## Problem of the week

### 2016-23 Inequality on complex numbers

Suppose that $$z_1, z_2, \dots, z_n$$ are complex numbers satisfying $$\sum_{k=1}^n z_k = 0$$. Prove that $\sum_{k=1}^n |z_{k+1} - z_k|^2 \geq 4 \sin^2 \left( \frac{\pi}{n} \right) \sum_{k=1}^n |z_k|^2,$ where we let $$z_{n+1} = z_1$$.