DK-GEORGMOHR 1996

In triangle $ABC$ , angle $C$ is right and the two catheti are both length $1$ . For one given the choice of the point $P$ on the cathetus $BC$ , the point $Q$ on the hypotenuse and the point $R$ are plotted on the second cathetus so that $PQ$ is parallel to $AC$ and $QR$ is parallel to $BC$ . Thereby the triangle is divided into three parts. Determine the locations of point $P$ for which the rectangular part has a larger area than each of the other two parts.

Determine all sets of real numbers $x,y,z$ which satisfy the system of equations $$\begin{cases} xy = z xz =y yz =x \end{cases}$$

Regarding a natural number $n$ , it is stated that the number $n^2$ has $7$ as the second to last digit. What is the last digit of $n^2$ ?

In a ballroom, seven gentlemen $A, B, C, D, E, F$ and $G$ sit directly across from seven queens $a, b, c, d, e, f$ and $g$ in random order. When the gentlemen rise and walks across the dance floor to bow to each of their ladies, someone notices that at least two of the men travel equally long distances. Will it always be like this? The figure shows an example. In the example, $|Bb| =|Ee|$ and $|Dd|=|Cc|$ .