IMO Shortlist 2015 problem C2

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Dodao/la: arhiva
30. kolovoza 2018.

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 centre-free if for any three different points A, B and C in \mathcal{S}, there is 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 centre-free set consisting of n points.