DE-BUNDESWETTBEWERB 2016

A number with $2016$ zeros that is written as $101010 \dots 0101$ is given, in which the zeros and ones alternate. Prove that this number is not prime.

A triangle $ABC$ with area $1$ is given. Anja and Bernd are playing the following game: Anja chooses a point $X$ on side $BC$ . Then Bernd chooses a point $Y$ on side $CA$ und at last Anja chooses a point $Z$ on side $AB$ . Also, $X,Y$ and $Z$ cannot be a vertex of triangle $ABC$ . Anja wants to maximize the area of triangle $XYZ$ and Bernd wants to minimize that area. What is the area of triangle $XYZ$ at the end of the game, if both play optimally?

Let $A,B,C$ and $D$ be points on a circle in this order. The chords $AC$ and $BD$ intersect in point $P$ . The perpendicular to $AC$ through C and the perpendicular to $BD$ through $D$ intersect in point $Q$ . Prove that the lines $AB$ and $PQ$ are perpendicular.