Problem

Source: VIII All-Ukrainian Tournament of Young Mathematicians, Qualifying p3

Tags: geometric inequality, geometry, Ukrainian TYM



Find the largest value of the expression $\frac{p}{R}\left( 1- \frac{r}{3R}\right)$ , where $p,R, r$ is, respectively, the perimeter, the radius of the circumscribed circle and the radius of the inscribed circle of a triangle.