We define the polynomial $$P (x) = 2014x^{2013} + 2013x^{2012} +... + 4x^3 + 3x^2 + 2x.$$Find the largest prime divisor of $P (2)$.

The positive integers $a, b, c$ are such that $$gcd \,\,\, (a, b, c) = 1,$$$$gcd \,\,\,(a, b + c) > 1,$$$$gcd \,\,\,(b, c + a) > 1,$$$$gcd \,\,\,(c, a + b) > 1.$$Determine the smallest possible value of $a + b + c$. Clarification: gcd stands for greatest common divisor.

Let $P$ be a point inside the equilateral triangle $ABC$ such that $6\angle PBC = 3\angle PAC = 2\angle PCA$. Find the measure of the angle $\angle PBC$ .

The next board is completely covered with dominoes in an arbitrary manner. a) Prove that we can paint $21$ dominoes in such a way that there are not two dominoes painted forming a $S$-tetramino. b) What is the largest positive integer $k$ for which it is always possible to paint $k$ dominoes (without matter how the board is filled) in such a way that there are not two painted dominoes forming a $S$-tetramine? Clarification: A domino is a $1 \times 2$ or $2 \times 1$ rectangle; the $S$-tetraminos are the figures of the following types: