IMO 1963 Problem B2 Prove that cos π/7 - cos 2π/7 + cos 3π/7 = 1/2.
Solution
Consider the roots of x 7 + 1 = 0. They are e iπ/7 , e i3π/7 , ... , e i13π/7 and must have sum zero since there is no x 6 term. Hence, in particular, their real parts sum to zero. But cos7π/7 = - 1 and the others are equal in pairs, because cos(2π - x) = cos x. So we get cos π/7 + cos 3π/7 + cos 5π/7 = 1/2. Finally since cos(π - x) = - cos x, cos 5π/7 = - cos 2π/7. 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. 5th IMO 1963 © John Scholes [email protected] 21 Sep 1998 Last corrected/updated 24 Sep 2003
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