Home > Olympiad > Balkan Mathematical Olympiad – 2015 Problems

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

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