IMO 1962

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Problem B1

Find all real solutions to cos2x + cos22x + cos23x = 1.

 

Solution

Put c = cos x, and use cos3x = 4c3 - 3c, cos 2x = 2 c2 - 1. We find the equation given is equivalent to c = 0, c2 = 1/2 or c2 = 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.

 

4th IMO 1962

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