Interesting

by imnotgoodatmathsorry, May 17, 2025, 4:45 PM

Problem 1. Let $x,y,z >0$ . Prove that:
$\frac{108(x^6+y^6)(y^6+z^6)(z^6+x^6)}{x^9y^9z^9} - (xy+yz+zx)^6 \le 135$
Problem 2. Let $a,b,c >0$ . Prove that:
$(a+b+c)^4(ab+bc+ca) - 9\sum{\frac{a}{c}} \ge 54[(a+b)(b+c)(c+a)+abc-1]$

inequalities

Easy geo

by kooooo, Jul 31, 2024, 4:34 PM

In triangle $ABC$ , let $O$ and $H$ be the circumcenter and orthocenter, respectively. Let $M$ and $N$ be the midpoints of $AC$ and $AB$ , respectively, and let $D$ and $E$ be the feet of the perpendiculars from $B$ and $C$ to the opposite sides, respectively. Show that if $X$ is the intersection of $MN$ and $DE$ , then $AX$ is perpendicular to $OH$ .

$n^{22}-1$ and $n^{40}-1$

by v_Enhance, Jan 16, 2024, 3:24 PM

Let $S$ denote the sum of all integers $n$ such that $1 \leq n \leq 2024$ and exactly one of $n^{22}-1$ and $n^{40}-1$ is divisible by $2024$ . Compute the remainder when $S$ is divided by $1000$ .

Raymond Zhu

CSMGO P3: A problem on the infamous line XH

by amar_04, Feb 16, 2021, 6:32 PM

Let $\triangle ABC$ be a scalene triangle with the orthocenter $H$ . Let $B'$ be the reflection of $B$ over $AC$ and $C'$ be the reflection of $C$ over $AB$ . Let the tangents to the circumcircle of $\triangle ABC$ at points $B$ and $C$ meet at a point $X$ . Suppose that the lines $B'C'$ and $BC$ meet at a point $T$ . Prove that $AT$ is perpendicular to $XH$ .

geometry

CSMGO

Hard Function

by johnlp1234, Jul 7, 2020, 3:40 PM

f:R+--->R+:
f(x^3+f(y))=y+(f(x))^3

function

Polish MO Finals 2014, Problem 5

by j___d, Jul 27, 2016, 10:12 PM

Find all pairs $(x,y)$ of positive integers that satisfy
$2^x+17=y^4$ .

contests

Diophantine equation

number theory

Line AT passes through either S_1 or S_2

by v_Enhance, Dec 21, 2015, 4:14 PM

Let $ABC$ be a scalene triangle with circumcircle $\Omega$ , and suppose the incircle of $ABC$ touches $BC$ at $D$ . The angle bisector of $\angle A$ meets $BC$ and $\Omega$ at $E$ and $F$ . The circumcircle of $\triangle DEF$ intersects the $A$ -excircle at $S_1$ , $S_2$ , and $\Omega$ at $T \neq F$ . Prove that line $AT$ passes through either $S_1$ or $S_2$ .

Proposed by Evan Chen

This post has been edited 1 time. Last edited by djmathman, Dec 21, 2015, 8:27 PM

Three mutually tangent circles

by math154, Jul 3, 2012, 3:50 AM

Let $\omega,\omega_1,\omega_2$ be three mutually tangent circles such that $\omega_1,\omega_2$ are externally tangent at $P$ , $\omega_1,\omega$ are internally tangent at $A$ , and $\omega,\omega_2$ are internally tangent at $B$ . Let $O,O_1,O_2$ be the centers of $\omega,\omega_1,\omega_2$ , respectively. Given that $X$ is the foot of the perpendicular from $P$ to $AB$ , prove that $\angle{O_1XP}=\angle{O_2XP}$ .

David Yang.

geometric transformation

IMO LongList 1985 CYP2 - System of Simultaneous Equations

by Amir Hossein, Sep 10, 2010, 10:57 PM

Solve the system of simultaneous equations
$\sqrt x - \frac 1y - 2w + 3z = 1,$ $x + \frac{1}{y^2} - 4w^2 - 9z^2 = 3,$ $x \sqrt x - \frac{1}{y^3} - 8w^3 + 27z^3 = -5,$ $x^2 + \frac{1}{y^4} - 16w^4 - 81z^4 = 15.$