IZhO 2015

Each point with integral coordinates in the plane is coloured white or blue. Prove that one can choose a colour so that for every positive integer $n$ there exists a triangle of area $n$ having its vertices of the chosen colour.

Inside the triangle $A B C$ a point $M$ is given. The line $B M$ meets the side $A C$ at $N$. The point $K$ is symmetrical to $M$ with respect to $A C$. The line $B K$ meets $A C$ at $P$. If $\angle A M P=\angle C M N$, prove that $\angle A B P=\angle C B N$.

Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that

$$f\left(x^{3}+y^{3}+x y\right)=x^{2} f(x)+y^{2} f(y)+f(x y)$$

for all $x, y \in \mathbb{R}$.

Determine the maximum integer $n$ with the property that for each positive integer $k \leq \frac{n}{2}$ there exist two positive divisors of $n$ with difference $k$.

Let $A_{n}$ be the set of partitions of the sequence $(1,2, \ldots, n)$ into several subsequences such that every two neighbouring terms of each subsequence have different parity, and let $B_{n}$ be the set of partitions of the sequence $(1,2, \ldots, n)$ into several subsequences such that all the terms of each subsequence have the same parity. 2

Prove that for every positive integer $n$ the sets $A_{n}$ and $B_{n+1}$ contain the same number of elements.

The area of a convex pentagon $ABCDE$ is $S$ , and the circumradii of the triangles $ABC$ , $BCD$ , $CDE$ , $DEA$ , $EAB$ are $R_1$ , $R_2$ , $R_3$ , $R_4$ , $R_5$ . Prove the inequality $$R_1^4+R_2^4+R_3^4+R_4^4+R_5^4\geq {4\over 5\sin^2 108^\circ}S^2.$$