DANUBE 2011

Let $ABCM$ be a quadrilateral and $D$ be an interior point such that $ABCD$ is a parallelogram. It is known that $\angle AMB =\angle CMD$ . Prove that $\angle MAD =\angle MCD$ .

Let S be a set of positive integers such that: min { lcm (x, y) : x, y ∈ S, $x \neq y$ } $\ge$ 2 + max S. Prove that $\sum\limits_{x \in S} \frac{1}{x} \le \frac{3}{2}$ .

Determine all positive integer numbers $n$ satisfying the following condition: the sum of the squares of any $n$ prime numbers greater than $3$ is divisible by $n$ .

Given a positive integer number $n$ , determine the maximum number of edges a triangle-free Hamiltonian simple graph on $n$ vertices may have.