Find a set of 8 points with no three collinear such that no subset of 5 forms a convex pentagon.

this is probably a cheat, but it does work according to the strict wording of the problem since it does not say that the points have to be coplanar.

Pick any three non-colinear points in 3-space.  Pick any point outside of their common plane that is not colinear with any two of them.  Pick the next point such that it is does not lie on any of the previous co-planes and is not co-linear with any 2 previous points. Continue this process and you will have 8 points such that no 4 of them are co-planar and no 3 of them are co-linear, and if no 4 of them are co-planar then obviously no 5 of them are either and thus there can not exist any pentagon convex or otherwise.

Edited on October 9, 2009, 1:24 pm

Posted by Daniel on 2009-10-09 13:21:31