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Problem 872

AIME late

Geometry Difficulty 5.7 Prove it

4. 23 The difference between $2 \pi$ and the sum of the face angles associated with a vertex in a convex polyhedron is called the solid angle deficit of that vertex. Prove that the sum of the solid angle deficits of all vertices of a convex polyhedron is $4 \pi$ .

This one wants a proof. Work it on paper, read the official solution, then mark yourself honestly — the ladder only means something if the record is true.

Official solution

None

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.