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Vertices of a convex polygon if and only if m(S) = f(n)

Source: IMO Shortlist 2000, C3

August 10, 2008
geometrycombinatoricscountingcombinatorial geometryIMO Shortlist

Problem Statement

Let n≥4 n \geq 4 be a fixed positive integer. Given a set S \equal{} \{P_1, P_2, \ldots, P_n\} of n n points in the plane such that no three are collinear and no four concyclic, let at, a_t, 1≤t≤n, 1 \leq t \leq n, be the number of circles PiPjPk P_iP_jP_k that contain Pt P_t in their interior, and let m(S)=a1+a2+⋯+an.m(S)=a_1+a_2+\cdots + a_n. Prove that there exists a positive integer f(n), f(n), depending only on n, n, such that the points of S S are the vertices of a convex polygon if and only if m(S)=f(n). m(S) = f(n).
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