IZhO 2020

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 $d$

Each of $2k+1$ distinct 7-element subsets of the 20 element set intersects with exactly $k$ of them. Find the maximum possible value of $k$ .

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

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 $AS=BS$ .

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 $x,y$

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.