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max of p/R ( 1- r/3R )

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

May 25, 2021
geometric inequalitygeometryUkrainian TYM

Problem Statement

Find the largest value of the expression pR(1−r3R)\frac{p}{R}\left( 1- \frac{r}{3R}\right) , where p,R,rp,R, r is, respectively, the perimeter, the radius of the circumscribed circle and the radius of the inscribed circle of a triangle.
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