# [Solutions] International Zhautykov Mathematical Olympiad 2012

1. An acute triangle $ABC$ is given. Let $D$ be an arbitrary inner point of the side $AB$. Let $M$ and $N$ be the feet of the perpendiculars from $D$ to $BC$ and $AC$, respectively. Let $H_1$ and $H_2$ be the orthocentres of triangles $MNC$ and $MND$, respectively. Prove that the area of the quadrilateral $AH_1BH_2$ does not depend on the position of $D$ on $AB$.
2. A set of (unit) squares of a $n\times n$ table is called convenient if each row and each column of the table contains at least two squares belonging to the set. For each $n\geq 5$ determine the maximum $m$ for which there exists a convenient set made of $m$ squares, which becomes inconvenient when any of its squares is removed.
3. Let $P, Q,R$ be three polynomials with real coefficients such that $P(Q(x)) + P(R(x))=\text{constant}$ for all $x$. Prove that $P(x)=\text{constant}$ or $Q(x)+R(x)=\text{constant}$ for all $x$.
4. Do there exist integers $m, n$ and a function $f\colon \mathbb R \to \mathbb R$ satisfying simultaneously the following two conditions?
• $f(f(x))=2f(x)-x-2$ for any $x \in \mathbb R$;
• $m \leq n$ and $f(m)=n$.
5. Equilateral triangles $ACB'$ and $BDC'$ are drawn on the diagonals of a convex quadrilateral $ABCD$ so that $B$ and $B'$ are on the same side of $AC$, and $C$ and $C'$ are on the same sides of $BD$. Find $\angle BAD + \angle CDA$ if $B'C' = AB+CD$.
6. Find all integer solutions of the equation the equation $2x^2-y^{14}=1$.
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