7th Iberoamerican Mathematical Olympiad Problems 1992

A1.  an is the last digit of 1 + 2 + ... + n. Find a1 + a2 + ... + a1992. A2.  Let f(x) = a1/(x + a1) + a2/(x + a2) + ... + an/(x + an), where ai are unequal positive reals. Find the sum of the lengths of the intervals in which f(x) ≥ 1. A3.  ABC is an equilateral triangle with side 2. Show that any point P on the incircle satisfies PA2 +