Question 55
If $m$ is a number from the set $\{1,2,3,4\}$ and each point of the plane is painted in red or blue, prove that in the plane there exists at least an equilateral triangle with the vertices of the same colour and with length side $m$.
Romanian version in the source (translated): Let $M=\{1,2,3,4\}$. Each point of the plane is colored red or blue. Prove that there exists an equilateral triangle of side length $m\in M$ whose vertices have the same color.
Romanian version in the source (translated): Let $M=\{1,2,3,4\}$. Each point of the plane is colored red or blue. Prove that there exists an equilateral triangle of side length $m\in M$ whose vertices have the same color.