IZhO 2018

Let $\alpha,\beta,\gamma$ measures of angles of opposite to the sides of triangle with measures $a,b,c$ respectively.Prove that $$2(cos^2\alpha+cos^2\beta+cos^2\gamma)\geq \frac{a^2}{b^2+c^2}+\frac{b^2}{a^2+c^2}+\frac{c^2}{a^2+b^2}$$

Let $N,K,L$ be points on $AB,BC,CA$ such that $CN$ bisector of angle $\angle ACB$ and $AL=BK$ .Let $BL\cap AK=P$ .If $I,J$ be incenters of triangles $\triangle BPK$ and $\triangle ALP$ and $IJ\cap CN=Q$ prove that $IQ=JP$

Prove that there exist infinitely pairs $(m,n)$ such that $m+n$ divides $(m!)^n+(n!)^m+1$

Crocodile chooses $1$ x $4$ tile from $2018$ x $2018$ square.The bear has tilometer that checks $3$ x $3$ square of $2018$ x $2018$ is there any of choosen cells by crocodile.Tilometer says "YES" if there is at least one choosen cell among checked $3$ x $3$ square.For what is the smallest number of such questions the Bear can certainly get an affirmative answer?

Find all real numbers $a$ such that there exist $f:\mathbb{R} \to \mathbb{R}$ with $$f(x-f(y))=f(x)+a[y]$$ for all $x,y\in \mathbb{R}$

In a circle with a radius $R$ a convex hexagon is inscribed. The diagonals $AD$ and $BE$ , $BE$ and $CF$ , $CF$ and $AD$ of the hexagon intersect at the points $M$ , $N$ and $K$ , respectively. Let $r_1,r_2,r_3,r_4,r_5,r_6$ be the radii of circles inscribed in triangles $ABM,BCN,CDK,DEM,EFN,AFK$ respectively. Prove that. $$r_1+r_2+r_3+r_4+r_5+r_6\leq R\sqrt{3}$$ .