β | Β¬(42 β£ j) β§ Β¬(46 β£ j)}
/--
Let $A$ be the set of all integers $n$ such that $1 \leq n \leq 2021$ and $\gcd(n,2021)=1$. For every nonnegative integer $j$, let $S(j)=\sum_{n \in A}n^j$. Determine all values of $j$ such that $S(j)$ is a multiple of $2021$.
-/
the... | import Mathlib
open Filter Topology
-- {j : β | Β¬(42 β£ j) β§ Β¬(46 β£ j)}
/--
Let $A$ be the set of all integers $n$ such that $1 \leq n \leq 2021$ and $\gcd(n,2021)=1$. For every nonnegative integer $j$, let $S(j)=\sum_{n \in A}n^j$. Determine all values of $j$ such that $S(j)$ is a multiple of $2021$.
-/
theorem putna... | import Mathlib
open Filter Topology
abbrev putnam_2021_a5_solution : Set β := sorry
-- {j : β | Β¬(42 β£ j) β§ Β¬(46 β£ j)}
/--
Let $A$ be the set of all integers $n$ such that $1 \leq n \leq 2021$ and $\gcd(n,2021)=1$. For every nonnegative integer $j$, let $S(j)=\sum_{n \in A}n^j$. Determine all values of $j$ such that ... | Let $A$ be the set of all integers $n$ such that $1 \leq n \leq 2021$ and $\gcd(n,2021)=1$. For every nonnegative integer $j$, let $S(j)=\sum_{n \in A}n^j$. Determine all values of $j$ such that $S(j)$ is a multiple of $2021$. | Show that the values of $j$ in question are those not divisible by either $42$ or $46$. | [
"number_theory"
] | null | null | |
putnam_1972_a3 | 580ea07d-bc3d-55e4-892c-31bc133fb1df | train | abbrev putnam_1972_a3_solution : Set (β β β) := sorry
-- {f | β A B : β, β x β Set.Icc 0 1, f x = A * x + B}
/--
We call a function $f$ from $[0,1]$ to the reals to be supercontinuous on $[0,1]$ if the Cesaro-limit exists for the sequence $f(x_1), f(x_2), f(x_3), \dots$ whenever it does for the sequence $x_1, x_2, x_3 ... | import Mathlib
open EuclideanGeometry Filter Topology Set
-- Note: uses (β β β) instead of (Set.Icc 0 1 β β)
-- {f | β A B : β, β x β Set.Icc 0 1, f x = A * x + B}
/--
We call a function $f$ from $[0,1]$ to the reals to be supercontinuous on $[0,1]$ if the Cesaro-limit exists for the sequence $f(x_1), f(x_2), f(x_3),... | import Mathlib
open EuclideanGeometry Filter Topology Set
-- Note: uses (β β β) instead of (Set.Icc 0 1 β β)
abbrev putnam_1972_a3_solution : Set (β β β) := sorry
-- {f | β A B : β, β x β Set.Icc 0 1, f x = A * x + B}
/--
We call a function $f$ from $[0,1]$ to the reals to be supercontinuous on $[0,1]$ if the Cesaro-... | We call a function $f$ from $[0,1]$ to the reals to be supercontinuous on $[0,1]$ if the Cesaro-limit exists for the sequence $f(x_1), f(x_2), f(x_3), \dots$ whenever it does for the sequence $x_1, x_2, x_3 \dots$. Find all supercontinuous functions on $[0,1]$. | Show that the solution is the set of affine functions. | [
"analysis"
] | null | null | |
putnam_1992_b5 | 09bbb283-72d4-5709-a22d-6620ff1e894c | train | abbrev putnam_1992_b5_solution : Prop := sorry
-- False
/--
Let $D_n$ denote the value of the $(n-1) \times (n-1)$ determinant
\[
\left[
\begin{array}{cccccc}
3 & 1 & 1 & 1 & \cdots & 1 \\
1 & 4 & 1 & 1 & \cdots & 1 \\
1 & 1 & 5 & 1 & \cdots & 1 \\
