SIMON-MARAIS 2025 B3

Let $n$ be a positive integer and let $\omega = e^{\frac{2\pi i}{n}}$. Suppose $z_1, z_2, \ldots, z_n$ are distinct complex numbers. Define $A$ to be the $n \times n$ matrix whose $(j, k)$-th entry is $(z_j - \omega^k)^n$, for $1 \leq j, k \leq n$.

Given that $\det A = 0$, prove that $z_1z_2 \cdots z_n = 1$.