VN-TST 1985 Problem 4

A convex polygon $A_1,A_2,\cdots ,A_n$ is inscribed in a circle with center $O$ and radius $R$ so that $O$ lies inside the polygon. Let the inradii of the triangles $A_1A_2A_3, A_1A_3A_4, \cdots , A_1A_{n - 1}A_n$ be denoted by $r_1,r_2,\cdots ,r_{n - 2}$ . Prove that $r_1 + r_2 + ... + r_{n - 2}\leq R(n\cos \frac {\pi}{n} - n + 2)$ .