IMO 1962 Problem B1 Find all real solutions to cos 2 x + cos 2 2x + cos 2 3x = 1.
Put c = cos x, and use cos3x = 4c 3 - 3c, cos 2x = 2 c 2 - 1. We find the equation given is equivalent to c = 0, c 2 = 1/2 or c 2 = 3/4. Hence x = π/2, 3π/2, π/4, 3π/4, π/6, 5π/6 or any multiple of π plus one of these. s are also available in: Samuel L Greitzer, International Mathematical Olympiads 1959-1977, MAA 1978, and in István Reiman, International Mathematical Olympiad 1959-1999, ISBN 189-8855-48- X. 4th IMO 1962 © John Scholes jscholes@kalva.demon.co.uk 19 Sep 1998 Last corrected/updated 24 Sep 2003