Bulletin Boards

Problem of the week

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 \( z_{n+1} = z_1 \).