19th Balkan 2002


A1. Show that a finite graph in which every point has at least three edges contains an even cycle.
A2. The sequence a_n is defined by a₁ = 20, a₂ = 30, a_n+1 = 3a_n - a_n-1. Find all n for which 5a_n+1a_n + 1 is a square.
A3. Two unequal circles intersect at A and B. The two common tangents touch one circle at P, Q and the other at R, S. Show that the orthocenters of APQ, BPQ, ARS, BRS form a rectangle.
A4. N is the set of positive integers. Find all functions f: N → N such that f( f(n) ) + f(n) = 2n + 2001 or 2n + 2002.

To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.