DK-GEORGMOHR 2011

Georg writes the numbers from $1$ to $15$ on different pieces of paper. He attempts to sort these pieces of paper into two stacks so that none of the stacks contains two numbers whose sum is a square number.Prove that this is impossible. (The square numbers are the numbers $0 = 0^2$ , $1 = 1^2$ , $4 = 2^2$ , $9 = 3^2$ etc.)

In the octagon below all sides have the length $1$ and all angles are equal. Determine the distance between the corners $A$ and $B$ .

Determine all the ways in which the fraction $\frac{1}{11}$ can be written as $\frac{1}{n}+\frac{1}{m}$ , where $n$ and $m$ are two different positive integers.

A function $f$ is given by $f(x) = x^2 - 2x$ . Prove that there exists a number a which satisfies $f(f(a)) = a$ without satisfying $f(a) = a$ .