each point of the 3-dimensional space is coloured
Source: Bundeswettbewerb Mathematik 1985, first stage, problem 4
May 14, 2004
geometry3D geometryspheregeometry solved
Problem Statement
Each point of the 3-dimensional space is coloured with exactly one of the colours red, green and blue. Let , and , respectively, be the sets of the lengths of those segments in space whose both endpoints have the same colour (which means that both are red, both are green and both are blue, respectively). Prove that at least one of these three sets includes all non-negative reals.