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}$

Each one of 2009 distinct points in the plane is coloured in blue or red, so that on every blue-centered unit circle there are exactly two red points. Find the gratest possible number of blue points.