IMO Shortlist 2015 problem C2
We say that a finite set of points in the plane is balanced if, for any two different points and in , there is a point in such that . We say that is centre-free if for any three different points , and in , there is no points in such that .
(a) Show that for all integers , there exists a balanced set consisting of points.
(b) Determine all integers for which there exists a balanced centre-free set consisting of points.