Diffuse $5$ points on a sphere. Prove there is a closed half-sphere that has at least $4$ points on it.
2026-03-31 10:04:26.1774951466
$5$ points on a sphere
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Connect two of the points on a great circle. By the pigeonhole principle, at least $2$ of the remaining $3$ points lie on the same side - giving a closed hemisphere with at least $4$ of the $5$ points.