RS-SMO 2008

Find all nonegative integers $x,y,z$ such that $12^x+y^4=2008^z$

Triangle $\triangle ABC$ is given. Points $D$ i $E$ are on line $AB$ such that $D - A - B - E, AD = AC$ and $BE = BC$ . Bisector of internal angles at $A$ and $B$ intersect $BC,AC$ at $P$ and $Q$ , and circumcircle of $ABC$ at $M$ and $N$ . Line which connects $A$ with center of circumcircle of $BME$ and line which connects $B$ and center of circumcircle of $AND$ intersect at $X$ . Prove that $CX \perp PQ$ .

In a convex pentagon $ABCDE$ , let $\angle EAB = \angle ABC = 120^{\circ}$ , $\angle ADB = 30^{\circ}$ and $\angle CDE = 60^{\circ}$ . Let $AB = 1$ . Prove that the area of the pentagon is less than $\sqrt {3}$ .