ISL 2015 C2

Let $\mathcal{V}$ be a finite set of points in the plane. We say that $\mathcal{V}$ is balanced if for any two distinct points $A, B \in \mathcal{V}$, there exists a point $C \in \mathcal{V}$ such that $A C=B C$. We say that $\mathcal{V}$ is center-free if for any distinct points $A, B, C \in \mathcal{V}$, there does not exist a point $P \in \mathcal{V}$ such that $P A=P B=P C$. (a) Show that for all $n \geqslant 3$, there exists a balanced set consisting of $n$ points. (b) For which $n \geqslant 3$ does there exist a balanced, center-free set consisting of $n$ points? (Netherlands)