# [Solutions] International Zhautykov Mathematical Olympiad 2016

1. A quadrilateral $ABCD$ is inscribed in a circle with center $O$. It's diagonals meet at $M$.The circumcircle of $ABM$ intersects the sides $AD$ and $BC$ at $N$ and $K$ respectively. Prove that areas of $NOMD$ and $KOMC$ are equal.
2. Let $a_1,a_2,...,a_{100}$ be permutation of $1,2,...,100$. Let $$S_1=a_1,\, S_2=a_1+a_2,...,S_{100}=a_1+a_2+...+a_{100}.$$ Find the maximum number of perfect squares from $S_i$.
3. There are $60$ towns in Graphland every two countries of which are connected by only a directed way. Prove that we can color four towns to red and four towns to green such that every way between green and red towns are directed from red to green
4. Find all $k>0$ for which a strictly decreasing function $g:(0;+\infty)\to(0;+\infty)$ exists such that $g(x)\geq kg(x+g(x))$ for all positive $x$.
5. A convex hexagon $ABCDEF$ is given such that $AB||DE$, $BC||EF$, $CD||FA$. The point $M, N, K$ are common points of the lines $BD$ and $AE$, $AC$ and $DF$, $CE$ and $BF$ respectively. Prove that perpendiculars drawn from $M$, $N$, $K$ to lines $AB$, $CD$, $EF$ respectively concurrent.
6. We call a positive integer $q$ a convenient denominator for a real number $\alpha$ if $$|\alpha - \dfrac{p}{q}|<\dfrac{1}{10q}$$ for some integer $p$. Prove that if two irrational numbers $\alpha$ and $\beta$ have the same set of convenient denominators then either $\alpha+\beta$ or $\alpha- \beta$ is an integer.
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