IZhO 2015 Problem 6

The area of a convex pentagon $ABCDE$ is $S$ , and the circumradii of the triangles $ABC$ , $BCD$ , $CDE$ , $DEA$ , $EAB$ are $R_1$ , $R_2$ , $R_3$ , $R_4$ , $R_5$ . Prove the inequality $$R_1^4+R_2^4+R_3^4+R_4^4+R_5^4\geq {4\over 5\sin^2 108^\circ}S^2.$$