PL-OM 2012

Decide, whether exists positive rational number $w$ , which isn't integer, such that $w^w$ is a rational number.

Determine all pairs $(m, n)$ of positive integers, for which cube $K$ with edges of length $n$ , can be build in with cuboids of shape $m \times 1 \times 1$ to create cube with edges of length $n + 2$ , which has the same center as cube $K$ .

Triangle $ABC$ with $AB = AC$ is inscribed in circle $o$ . Circles $o_1$ and $o_2$ are internally tangent to circle $o$ in points $P$ and $Q$ , respectively, and they are tangent to segments $AB$ and $AC$ , respectively, and they are disjoint with the interior of triangle $ABC$ . Let $m$ be a line tangent to circles $o_1$ and $o_2$ , such that points $P$ and $Q$ lie on the opposite side than point $A$ . Line $m$ cuts segments $AB$ and $AC$ in points $K$ and $L$ , respectively. Prove, that intersection point of lines $PK$ and $QL$ lies on bisector of angle $BAC$ .

$n$ players ( $n \ge 4$ ) took part in the tournament. Each player played exactly one match with every other player, there were no draws. There was no four players $(A, B, C, D)$ , such that $A$ won with $B$ , $B$ won with $C$ , $C$ won with $D$ and $D$ won with $A$ . Determine, depending on $n$ , maximum number of trios of players $(A, B, C)$ , such that $A$ won with $B$ , $B$ won with $C$ and $C$ won with $A$ . (Attention: Trios $(A, B, C)$ , $(B, C, A)$ and $(C, A, B)$ are the same trio.)

Point $O$ is a center of circumcircle of acute triangle $ABC$ , bisector of angle $BAC$ cuts side $BC$ in point $D$ . Let $M$ be a point such that, $MC \perp BC$ and $MA \perp AD$ . Lines $BM$ and $OA$ intersect in point $P$ . Show that circle of center in point $P$ passing through a point $A$ is tangent to line $BC$ .

Show that for any positive real numbers $a, b, c$ true is inequality: $\left(\frac{a - b}{c}\right)^2 + \left(\frac{b - c}{a}\right)^2 + \left(\frac{c - a}{b}\right)^2 \ge 2\sqrt{2}\left(\frac{a - b}{c} + \frac{b - c}{a} + \frac{c - a}{b} \right)$ .