# [Solutions] International Zhautykov Mathematical Olympiad 2015

1. 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.
2. Inside the triangle $ABC$ a point $M$ is given. The line $BM$ meets the side $AC$ at $N$. The point $K$ is symmetrical to $M$ with respect to $AC$. The line $BK$ meets $AC$ at $P$. Prove that if $\angle AMP = \angle CMN$ then $$\angle ABP=\angle CBN.$$
3. Find all functions $f\colon \mathbb{R} \to \mathbb{R}$ such that $$f(x^3+y^3+xy)=x^2f(x)+y^2f(y)+f(xy),\,\forall x,y \in \mathbb{R}.$$
4. Determine the maximum integer $n$ such that for each positive integer $k \le \frac{n}{2}$ there are two positive divisors of $n$ with difference $k$.
5. Let $A_n$ be the set of partitions of the sequence $1,2,..., n$ into several subsequences such that every two neighbouring terms of each subsequence have different parity,and $B_n$ the set of partitions of the sequence $1,2,..., n$ into several subsequences such that all the terms of each subsequence have the same parity (for example,the partition ${(1,4,5,8),(2,3),(6,9),(7)}$ is an element of $A_9$, and the partition ${(1,3,5),(2,4),(6)}$ is an element of $B_6$ ). Prove that for every positive integer $n$ the sets $A_n$ and $B_{n+1}$ contain the same number of elements.
6. 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 $R_1^4+R_2^4+R_3^4+R_4^4+R_5^4\geq {4\over 5\sin^2 108^\circ}S^2.$
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