Problem

Source: Serbian Mathematical Olympiad 2007

Tags: combinatorics proposed, combinatorics



Triangle $\Delta GRB$ is dissected into $25$ small triangles as shown. All vertices of these triangles are painted in three colors so that the following conditions are satisfied: Vertex $G$ is painted in green, vertex $R$ in red, and $B$ in blue; Each vertex on side $GR$ is either green or red, each vertex on $RB$ is either red or blue, and each vertex on $GB$ is either green or blue. The vertices inside the big triangle are arbitrarily colored. Prove that, regardless of the way of coloring, at least one of the $25$ small triangles has vertices of three different colors.


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