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Red geometry!?

Source: Iberoamerican MO 2024 Day 2 P4

September 23, 2024
geometryColoring

Problem Statement

We color some points in the plane with red, in such way that if P,QP,Q are red and XX is a point such that triangle △PQX\triangle PQX has angles 30º,60º,90º30º, 60º, 90º in some order, then XX is also red. If we have vertices A,B,CA, B, C all red, prove that the barycenter of triangle △ABC\triangle ABC is also red.
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