IMO 1973 SL 6

Origin: POL

Problem

Does there exist a finite set M of points in space, not all in the same plane, such that for each two points A, B \inM there exist two other points C, D \inM such that lines AB and CD are parallel?

Solution

Yes. Take for M the set of vertices of a cube ABCDEFGH and two points I, J symmetric to the center O of the cube with respect to the laterals ABCD and EFGH. Remark. We prove a stronger result: Given an arbitrary finite set of points S, then there is a finite set M \supsetS with the described property. Choose a point A \inS and any point O such that AO \parallelBC for some two points B, C \inS. Now let X′ be the point symmetric to X with respect to O, and S′ = {X, X′ | X \inS}. Finally, take M = {X, X | X \inS′}, where X denotes the point symmetric to X with respect to A. This M has the desired property: If X, Y \inM and Y ̸= X, then XY \parallelXY ; otherwise, XX, i.e., XA is parallel to X′A′ if X ̸= A′, or to BC otherwise.