Given five points on a sphere, can you always draw an equator such that four or more points lie on one hemisphere? How?
Points on the equator count as being on either side.
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$\begingroup$ Does this answer your question? 5 points on a ball, divide the ball into 2 halves so that one half as exactly 4 points $\endgroup$– BassCommented Mar 29, 2023 at 8:22
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This is a standard application of the pigeonhole principle.
Given 3 points in $\mathbb R^3$, you have a plane that passes through them. Choose 2 points on the sphere and the third point to be the center of the sphere. The plane passing through these three points divides the sphere into two hemispheres. By the PHP, there is a hemisphere with at least 2 points, out of the remaining 3, on it. These two points along with the two points on the plane are 4 points that lie on a closed hemisphere.