If there are five points on the surface of a sphere, then no matter how they are arranged at least four of them lie in the same hemisphere.
Proof: Pick any two of the points. These two points, taken with the centre of the sphere, define a plane that cuts the sphere into two hemispheres, both containing the points we picked.
Three points remain, and of these at least two must lie in the same hemisphere, which will then contain the required four points.
Solution to "Five Points of a Sphere"
If there are five points on the surface of a sphere, then no matter how they are arranged at least four of them lie in the same hemisphere.
Proof: Pick any two of the points. These two points, taken with the centre of the sphere, define a plane that cuts the sphere into two hemispheres, both containing the points we picked.
Three points remain, and of these at least two must lie in the same hemisphere, which will then contain the required four points.