Problem 1. A finite set S of points in a plane is BALANCED if, for any two points A and B in S, there exists C in S such that AC = BC. We say S is CENTER-FREE if for any different A,B,C in S, there is no point P in S such that PA = PB = PC.
Prove a). for any n > 2, there exists a balanced set of n points.
b). Determine all n > 2 for which there exists a balanced and center-free set of n points.

This problem is not too hard. Just use equilateral triangles and a circle.