CMO 2005 Problem 3

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

a) 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 $S$.

b) Show that for no value of $n$ can four such points in $S$ (and corresponding points on the circle) be guaranteed.