Question 177
A convex hexagon $A B C D E F$ is inscribed in a circle with radius $R$. Diagonals $A D$ and $B E$, $B E$ and $C F$, $A D$ and $C F$ of the hexagon meet at points $M$, $N$, $K$ respectively. Let $r_{1}, r_{2}, r_{3}, r_{4}, r_{5}, r_{6}$ be the inradii of the triangles $A B M$, $B C N$, $C D K$, $D E M$, $E F N$, $A F K$ respectively. Prove that $r_{1}+r_{2}+r_{3}+r_{4}+r_{5}+r_{6} \leqslant R \sqrt{3}$.