BALKAN 2014

Let $x, y$ and $z$ be positive real numbers such that $x y+y z+z x=3 x y z$. Prove that

$$x^{2} y+y^{2} z+z^{2} x \geq 2(x+y+z)-3$$

and determine when equality holds.

A special number is a positive integer $n$ for which there exist positive integers $a, b, c$ and $d$ with

$$n=\frac{a^{3}+2 b^{3}}{c^{3}+2 d^{3}}$$

Prove that: (a) there are infinitely many special numbers; (b) 2014 is not a special number.

Let $A B C D$ be a trapezium inscribed in a circle $\Gamma$ with diameter $A B$. Let $E$ be the intersection point of the diagonals $A C$ and $B D$. The circle with center $B$ and radius $B E$ meets $\Gamma$ at the points $K$ and $L$, where $K$ is on the same side of $A B$ as $C$. The line perpendicular to $B D$ at $E$ intersects $C D$ at $M$.

Prove that $K M$ is perpendicular to $D L$.

Let $n$ be a positive integer. A regular hexagon with side length $n$ is divided into equilateral triangles with side length 1 by lines parallel to its sides. Find the number of regular hexagons all of whose vertices are among the vertices of the equilateral triangles.