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## Balkan Mathematical Olympiad – 2015 Problems

Problem 1. If ${{a, b}}$ and ${c}$ are positive real numbers, prove that

$\displaystyle a ^ 3b ^ 6 + b ^ 3c ^ 6 + c ^ 3a ^ 6 + 3a ^ 3b ^ 3c ^ 3 \ge{ abc \left (a ^ 3b ^ 3 + b ^ 3c ^ 3 + c ^ 3a ^ 3 \right) + a ^ 2b ^ 2c ^ 2 \left (a ^ 3 + b ^ 3 + c ^ 3 \right)}.$

(Montenegro)

Problem 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. (Cyprus)

Problem 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. (Cyprus)

Problem 4. Prove that among ${20}$ consecutive positive integers there is an integer ${d}$ such that for every positive integer ${n}$ the following inequality holds

$\displaystyle n \sqrt{d} \left\{n \sqrt {d} \right \} > \dfrac{5}{2}$

where by ${\left \{x \right \}}$ denotes the fractional part of the real number ${x}$. The fractional part of the real number ${x}$ is defined as the difference between the largest integer that is less than or equal to ${x}$ to the actual number ${x}$. (Serbia)

Source: AoPS