5th Iberoamerican Mathematical Olympiad Problems 1990

A1.  The function f is defined on the non-negative integers. f(2n - 1) = 0 for n = 0, 1, 2, ... . If m is not of the form 2n - 1, then f(m) = f(m+1) + 1. Show that f(n) + n = 2k - 1 for some k, and find f(21990). A2.  I is the incenter of the triangle ABC and the incircle touches BC, CA, AB at D, E, F respectively. AD meets the incircle again at P. M