Combinatorics · Colourings · Monochromatic configurations · Geometry in space · Midpoints · Pigeonhole
Every point of three-dimensional space is coloured with one of two colours, red or blue, in such a way that
whenever three points \(A\), \(B\), \(C\) have the same colour and \(AB = AC\), the midpoint of the segment \(BC\) has that colour as well.
Prove that there exists a rectangular box all eight of whose vertices have the same colour.
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Serbian National Competition (Drzavno takmicenje) 2016, high school grade I, category A, problem 3. Organized by the Mathematical Society of Serbia (DMS). Source