Nemotron-IMO-Bench 2026 G39

Let $P_1,\ldots, P_{2n}$ be cyclically ordered points on a circle and let $$R_i=\prod_{j\ne i} P_iP_j.$$ Prove that $$\sum_{i=1}^n \frac{1}{R_{2i}}=\sum_{i=1}^n \frac{1}{R_{2i-1}}.$$