Question 154
Complex numbers $\omega_{1}$, $\ldots$, $\omega_{n}$ each have magnitude $1$. Let $z$ be a complex number distinct from $\omega_{1}$, $\ldots$, $\omega_{n}$ such that
$$
\frac{z + \omega_{1}}{z - \omega_{1}} + \dots + \frac{z + \omega_{n}}{z - \omega_{n}} = 0.
$$
Prove that $|z| = 1$.
$$
\frac{z + \omega_{1}}{z - \omega_{1}} + \dots + \frac{z + \omega_{n}}{z - \omega_{n}} = 0.
$$
Prove that $|z| = 1$.