The plane is colored with two colors so that the following property holds: for each real $a>0$ there is an equilateral triangle of side length $a$ whose $3$ vertices are of the same color. Prove that for any three numbers $a,b,c>0$ for which the sum of any two is greater than the third there is a triangle with sides $a$, $b$, and $c$ whose $3$ vertices are of the same color.
Problem
Source: 2023 Grosman MO, P7
Tags: combinatorical geometry, Coloring, Equilateral Triangle, combinatorics, geometry