Nemotron-IMO-Bench 2026 G24

The convex polygon $A_1A_2\ldots A_n$ with centroid $G$ is inscribed in a circle with center $O$ and radius $R$. Prove that $$MA_1+\ldots+MA_n\geq nR$$ for any point $M$ in the plane of the polygon if and only if $O=G$.