Find all real solutions to cos^{2}x + cos^{2}2x + cos^{2}3x = 1.

**Solution**

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.

Solutions 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.

© John Scholes

jscholes@kalva.demon.co.uk

19 Sep 1998

Last corrected/updated 24 Sep 2003