1 & 1 & 1 & 6 & \cdots & 1 \\
\vdots & \vdots & \vdots & \vdots & \ddot... | import Mathlib
open Topology Filter Nat Function Polynomial
-- False
/--
Let $D_n$ denote the value of the $(n-1) \times (n-1)$ determinant
\[
\left[
\begin{array}{cccccc}
3 & 1 & 1 & 1 & \cdots & 1 \\
1 & 4 & 1 & 1 & \cdots & 1 \\
1 & 1 & 5 & 1 & \cdots & 1 \\
1 & 1 & 1 & 6 & \cdots & 1 \\
\vdots & \vdots & \vdots &... | import Mathlib
open Topology Filter Nat Function Polynomial
abbrev putnam_1992_b5_solution : Prop := sorry
-- False
/--
Let $D_n$ denote the value of the $(n-1) \times (n-1)$ determinant
\[
\left[
\begin{array}{cccccc}
3 & 1 & 1 & 1 & \cdots & 1 \\
1 & 4 & 1 & 1 & \cdots & 1 \\
1 & 1 & 5 & 1 & \cdots & 1 \\
1 & 1 & 1... | Let $D_n$ denote the value of the $(n-1) \times (n-1)$ determinant
\[
\left[
\begin{array}{cccccc}
3 & 1 & 1 & 1 & \cdots & 1 \\
1 & 4 & 1 & 1 & \cdots & 1 \\
1 & 1 & 5 & 1 & \cdots & 1 \\
1 & 1 & 1 & 6 & \cdots & 1 \\
\vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\
1 & 1 & 1 & 1 & \cdots & n+1
\end{array}
\righ... | Prove that the set is not bounded. | [
"linear_algebra",
"analysis"
] | null | null | |
putnam_1971_b1 | d2e47eb7-e0d1-5dbb-a0ff-7866558a4379 | train | theorem putnam_1971_b1
(S : Type*) [Mul S]
(hself : β x : S, x * x = x)
(h2 : β x y z : S, (x * y) * z = (y * z) * x)
: (β x y z : S, (x * y) * z = x * (y * z)) β§ (β x y : S, x * y = y * x) :=
sorry | import Mathlib
open Set MvPolynomial
/--
Let $S$ be a set and let $\cdot$ be a binary operation on $S$ satisfying the two following laws: (1) for all $x$ in $S$, $x = x \cdot x$, (2) for all $x,y,z$ in $S$, $(x \cdot y) \cdot z) = (y \cdot z) \cdot x$. Show that $\cdot$ is associative and commutative.
-/
theorem putn... | import Mathlib
open Set MvPolynomial
/--
Let $S$ be a set and let $\cdot$ be a binary operation on $S$ satisfying the two following laws: (1) for all $x$ in $S$, $x = x \cdot x$, (2) for all $x,y,z$ in $S$, $(x \cdot y) \cdot z) = (y \cdot z) \cdot x$. Show that $\cdot$ is associative and commutative.
-/
theorem putn... | Let $S$ be a set and let $\cdot$ be a binary operation on $S$ satisfying the two following laws: (1) for all $x$ in $S$, $x = x \cdot x$, (2) for all $x,y,z$ in $S$, $(x \cdot y) \cdot z) = (y \cdot z) \cdot x$. Show that $\cdot$ is associative and commutative. | null | [
"abstract_algebra"
] | null | null | |
putnam_1988_a3 | 6738c846-ceb6-5047-adf2-34006728d663 | train | abbrev putnam_1988_a3_solution : Set β := sorry
-- {x | x > 1 / 2}
/--
Determine, with proof, the set of real numbers $x$ for which
\[
\sum_{n=1}^\infty \left( \frac{1}{n} \csc \frac{1}{n} - 1 \right)^x
\]
converges.
-/
theorem putnam_1988_a3
: {x : β | β L : β, Tendsto (fun t β¦ β n in Finset.Icc (1 : β) t, (((1 / n) /... | import Mathlib
open Set Filter Topology
-- {x | x > 1 / 2}
/--
Determine, with proof, the set of real numbers $x$ for which
\[
\sum_{n=1}^\infty \left( \frac{1}{n} \csc \frac{1}{n} - 1 \right)^x
\]
converges.
