23rd International Mathematical Olympiad 1982 Problems & Solutions

A1.  The function f(n) is defined on the positive integers and takes non-negative integer values. f(2) = 0, f(3) > 0, f(9999) = 3333 and for all m, n:       f(m+n) - f(m) - f(n) = 0 or 1. Determine f(1982). A2.  A non-isosceles triangle A1A2A3 has sides a1, a2, a3 with ai opposite Ai. Mi is the midpoint of side ai and Ti is the point where the incircle touches side ai. Denote