Given arbitrary complex numbers $w_1,w_2,\ldots,w_n$, show that there exists a positive integer $k\le 2n+1$ for which $\text{Re} (w_1^k+w_2^k+\cdots+w_n^k)\ge 0$.
Problem
Source: India Postal Coaching 2014 Set 1 Problem 4
Tags: complex numbers, algebra unsolved, algebra