Mathematics

Problem

Let

w = e^{i\pi/3}

, where

i = \sqrt{-1}

, and let

a, b, c, x, y,

and

z

be non-zero complex numbers such that a + b + c = x a + bw + cw^2 = y a + bw^2 + cw = z. Find the value of

\frac{|x|^2 + |y|^2 + |z|^2}{|a|^2 + |b|^2 + |c|^2}