# [Solutions] Balkan Mathematical Olympiad 2015

1. Let $a, b,c$ be positive real numbers. Prove that  $$a ^ 3b ^ 6 + b ^ 3c ^ 6 + c ^ 3a ^ 6 + 3a ^ 3b ^ 3c ^ 3 \geq abc \left (a ^ 3b ^ 3 + b ^ 3c ^ 3 + c ^ 3a ^ 3 \right) + a ^ 2b ^ 2c ^ 2 \left (a ^ 3 + b ^ 3 + c ^ 3 \right).$$
2. Let $\triangle{ABC}$ be a scalene triangle with incentre $I$ and circumcircle $\omega$. Lines $AI$, $BI$, $CI$ intersect $\omega$ for the second time at points $D$, $E$, $F$, respectively. The parallel lines from $I$ to the sides $BC$, $AC$, $AB$ intersect $EF$, $DF$, $DE$ at points $K$, $L$, $M$, respectively. Prove that the points $K$, $L$, $M$ are collinear.
3. A committee of $3366$ film critics are voting for the Oscars. Every critic voted just an actor and just one actress. After the voting, it was found that for every positive integer $n \in \left \{1, 2, \ldots, 100 \right \}$, there is some actor or some actress who was voted exactly $n$ times. Prove that there are two critics who voted the same actor and the same actress.
4. Prove that among $20$ consecutive positive integers there is an integer $d$ such that for every positive integer $n$ the following inequality holds
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