If there are 5 points on the surface of a sphere, then there is a closed half sphere, containing at least 4 of them.
It's in a pigeonhole list of problems. But, I think I have to use rotations in more than 1 dimension.
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Pick two distinct points out of your 5 (if all 5 are identical then they clearly all lie in a single hemisphere). These two points define at least one great circle (if they're antipodal, they define infinitely many); pick a great circle they define. This circle then cuts the sphere into two hemispheres. Now pigeonhole the other three points between these two hemispheres.