Tri-Colour
Six points are arranged in space so that no three are collinear.
How many line segments can be formed by joining the points in
pairs?
If all the line segments are coloured either red or green prove
that there is a triangle all of whose sides are of the same
colour.
If each point is joined to every other point then this makes (6 x 5)/2 = 15 line segments. All the line segments are coloured either red or green.
Joel of ACS (Barker), Singapore proved that there must then be a triangle all of whose sides are the same colour. This is his solution:
Label the points from A to F. Out of the line segments connected to A, at least 3 have to be the same colour, (let's say AB, AC and AD are red). Then BC has to be green, because otherwise ABC will be all red. Ditto for CD and BD. So BCD will be all green.