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Points and triangles in the plane

Source: Brazilian Mathematical Olympiad 2018 - Q6

November 16, 2018

combinatorial geometrycombinatoricsBrazilian Math OlympiadBrazilian Math Olympiad 2018geometry

Problem Statement

Consider

4n

points in the plane, with no three points collinear. Using these points as vertices, we form

\binom{4n}{3}

triangles. Show that there exists a point

X

of the plane that belongs to the interior of at least

2n^3

of these triangles.