# [Solutions] International Zhautykov Mathematical Olympiad 2014

1. Points $M$, $N$, $K$ lie on the sides $BC$, $CA$, $AB$ of a triangle $ABC$, respectively, and are different from its vertices. The triangle $MNK$ is called beautiful if $\angle BAC=\angle KMN$ and $\angle ABC=\angle KNM$. If in the triangle $ABC$ there are two beautiful triangles with a common vertex, prove that the triangle $ABC$ is right-angled.
2. Does there exist a function $f: \mathbb R \to \mathbb R$ satisfying the following conditions
• for each real $y$ there is a real $x$ such that $f(x)=y$, and
• $f(f(x)) = (x - 1)f(x) + 2$ for all real $x$ ?
3. Given are $100$ different positive integers. We call a pair of numbers good if the ratio of these numbers is either $2$ or $3$. What is the maximum number of good pairs that these $100$ numbers can form? (A number can be used in several pairs.)
4. Does there exist a polynomial $P(x)$ with integral coefficients such that $P(1+\sqrt 3) = 2+\sqrt 3$ and $P(3+\sqrt 5) = 3+\sqrt 5$?.
5. Let $U=\{1, 2,\ldots, 2014\}$. For positive integers $a$, $b$, $c$ we denote by $f(a, b, c)$ the number of ordered 6-tuples of sets $(X_1,X_2,X_3,Y_1,Y_2,Y_3)$ satisfying the following conditions
• $Y_1 \subseteq X_1 \subseteq U$ and $|X_1|=a$;
• $Y_2 \subseteq X_2 \subseteq U\setminus Y_1$ and $|X_2|=b$;
• $Y_3 \subseteq X_3 \subseteq U\setminus (Y_1\cup Y_2)$ and $|X_3|=c$.
Prove that $f(a,b,c)$ does not change when $a$, $b$, $c$ are rearranged.
6. Four segments divide a convex quadrilateral into nine quadrilaterals. The points of intersections of these segments lie on the diagonals of the quadrilateral. It is known that the quadrilaterals $1, 2, 3, 4$ admit inscribed circles. Prove that the quadrilateral $5$ also has an inscribed circle.
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