IMO 1959 Problem A3 Let a, b, c be real numbers. Given the equation for cos x: a cos 2 x + b cos x + c = 0, form a quadratic equation in cos 2x whose roots are the same values of x. Compare the equations in cos x and cos 2x for a=4, b=2, c=-1.
You need that cos 2x = 2 cos 2 x - 1. Some easy manipulation then gives: a 2 cos 2 2x + (2a 2 + 4ac - 2b 2 ) cos 2x + (4c 2 + 4ac - 2b 2 + a 2 ) = 0. The equations are the same for the values of a, b, c given. The angles are 2π/5 (or 8π/5) and 4π/5 (or 6π/5). 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. 1st IMO 1959 home © John Scholes jscholes@kalva.demon.co.uk 17 Sep 1998 Last corrected/updated 24 Sep 2003