DANUBE 2015

Let $ABCD$ be a rectangle with $AB\ge BC$ Point $M$ is located on the side $(AD)$ , and the perpendicular bisector of $[MC]$ intersects the line $BC$ at the point $N$ . Let ${Q} =MN\cup AB$ . Knowing that $\angle MQA= 2\cdot \angle BCQ$ , show that the quadrilateral $ABCD$ is a square.

A lantern needs exactly $2$ charged batteries in order to work.We have available $n$ charged batteries and $n$ uncharged batteries, $n\ge 4$ (all batteries look the same). A try consists in introducing two batteries in the lantern and verifying if the lantern works.Prove that we can find a pair of charged batteries in at most $n+2$ tries.