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Beautiful Triangles

Source: 2023 RMM, Problem 2

March 1, 2023
combinatoricsRMM 2023

Problem Statement

Fix an integer n≥3n \geq 3. Let S\mathcal{S} be a set of nn points in the plane, no three of which are collinear. Given different points A,B,CA,B,C in S\mathcal{S}, the triangle ABCABC is nice for ABAB if [ABC]≤[ABX][ABC] \leq [ABX] for all XX in S\mathcal{S} different from AA and BB. (Note that for a segment ABAB there could be several nice triangles). A triangle is beautiful if its vertices are all in S\mathcal{S} and is nice for at least two of its sides.
Prove that there are at least 12(n−1)\frac{1}{2}(n-1) beautiful triangles.
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