Nemotron-IMO-Bench 2026 G18

In a convex pentagon $ABCDE$ with area $S$ triangles $$ABC, BCD, CDE, DEA, EAB$$ have circumradii $R_1,\ldots, R_5$. Prove that $$\sum_{i=1}^5 R_i^4\geq \frac{4}{5\sin^2 108^0} S^2.$$