if there are 5 points on a sphere then 4 of them belong to a half-sphere.

geometry pigeonhole-principle

Solution 1:

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.

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