IMO 1959

Let a, b, c be real numbers. Given the equation for cos x:

        a cos²x + 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.

 
Solution

You need that cos 2x = 2 cos²x - 1. Some easy manipulation then gives:

a²cos²2x + (2a² + 4ac - 2b²) cos 2x + (4c² + 4ac - 2b² + a²) = 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).

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.

 
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© John Scholes
jscholes@kalva.demon.co.uk
17 Sep 1998
Last corrected/updated 24 Sep 2003