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Problem 1412

AIME late

Combinatorics Difficulty 5.3 Prove it Thailand Mathematical Olympiad · Thailand

Each point of the plane is colored either red or blue. Show that there exists a triangle with side lengths $1$ , $2$ , $\sqrt{3}$ , and its three vertices are of the same color.

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

Assume on the contrary that there is a coloring for which any triangle with side lengths $1$ , $2$ , $\sqrt{3}$ has at least one red vertex and one blue vertex. Consider an equilateral triangle $ABC$ with side length $2$ .

There are at least two vertices among $A$ , $B$ , $C$ with the same color, let them be $B$ , $C$ with red color.
Let $D$ , $E$ be the midpoints of $AB$ , $AC$ , respectively, and let $D'$ , $E'$ be their reflections with respect to the line $BC$ . The triangles $BDC$ , $BEC$ , $BD'C$ , $BE'C$ all have side lengths $1$ , $2$ , $\sqrt{3}$ , and so the vertices $D$ , $E$ , $D'$ , $E'$ are of blue color. However, the triangle $DD'E$ has side lengths $1$ , $2$ , $\sqrt{3}$ but all of its vertices are of blue color, which is a contradiction.

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.