OME 1992 Problem 6

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