Let $ABCD$ be a trapezoid such that $AD\parallel BC$. The interior angle bisectors of the corners $A$ and $B$ meet on $[DC]$. If $|BC|=9$ and $|AD|=4$, find $|AB|$.

Find all integer triples $(x,y,z)$ such that \[ \begin{array}{rcl} x-yz &=& 11 \\ xz+y &=& 13. \end{array}\]

In the beginnig, all nine squares of $3\times 3$ chessboard contain $0$. At each step, we choose two squares sharing a common edge, then we add $1$ to them or $-1$ to them. Show that it is not possible to make all squares $2$, after a finite number of steps.