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Problem 720

AIME late

Geometry Difficulty 5.3 Prove it

Example 2. For any tetrahedron $ABCD$ , there always exists a vertex such that the three edges meeting at this vertex can form a triangle. (10th IMO Problem)

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

Let's use proof by contradiction. Suppose $AB$ is the largest chord, then one of the points $A$ , $B$ must meet the requirement. Otherwise, we have $AB \geqslant AC + AD, \quad AB \geqslant BC + BD$ , but $2AB \geqslant (AC + BC) + (AD + BD)$ $> AB + AB = 2AB$ . This is impossible.

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