Competition problem — Combinatorics

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Combinatorics Difficulty 4.8 AIME Prove it United States

Problem:
Let $ABC$ be a triangle. A line is drawn not passing through any vertex of $ABC$ . Prove that some side of $ABC$ is not cut by the line.

Solution

Solution:
Consider the two sides of the line $\ell$ . By the pigeonhole principle on the three vertices of $ABC$ , two of these vertices, say $A$ and $B$ , lie on the same side of $\ell$ . Then segment $AB$ does not intersect $\ell$ .

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