DK-GEORGMOHR 2009

In the figure, triangle $ADE$ is produced from triangle $ABC$ by a rotation by $90^o$ about the point $A$ . If angle $D$ is $60^o$ and angle $E$ is $40^o$ , how large is then angle $u$ ?

Solve the system of equations $$\begin{cases} \dfrac{1}{x+y}+ x = 3 \dfrac{x}{x+y}=2 \end{cases}$$

Georg has bought lots of filled chocolates for a party, and when he counts how many he has, he discovers that the number is a prime number. He distributes so many of the chocolates as possible on $60$ trays with an equal number on each. He notes then that he has more than one piece left and that the number left pieces is not a prime number. How many pieces of chocolate does Georg have left?

Let $E$ be an arbitrary point different from $A$ and $B$ on the side $AB$ of a square $ABCD$ , and let $F$ and $G$ be points on the segment $CE$ so that $BF$ and $DG$ are perpendicular to $CE$ . Prove that $DF = AG$ .

Imagine a square scheme consisting of $n\times n$ fields with edge length $1$ , where $n$ is an arbitrary positive integer. What is the maximum possible length of a route you can follow along the edges of the fields from point $A$ in the lower left corner to point $B$ in the upper right corner if you must never return to one point where you have been before? (The figure shows for $n = 5$ an example of a permitted route and an example of a not permitted route).