Olympiad problems

2020 Caucasus Mathematical Olympiad, 7

A regular triangle $ABC$ is given. Points $K$ and $N$ lie in the segment $AB$, a point $L$ lies in the segment $AC$, and a point $M$ lies in the segment $BC$ so that $CL=AK$, $CM=BN$, $ML=KN$. Prove that $KL \parallel MN$.

2000 All-Russian Olympiad, 6

Tags: number theory unsolved , number theory , easy problem

A perfect number, greater than $6$, is divisible by $3$. Prove that it is also divisible by $9$.

2010 Middle European Mathematical Olympiad, 4

Tags: inequalities , number theory proposed , number theory , easy problem

Find all positive integers $n$ which satisfy the following tow conditions: (a) $n$ has at least four different positive divisors; (b) for any divisors $a$ and $b$ of $n$ satisfying $1<a<b<n$, the number $b-a$ divides $n$. [i](4th Middle European Mathematical Olympiad, Individual Competition, Problem 4)[/i]

2018 Middle European Mathematical Olympiad, 1

Tags: inequalities , am-gm inequality , easy problem

Let $a,b$ and $c$ be positive real numbers satisfying $abc=1.$ Prove that$$\frac{a^2-b^2}{a+bc}+\frac{b^2-c^2}{b+ca}+\frac{c^2-a^2}{c+ab}\leq a+b+c-3.$$

2010 Contests, 4

Tags: inequalities , number theory proposed , number theory , easy problem

Find all positive integers $n$ which satisfy the following tow conditions: (a) $n$ has at least four different positive divisors; (b) for any divisors $a$ and $b$ of $n$ satisfying $1<a<b<n$, the number $b-a$ divides $n$. [i](4th Middle European Mathematical Olympiad, Individual Competition, Problem 4)[/i]