MK-MO 2009

Find all natural numbers $x,y,z$ such that $a+2^x3^y=z^2$ .

Let $O$ be the centre of the incircle of $\triangle ABC$ . Points $K,L$ are the intersection points of the circles circumscribed about triangles $BOC,AOC$ respectively with the bisectors of the angles at $A,B$ respectively $(K,L\not= O)$ . Also $P$ is the midpoint of segment $KL$ , $M$ is the reflection of $O$ with respect to $P$ and $N$ is the reflection of $O$ with respect to line $KL$ . Prove that the points $K,L,M$ and $N$ lie on the same circle.

Let $a,b,c$ be positive real numbers for which $ab+bc+ca=\frac{1}{3}$ . Prove the inequality $$\frac{a}{a^2-bc+1}+\frac{b}{b^2-ca+1}+\frac{c}{c^2-ab+1}\ge\frac{1}{a+b+c}$$