College Math linear algebra

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Hardest in ARO 2008

discredit 26

N 22 minutes ago by JARP091

Source: ARO 2008, Problem 11.8

In a chess tournament $2n+3$ players take part. Every two play exactly one match. The schedule is such that no two matches are played at the same time, and each player, after taking part in a match, is free in at least $n$ next (consecutive) matches. Prove that one of the players who play in the opening match will also play in the closing match.

26 replies

discredit

Jun 11, 2008

JARP091

22 minutes ago

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Inequality

Kei0923 2

N an hour ago by Kei0923

Source: Own.

Let $k\leq 1$ be a fixed positive real number. Find the minimum possible value $M$ such that for any positive reals $a$ , $b$ , $c$ , $d$ , we have
$\sqrt{\frac{ab}{(a+b)(b+c)}}+\sqrt{\frac{cd}{(c+d)(d+ka)}}\leq M.$

2 replies

Kei0923

Jul 25, 2023

Kei0923

an hour ago

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PAMO 2023 Problem 2

kerryberry 6

N an hour ago by justaguy_69

Source: 2023 Pan African Mathematics Olympiad Problem 2

Find all positive integers $m$ and $n$ with no common divisor greater than 1 such that $m^3 + n^3$ divides $m^2 + 20mn + n^2$ . (Professor Yongjin Song)

6 replies

kerryberry

May 17, 2023

justaguy_69

an hour ago

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My Unsolved Problem

ZeltaQN2008 0

an hour ago

Source: IDK

Given a positive integer $m$ and $a > 1$ . Prove that there always exists a positive integer $n$ such that $m \mid (a^n + n)$ .

P/s: I can prove the problem if $m$ is a power of a prime number, but for arbitrary $m$ then well.....

0 replies

ZeltaQN2008

an hour ago

0 replies

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Computing functions

BBNoDollar 3

N an hour ago by wh0nix

Let $f : [0, \infty) \to [0, \infty)$ , $f(x) = \dfrac{ax + b}{cx + d}$ , with $a, d \in (0, \infty)$ , $b, c \in [0, \infty)$ . Prove that there exists $n \in \mathbb{N}^*$ such that for every $x \geq 0$
$f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0.$ (For $n \in \mathbb{N}^*$ and $x \geq 0$ , the notation $f_n(x)$ represents $\underbrace{(f \circ f \circ \dots \circ f)}_{n \text{ times}}(x)$ . )

3 replies

BBNoDollar

May 21, 2025

wh0nix

an hour ago

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Computing functions

BBNoDollar 8

N an hour ago by wh0nix

Let $f : [0, \infty) \to [0, \infty)$ , $f(x) = \dfrac{ax + b}{cx + d}$ , with $a, d \in (0, \infty)$ , $b, c \in [0, \infty)$ . Prove that there exists $n \in \mathbb{N}^*$ such that for every $x \geq 0$
$f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0.$ (For $n \in \mathbb{N}^*$ and $x \geq 0$ , the notation $f_n(x)$ represents $\underbrace{(f \circ f \circ \dots \circ f)}_{n \text{ times}}(x)$ . )

8 replies

BBNoDollar

May 18, 2025

wh0nix

an hour ago

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Find the remainder

Jackson0423 1

N 2 hours ago by wh0nix

Find the remainder when
$\frac{5^{2000} - 1}{4}$ is divided by $64$ .

1 reply

Jackson0423

3 hours ago

wh0nix

2 hours ago

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IMO 2018 Problem 1

juckter 170

N 2 hours ago by Adywastaken

Let $\Gamma$ be the circumcircle of acute triangle $ABC$ . Points $D$ and $E$ are on segments $AB$ and $AC$ respectively such that $AD = AE$ . The perpendicular bisectors of $BD$ and $CE$ intersect minor arcs $AB$ and $AC$ of $\Gamma$ at points $F$ and $G$ respectively. Prove that lines $DE$ and $FG$ are either parallel or they are the same line.

Proposed by Silouanos Brazitikos, Evangelos Psychas and Michael Sarantis, Greece

170 replies

juckter

Jul 9, 2018

Adywastaken

2 hours ago

J

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Nice "if and only if" function problem

ICE_CNME_4 2

N 2 hours ago by wh0nix

Let $f : [0, \infty) \to [0, \infty)$ , $f(x) = \dfrac{ax + b}{cx + d}$ , with $a, d \in (0, \infty)$ , $b, c \in [0, \infty)$ . Prove that there exists $n \in \mathbb{N}^*$ such that for every $x \geq 0$
$f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0.$ (For $n \in \mathbb{N}^*$ and $x \geq 0$ , the notation $f_n(x)$ represents $\underbrace{(f \circ f \circ \dots \circ f)}_{n \text{ times}}(x)$ . )

Please do it at 9th grade level. Thank you!

2 replies