MathDB
colouring

Source: Austria 1988

July 8, 2009
combinatorics proposedcombinatorics

Problem Statement

An equilateral triangle A1A2A3 A_1 A_2 A_3 is divided into four smaller equilateral triangles by joining the midpoints A4,A5,A6 A_4,A_5,A_6 of its sides. Let A7,...,A15 A_7,...,A_{15} be the midpoints of the sides of these smaller triangles. The 15 15 points A1,...,A15 A_1,...,A_{15} are colored either green or blue. Show that with any such colouring there are always three mutually equidistant points Ai,Aj,Ak A_i,A_j,A_k having the same color.
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