Problem

Source: ILL 1980-20

Tags: inequalities, geometry, inradius, circumcircle, geometry unsolved



The radii of the circumscribed circle and the inscribed circle of a regular $n$-gon, $n\ge 3$ are denoted by $R_n$ and $r_n$, respectively. Prove that \[\frac{r_n}{R_n}\ge\left(\frac{r_{n+1}}{R_{n+1}}\right)^2.\]