Problem

Source: Romania IMO TST 1990 p5

Tags: geometry, circumcircle, circles, area, Geometric Inequalities, inequalities



Let $O$ be the circumcenter of an acute triangle $ABC$ and $R$ be its circumcenter. Consider the disks having $OA,OB,OC$ as diameters, and let $\Delta$ be the set of points in the plane belonging to at least two of the disks. Prove that the area of $\Delta$ is greater than $R^2/8$.