8th Australian Mathematical Olympiad Problems 1987

A1.  ABC is an isosceles triangle with AB = AC. M is the midpoint of AC. D is a point on the arc BC of the circumcircle of BMC not containing M, and the ray BD meets the ray AC at E so that DE = MC. Show that MD2 = AC·CE and CE2 = BC·MD/2. A2.  Show that (2p)!/(p! p!) - 2 is a multiple of p if p is prime. A3.  A graph has 20 points and there is an edge between