-/
theorem putnam_1988_a3
: {x : β | β L : β, Tendsto (fun t β¦ β n in Finset.Icc (1 : β) t, (((1 / n) / Real.... | import Mathlib
open Set Filter Topology
abbrev putnam_1988_a3_solution : Set β := sorry
-- {x | x > 1 / 2}
/--
Determine, with proof, the set of real numbers $x$ for which
\[
\sum_{n=1}^\infty \left( \frac{1}{n} \csc \frac{1}{n} - 1 \right)^x
\]
converges.
-/
theorem putnam_1988_a3
: {x : β | β L : β, Tendsto (fun t ... | Determine, with proof, the set of real numbers $x$ for which
\[
\sum_{n=1}^\infty \left( \frac{1}{n} \csc \frac{1}{n} - 1 \right)^x
\]
converges. | Show that the series converges if and only if $x > \frac{1}{2}$. | [
"analysis"
] | null | null | |
putnam_2004_b6 | 1265fcdb-07f6-5762-bc4f-216d7e6c660a | train | theorem putnam_2004_b6
(A B : Set β)
(N : β β β)
(b : β β β)
(Anempty : A.Nonempty)
(Apos : β a β A, a > 0)
(hN : β x : β, N x = Set.encard {a : A | a β€ x})
(hB : B = {b' > 0 | β a β A, β a' β A, b' = a - a'})
(hbB : Set.range b = B β§ β i : β, b i < b (i + 1)) :
(β r : β, β i : β, (b (i + 1) - b i) β₯ ... | import Mathlib
open Nat Topology Filter
/--
Let $\mathcal{A}$ be a non-empty set of positive integers, and let $N(x)$ denote the number of elements of $\mathcal{A}$ not exceeding $x$. Let $\mathcal{B}$ denote the set of positive integers $b$ that can be written in the form $b=a-a'$ with $a \in \mathcal{A}$ and $a' \i... | import Mathlib
open Nat Topology Filter
/--
Let $\mathcal{A}$ be a non-empty set of positive integers, and let $N(x)$ denote the number of elements of $\mathcal{A}$ not exceeding $x$. Let $\mathcal{B}$ denote the set of positive integers $b$ that can be written in the form $b=a-a'$ with $a \in \mathcal{A}$ and $a' \i... | Let $\mathcal{A}$ be a non-empty set of positive integers, and let $N(x)$ denote the number of elements of $\mathcal{A}$ not exceeding $x$. Let $\mathcal{B}$ denote the set of positive integers $b$ that can be written in the form $b=a-a'$ with $a \in \mathcal{A}$ and $a' \in \mathcal{A}$. Let $b_1<b_2<\cdots$ be the me... | null | [
"analysis"
] | null | null | |
putnam_2002_b3 | 952c4a96-1ce1-5ece-9077-347c15abd161 | train | theorem putnam_2002_b3
(e : β)
(he : e = Real.exp 1)
(f : β€ β β)
(hf : f = fun n : β€ => 1/e - (1 - 1/n)^n)
: β n : β€, n > 1 β 1/(2*n*e) < f n β§ f n < 1/(n*e) :=
sorry | import Mathlib
open Nat Set Topology Filter
/--
Show that, for all integers $n > 1$,
\[
\frac{1}{2ne} < \frac{1}{e} - \left( 1 - \frac{1}{n} \right)^n
< \frac{1}{ne}.
\]
-/
theorem putnam_2002_b3
(e : β)
(he : e = Real.exp 1)
(f : β€ β β)
(hf : f = fun n : β€ => 1/e - (1 - 1/n)^n)
: β n : β€, n > 1 β 1/(2*n*e) < f n β§ f... | import Mathlib
open Nat Set Topology Filter
/--
Show that, for all integers $n > 1$,
\[
\frac{1}{2ne} < \frac{1}{e} - \left( 1 - \frac{1}{n} \right)^n
< \frac{1}{ne}.
