BALKAN 2002

Consider $n$ points $A_1,A_2,A_3,\ldots, A_n$ ( $n\geq 4$ ) in the plane, such that any three are not collinear. Some pairs of distinct points among $A_1,A_2,A_3,\ldots, A_n$ are connected by segments, such that every point is connected with at least three different points. Prove that there exists $k>1$ and the distinct points $X_1,X_2,\ldots, X_{2k}$ in the set $\{A_1,A_2,A_3,\ldots, A_n\}$ , such that for every $i\in \overline{1,2k-1}$ the point $X_i$ is connected with $X_{i+1}$ , and $X_{2k}$ is connected with $X_1$ .

Let the sequence $\{a_n\}_{n\geq 1}$ be defined by $a_1 = 20$ , $a_2 = 30$ and $a_{n + 2} = 3a_{n + 1} - a_n$ for all $n\geq 1$ . Find all positive integers $n$ such that $1 + 5a_n a_{n + 1}$ is a perfect square.

Two circles with different radii intersect in two points $A$ and $B$ . Let the common tangents of the two circles be $MN$ and $ST$ such that $M,S$ lie on the first circle, and $N,T$ on the second. Prove that the orthocenters of the triangles $AMN$ , $AST$ , $BMN$ and $BST$ are the four vertices of a rectangle.

Determine all functions $f: \mathbb N\to \mathbb N$ such that for every positive integer $n$ we have: $$2n+2001\leq f(f(n))+f(n)\leq 2n+2002.$$