Can we find a finite set of non-coplanar points, such that given any two points, A and B, there are two others, C and D, with the lines AB and CD parallel and distinct?
Solution
To warm up, we may notice that a regular hexagon is a planar set satisfying the condition.
Take two regular hexagons with a common long diagonal and their planes perpendicular. Now if we take A, B in the same hexagon, then we can find C, D in the same hexagon. If we take A in one and B in the other, then we may take C at the opposite end of a long diagonal from A, and D at the opposite end of a long diagonal from B.
Solutions are also available in: Samuel L Greitzer, International Mathematical Olympiads 1959-1977, MAA 1978, and in István Reiman, International Mathematical Olympiad 1959-1999, ISBN 189-8855-48-X.
© John Scholes
jscholes@kalva.demon.co.uk
10 Oct 1998