2005 Canadian MO Problems/Problem 3

Problem

Let $S$ be a set of $n\ge 3$ points in the interior of a circle.

Show that there are three distinct points $a,b,c\in S$ and three distinct points $A,B,C$ on the circle such that $a$ is (strictly) closer to $A$ than any other point in $S$ , $b$ is closer to $B$ than any other point in $S$ and $c$ is closer to $C$ than any other point in $S$ .
Show that for no value of $n$ can four such points in $S$ (and corresponding points on the circle) be guaranteed.