DK-GEORGMOHR 2004

The width of rectangle $ABCD$ is twice its height, and the height of rectangle $EFCG$ is twice its width. The point $E$ lies on the diagonal $BD$ . Which fraction of the area of the big rectangle is that of the small one?

Show that if $a$ and $b$ are integer numbers, and $a^2 + b^2 + 9ab$ is divisible by $11$ , then $a^2-b^2$ divisible by $11$ .

The digits from $1$ to $9$ are placed in the figure below with one digit in each square. The sum of three numbers placed in the same horizontal or vertical line is $13$ . Show that the marked place says $4$ .

Find all sets $x,y,z$ of real numbers that satisfy $$\begin{cases} x^3 - y^2 = z^2 - x y^3 -z^2 =x^2 -y z^3 -x^2 = y^2 -z \end{cases}$$