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Sum of f(P) where P is on circle C and in the set M is 0

Source: Romanian TST 1997

September 17, 2011
functioncombinatorics proposedcombinatorics

Problem Statement

Let n≥4n\ge 4 be a positive integer and let MM be a set of nn points in the plane, where no three points are collinear and not all of the nn points being concyclic. Find all real functions f:M→Rf:M\to\mathbb{R} such that for any circle C\mathcal{C} containing at least three points from MM, the following equality holds: ∑P∈C∩Mf(P)=0\sum_{P\in\mathcal{C}\cap M} f(P)=0 Dorel Mihet
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