\]
-/
theorem putnam_2002_b3
(e : β)
(he : e = Real.exp 1)
(f : β€ β β)
(hf : f = fun n : β€ => 1/e - (1 - 1/n)^n)
: β n : β€, n > 1 β 1/(2*n*e) < f n β§ f... | Show that, for all integers $n > 1$,
\[
\frac{1}{2ne} < \frac{1}{e} - \left( 1 - \frac{1}{n} \right)^n
< \frac{1}{ne}.
\] | null | [
"algebra"
] | null | null | |
putnam_1997_a5 | eb42a57e-312a-584d-9ba3-29acb484c31d | train | abbrev putnam_1997_a5_solution : Prop := sorry
-- True
/--
Let $N_n$ denote the number of ordered $n$-tuples of positive integers $(a_1,a_2,\ldots,a_n)$ such that $1/a_1 + 1/a_2 +\ldots + 1/a_n=1$. Determine whether $N_{10}$ is even or odd.
-/
theorem putnam_1997_a5
(N : (n : β+) β Set (Fin n β β+))
(hN : N = fun (n :... | import Mathlib
open Filter Topology
-- True
/--
Let $N_n$ denote the number of ordered $n$-tuples of positive integers $(a_1,a_2,\ldots,a_n)$ such that $1/a_1 + 1/a_2 +\ldots + 1/a_n=1$. Determine whether $N_{10}$ is even or odd.
-/
theorem putnam_1997_a5
(N : (n : β+) β Set (Fin n β β+))
(hN : N = fun (n : β+) => {... | import Mathlib
open Filter Topology
abbrev putnam_1997_a5_solution : Prop := sorry
-- True
/--
Let $N_n$ denote the number of ordered $n$-tuples of positive integers $(a_1,a_2,\ldots,a_n)$ such that $1/a_1 + 1/a_2 +\ldots + 1/a_n=1$. Determine whether $N_{10}$ is even or odd.
-/
theorem putnam_1997_a5
(N : (n : β+) ... | Let $N_n$ denote the number of ordered $n$-tuples of positive integers $(a_1,a_2,\ldots,a_n)$ such that $1/a_1 + 1/a_2 +\ldots + 1/a_n=1$. Determine whether $N_{10}$ is even or odd. | Show that $N_{10}$ is odd. | [
"number_theory"
] | null | null | |
putnam_1985_b5 | c8adf7a3-ec50-514c-947d-01b392fc2971 | train | abbrev putnam_1985_b5_solution : β := sorry
-- sqrt (Real.pi / 1985) * exp (-3970)
/--
Evaluate $\int_0^\infty t^{-1/2}e^{-1985(t+t^{-1})}\,dt$. You may assume that $\int_{-\infty}^\infty e^{-x^2}\,dx = \sqrt{\pi}$.
-/
theorem putnam_1985_b5
(fact : β« x in univ, exp (- x ^ 2) = sqrt (Real.pi))
: (β« t in Set.Ioi 0, t ^ ... | import Mathlib
open Set Filter Topology Real Polynomial Function
-- sqrt (Real.pi / 1985) * exp (-3970)
/--
Evaluate $\int_0^\infty t^{-1/2}e^{-1985(t+t^{-1})}\,dt$. You may assume that $\int_{-\infty}^\infty e^{-x^2}\,dx = \sqrt{\pi}$.
-/
theorem putnam_1985_b5
(fact : β« x in univ, exp (- x ^ 2) = sqrt (Real.pi))
: ... | import Mathlib
open Set Filter Topology Real Polynomial Function
noncomputable abbrev putnam_1985_b5_solution : β := sorry
-- sqrt (Real.pi / 1985) * exp (-3970)
/--
Evaluate $\int_0^\infty t^{-1/2}e^{-1985(t+t^{-1})}\,dt$. You may assume that $\int_{-\infty}^\infty e^{-x^2}\,dx = \sqrt{\pi}$.
