BALKAN 2007

Let $ABCD$ a convex quadrilateral with $AB=BC=CD$ , with $AC$ not equal to $BD$ and $E$ be the intersection point of it's diagonals. Prove that $AE=DE$ if and only if $\angle BAD+\angle ADC = 120$ .

Find all real functions $f$ defined on $\mathbb R$ , such that $$f(f(x)+y) = f(f(x)-y)+4f(x)y ,$$ for all real numbers $x,y$ .

Find all positive integers $n$ such that there exist a permutation $\sigma$ on the set $\{1,2,3, \ldots, n\}$ for which $$\sqrt{\sigma(1)+\sqrt{\sigma(2)+\sqrt{\ldots+\sqrt{\sigma(n-1)+\sqrt{\sigma(n)}}}}}$$ is a rational number.