Problem

Source: II International Festival of Young Mathematicians Sozopol 2011, Theme for 10-12 grade

Tags: geometry, Plane, points, lines



There are $n$ points in a plane. Prove that there exist a point $O$ (not necessarily from the given $n$) such that on each side of an arbitrary line, through $O$, lie at least $\frac{n}{3}$ points (including the points on the line).