Question 1
We say that a finite set $\mathcal{S}$ of points in the plane is *balanced* if, for any two different points $A$ and $B$ in $\mathcal{S}$, there is a point $C$ in $\mathcal{S}$ such that $AC=BC$. We say that $\mathcal{S}$ is *center-free* if for any three different points $A$, $B$ and $C$ in $\mathcal{S}$, there are no points $P$ in $\mathcal{S}$ such that $PA=PB=PC$.
(a) Show that for all integers $n\ge 3$, there exists a balanced set consisting of $n$ points.
(b) Determine all integers $n\ge 3$ for which there exists a balanced center-free set consisting of $n$ points.
(a) Show that for all integers $n\ge 3$, there exists a balanced set consisting of $n$ points.
(b) Determine all integers $n\ge 3$ for which there exists a balanced center-free set consisting of $n$ points.