OLYMPIC-REVENGE 2016 Problem 3

Let $\Gamma$ a fixed circunference. Find all finite sets $S$ of points in $\Gamma$ such that: For each point $P\in \Gamma$ , there exists a partition of $S$ in sets $A$ and $B$ ( $A\cup B=S$ , $A\cap B=\phi$ ) such that $\sum_{X\in A}PX = \sum_{Y\in B}PY$ .