Problem

Source: Indonesia TST 2016 Round 3

Tags: geometry, geometric inequality, perimeter, inequalities



Given a convex polygon with $n$ sides and perimeter $S$, which has an incircle $\omega$ with radius $R$. A regular polygon with $n$ sides, whose vertices lie on $\omega$, has a perimeter $s$. Determine whether the following inequality holds: \[ S \ge \frac{2sRn}{\sqrt{4n^2R^2-s^2}}. \]