Nemotron-IMO-Bench 2026 G22

Let $A_1\ldots A_n$ be a cyclic convex polygon whose circumcenter lies strictly inside the polygon and let $B_1,\ldots, B_n$ points on the sides $A_1A_2,\ldots, A_nA_1$ other than the vertices of the polygon. Prove that $$\sum_{i=1}^n \frac{B_iB_{i+1}}{A_iA_{i+2}}>1.$$