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Romanian National Olympiad 2018 - Grade 10 - problem 2

Source: Romania NMO - 2018

April 12, 2018
geometrycircumcirclecomplex numbers

Problem Statement

Let ABCABC be a triangle, OO its circumcenter and R=1R=1 its circumradius. Let G1,G2,G3G_1,G_2,G_3 be the centroids of the triangles OBC,OACOBC, OAC and OAB.OAB. Prove that the triangle ABCABC is equilateral if and only if AG1+BG2+CG3=4AG_1+BG_2+CG_3=4
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