exists a circle with only one rational point ?

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0 replies

jlacosta

Apr 2, 2025

0 replies

1 line solution to Inequality

ItzsleepyXD 2

N 2 minutes ago by Vivaandax

Source: Own , Mock Thailand Mathematic Olympiad P8

Let $x_1,x_2,\dots,x_n$ be positive real integer such that $x_1^2+x_2^2+\cdots+x_n^2=2$ Prove that
$\sum_{i=1}^{n}\frac{1}{x_i^3(x_{i-1}+x_{i+1})}\geqslant \left(\sum_{i=1}^{n}\frac{x_i}{x_{i-1}+x_{i+1}}\right)^3$ such that $x_{n+1}=x_1$ and $x_0=x_n$

2 replies

ItzsleepyXD

Today at 9:27 AM

Vivaandax

2 minutes ago

Cool functional equation

Rayanelba 1

N 6 minutes ago by NO_SQUARES

Source: Own

Find all functions $f:\mathbb{Z}_{>0}\to \mathbb{Z}_{>0}$ that verify the following equation for all $x,y\in \mathbb{Z}_{>0}$ :
$max(f^{f(y)}(x),f^{f(y)}(y))|min(x,y)$

1 reply

Rayanelba

14 minutes ago

NO_SQUARES

6 minutes ago

primes,exponentials,factorials

skellyrah 2

N 35 minutes ago by CM1910

find all primes p,q such that $\frac{p^q+q^p-p-q}{p!-q!}$ is a prime number

2 replies

skellyrah

an hour ago

CM1910

35 minutes ago

Queue geo

vincentwant 1

N an hour ago by hukilau17

Let $ABC$ be an acute scalene triangle with circumcenter $O$ . Let $Y, Z$ be the feet of the altitudes from $B, C$ to $AC, AB$ respectively. Let $D$ be the midpoint of $BC$ . Let $\omega_1$ be the circle with diameter $AD$ . Let $Q\neq A$ be the intersection of $(ABC)$ and $\omega$ . Let $H$ be the orthocenter of $ABC$ . Let $K$ be the intersection of $AQ$ and $BC$ . Let $l_1,l_2$ be the lines through $Q$ tangent to $\omega,(AYZ)$ respectively. Let $I$ be the intersection of $l_1$ and $KH$ . Let $P$ be the intersection of $l_2$ and $YZ$ . Let $l$ be the line through $I$ parallel to $HD$ and let $O'$ be the reflection of $O$ across $l$ . Prove that $O'P$ is tangent to $(KPQ)$ .

1 reply

vincentwant

4 hours ago