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Harvard-MIT Mathematics Tournament

Geometry Difficulty 4.1 AIME Prove it United States

Problem:
How many different values can $\angle ABC$ take, where $A, B, C$ are distinct vertices of a cube?

Solution

Solution:
Answer: 5. In a unit cube, there are 3 types of triangles, with side lengths $(1, 1, \sqrt{2})$ , $(1, \sqrt{2}, \sqrt{3})$ and $(\sqrt{2}, \sqrt{2}, \sqrt{2})$ . Together they generate 5 different angle values.

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.