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(x+iy)^n+(x-iy)^n = 2x^n , z = x+iy, |z| = 1

Source: Spanish Mathematical Olympiad 1992 P6

August 1, 2018
complex numbersalgebra

Problem Statement

For a positive integer nn, let S(n)S(n) be the set of complex numbers z=x+iyz = x+iy (x,y∈Rx,y \in R) with ∣z∣=1 |z| = 1 satisfying (x+iy)n+(x−iy)n=2xn(x+iy)^n+(x-iy)^n = 2x^n . (a) Determine S(n)S(n) for n=2,3,4n = 2,3,4. (b) Find an upper bound (depending on nn) of the number of elements of S(n)S(n) for n>5n > 5.
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