13th International Mathematical Olympiad 1971 Problems & Solutions

A1.  Let En = (a1 - a2)(a1 - a3) ... (a1 - an) + (a2 - a1)(a2 - a3) ... (a2 - an) + ... + (an - a1)(an - a2) ... (an - an-1). Let Sn be the proposition that En ≥ 0 for all real ai. Prove that Sn is true for n = 3 and 5, but for no other n > 2. A2.  Let P1 be a convex polyhedron with vertices A1, A2, ... , A9. Let Pi be the polyhedron obtained from P1 by a translation that moves