Let $(a_{n})_{n=1}^{\infty}$ be a sequence of positive real numbers defined by $a_{1}=1$, $a_{2}=2$ and $$\frac{a_{n+1}^{4}}{a_{n}^3} = 2a_{n+2}-a_{n+1}.$$Prove that the following inequality holds for every positive integer $N>1$: $$\sum_{k=1}^{N}\frac{a_{k}^{2}}{a_{k+1}}<3.$$Note: The bound is not sharp. Proposed by Nikola Velov

At a chess tournament, every pair of contestants played each other at most once. If any two con- testants, $A$ and $B$, failed to play each other, then exactly two other contestants, $C$ and $D$, played against both $A$ and $B$ during the tournament. Moreover, no $4$ contestants played exactly $5$ games between them. Prove that every contestant played the same number of games. Proposed by Mirko Petrushevski

Let $ABC$ be a triangle such that $AB<AC$. Let $D$ be a point on the segment $BC$ such that $BD<CD$. The angle bisectors of $\angle ADB$ and $\angle ADC$ meet the segments $AB$ and $AC$ at $E$ and $F$ respectively. Let $\omega$ be the circumcircle of $AEF$ and $M$ be the midpoint of $EF$. The ray $AD$ meets $\omega$ at $X$ and the line through $X$ parallel to $EF$ meets $\omega$ again at $Y$. If $YM$ meets $\omega$ at $T$, show that $AT$, $EF$ and $BC$ are concurrent. Proposed by Nikola Velov

Let $f$ be a non-zero function from the set of positive integers to the set of non-negative integers such that for all positive integers $a$ and $b$ we have $$2f(ab)=(b+1)f(a)+(a+1)f(b).$$Prove that for every prime number $p$ there exists a prime $q$ and positive integers $x_{1}$, ..., $x_{n}$ and $m \geq 0$ so that $$\frac{f(q^{p})}{f(q)} = (px_{1}+1) \cdot ... \cdot (px_{n}+1) \cdot p^{m},$$where the integers $px_{1}+1$,..., $px_{n}+1$ are all prime. Proposed by Nikola Velov