12th International Mathematical Olympiad 1970 Problems & Solutions

A1.  M is any point on the side AB of the triangle ABC. r, r1, r2 are the radii of the circles inscribed in ABC, AMC, BMC. q is the radius of the circle on the opposite side of AB to C, touching the three sides of AB and the extensions of CA and CB. Similarly, q1 and q2. Prove that r1r2q = rq1q2. A2.  We have 0 ≤ xi < b for i = 0, 1, ... , n and xn > 0, xn-1 > 0. If a > b, and