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M is a set of n points with should lie on a circle

Source: Romanian IMO Team Selection Test TST 1996, problem 12

September 27, 2005
functionmodular arithmeticalgebradomaincombinatorics proposedcombinatorics

Problem Statement

Let n≥3 n\geq 3 be an integer and let p≥2n−3 p\geq 2n-3 be a prime number. For a set M M of n n points in the plane, no 3 collinear, let f:M→{0,1,…,p−1} f: M\to \{0,1,\ldots, p-1\} be a function such that (i) exactly one point of M M maps to 0, (ii) if a circle C \mathcal{C} passes through 3 distinct points of A,B,C∈M A,B,C\in M then ∑P∈M∩Cf(P)≡0(modp) \sum_{P\in M\cap \mathcal{C}} f(P) \equiv 0 \pmod p . Prove that all the points in M M lie on a circle.
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