IMO Shortlist 2000 problem C3

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Dodao/la: arhiva
2. travnja 2012.
Let n \geq 4 be a fixed positive integer. Given a set S = \{P_1, P_2, \ldots, P_n\} of n points in the plane such that no three are collinear and no four concyclic, let a_t, 1 \leq t \leq n, be the number of circles P_iP_jP_k that contain P_t in their interior, and let m(S) = \sum^n_{i=1} a_i. Prove that there exists a positive integer f(n), depending only on n, such that the points of S are the vertices of a convex polygon if and only if m(S) = f(n).
Izvor: Međunarodna matematička olimpijada, shortlist 2000