-/
theorem putnam_1985_... | Evaluate $\int_0^\infty t^{-1/2}e^{-1985(t+t^{-1})}\,dt$. You may assume that $\int_{-\infty}^\infty e^{-x^2}\,dx = \sqrt{\pi}$. | Show that the integral evaluates to $\sqrt{\frac{\pi}{1985}}e^{-3970}$. | [
"analysis"
] | null | null | |
putnam_1995_a5 | 087c9c7e-b687-5f72-a5ab-bc1edf104827 | train | abbrev putnam_1995_a5_solution : Prop := sorry
-- True
/--
Let $x_{1},x_{2},\dots,x_{n}$ be differentiable (real-valued) functions of a single variable $f$ which satisfy \begin{align*} \frac{dx_{1}}{dt} &= a_{11}x_{1} + a_{12}x_{2} + \cdots + a_{1n}x_{n} \ \frac{dx_{2}}{dt} &= a_{21}x_{1} + a_{22}x_{2} + \cdots + a_{2n... | import Mathlib
open Filter Topology Real
-- True
/--
Let $x_{1},x_{2},\dots,x_{n}$ be differentiable (real-valued) functions of a single variable $f$ which satisfy \begin{align*} \frac{dx_{1}}{dt} &= a_{11}x_{1} + a_{12}x_{2} + \cdots + a_{1n}x_{n} \ \frac{dx_{2}}{dt} &= a_{21}x_{1} + a_{22}x_{2} + \cdots + a_{2n}x_{... | import Mathlib
open Filter Topology Real
abbrev putnam_1995_a5_solution : Prop := sorry
-- True
/--
Let $x_{1},x_{2},\dots,x_{n}$ be differentiable (real-valued) functions of a single variable $f$ which satisfy \begin{align*} \frac{dx_{1}}{dt} &= a_{11}x_{1} + a_{12}x_{2} + \cdots + a_{1n}x_{n} \ \frac{dx_{2}}{dt} &=... | Let $x_{1},x_{2},\dots,x_{n}$ be differentiable (real-valued) functions of a single variable $f$ which satisfy \begin{align*} \frac{dx_{1}}{dt} &= a_{11}x_{1} + a_{12}x_{2} + \cdots + a_{1n}x_{n} \ \frac{dx_{2}}{dt} &= a_{21}x_{1} + a_{22}x_{2} + \cdots + a_{2n}x_{n} \ \vdots && \vdots \ \frac{dx_{n}}{dt} &= a_{n1}x_{1... | Show that the answer is yes, the functions must be linearly dependent. | [
"linear_algebra",
"analysis"
] | null | null | |
putnam_1966_a3 | bdecfaad-b681-566f-baa4-9101bbc7c3e5 | train | theorem putnam_1966_a3
(x : β β β)
(hx1 : 0 < x 1 β§ x 1 < 1)
(hxi : β n β₯ 1, x (n + 1) = (x n) * (1 - (x n)))
: Tendsto (fun n : β => n * (x n)) atTop (π 1) :=
sorry | import Mathlib
open Topology Filter
/--
If $0 < x_1 < 1$ and $x_{n+1} = x_n(1 - x_n)$ for all $n \ge 1$, prove that $\lim_{n \to \infty} nx_n = 1$.
-/
theorem putnam_1966_a3
(x : β β β)
(hx1 : 0 < x 1 β§ x 1 < 1)
(hxi : β n β₯ 1, x (n + 1) = (x n) * (1 - (x n)))
: Tendsto (fun n : β => n * (x n)) atTop (π 1) := by
| import Mathlib
open Topology Filter
/--
If $0 < x_1 < 1$ and $x_{n+1} = x_n(1 - x_n)$ for all $n \ge 1$, prove that $\lim_{n \to \infty} nx_n = 1$.
-/
theorem putnam_1966_a3
(x : β β β)
(hx1 : 0 < x 1 β§ x 1 < 1)
(hxi : β n β₯ 1, x (n + 1) = (x n) * (1 - (x n)))
: Tendsto (fun n : β => n * (x n)) atTop (π 1) :=
sorry
| If $0 < x_1 < 1$ and $x_{n+1} = x_n(1 - x_n)$ for all $n \ge 1$, prove that $\lim_{n \to \infty} nx_n = 1$. | null | [
"analysis"
] | null | null |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.