JBMO 2009

Let $ABCDE$ be a convex pentagon such that $AB+CD=BC+DE$ and $k$ a circle with center on side $AE$ that touches the sides $AB$ , $BC$ , $CD$ and $DE$ at points $P$ , $Q$ , $R$ and $S$ (different from vertices of the pentagon) respectively. Prove that lines $PS$ and $AE$ are parallel.

Solve in non-negative integers the equation $2^{a}3^{b} + 9 = c^{2}$

Let $x$ , $y$ , $z$ be real numbers such that $0 < x,y,z < 1$ and $xyz = (1 - x)(1 - y)(1 - z)$ . Show that at least one of the numbers $(1 - x)y,(1 - y)z,(1 - z)x$ is greater than or equal to $\frac {1}{4}$