Olympiad problems

2020 International Zhautykov Olympiad, 4

In a scalene triangle $ABC$ $I$ is the incentr and $CN$ is the bisector of angle $C$. The line $CN$ meets the circumcircle of $ABC$ again at $M$. The line $l$ is parallel to $AB$ and touches the incircle of $ABC$. The point $R$ on $l$ is such. That $CI \bot IR$. The circumcircle of $MNR$ meets the line $IR$ again at S. Prpve that $AS=BS$.

2020 International Zhautykov Olympiad, 5

Tags: IZHO 2020 , algebra , functional equation

Let $Z$ be the set of all integers. Find all the function $f: Z->Z$ such that $f(4x+3y)=f(3x+y)+f(x+2y)$ For all integers $x,y$

2020 International Zhautykov Olympiad, 6

Tags: combinatorics , IZHO 2020

Some squares of a $n \times n$ tabel ($n>2$) are black, the rest are withe. In every white square we write the number of all the black squares having at least one common vertex with it. Find the maximum possible sum of all these numbers.

2020 International Zhautykov Olympiad, 1

Tags: number theory , zhautykov , IZHO 2020 , Euler s Phi Function , Order , prime numbers

Given natural number n such that, for any natural $a,b$ number $2^a3^b+1$ is not divisible by $n$.Prove that $2^c+3^d$ is not divisible by $n$ for any natural $c$ and $d$

2020 International Zhautykov Olympiad, 3

Tags: geometry , hexagon , Inequality , IZHO 2020 , Inversion

Given convex hexagon $ABCDEF$, inscribed in the circle. Prove that $AC*BD*DE*CE*EA*FB \geq 27 AB * BC * CD * DE * EF * FA$