DK-GEORGMOHR 2015

The numbers $a, b, c, d$ and $e$ satisfy $$a + b < c + d < e + a < b + c < d + e .$$ Which of the numbers is the smallest, and which is the largest?

The numbers $1, 2, 3, . . . , 624$ are paired in such a way that the sum of the two numbers in each pair is $625$ . For example $1$ and $624$ form a pair, and $30$ and $595$ form a pair. In how many of the $312$ pairs does the smaller number evenly divide the larger?

Triangle $ABC$ is equilateral. The point $D$ lies on the extension of $AB$ beyond $B$ , the point $E$ lies on the extension of $CB$ beyond $B$ , and $|CD| = |DE|$ . Prove that $|AD| = |BE|$ .

Determine all numbers $x, y$ and $z$ satisfying the system of equations $$\begin{cases} x^2 + yz = 1 y^2 - xz = 0 z^2 + xy = 1\end{cases}$$

For which numbers $n$ is it possible to put marks on a stick such that all distances $1$ cm, $2$ cm, . . . , $n$ cm each appear exactly once as the distance between two of the marks, and no other distance appears as such a distance?