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Problem 786

AMC 12 late, AIME early

Geometry Difficulty 4.3 Prove it Berkeley Math Circle: Monthly Contest 1 · United States

A scalene triangle has side lengths which are all prime numbers. What is the smallest possible perimeter it could have?

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

Solution:
The answer is $3 + 5 + 7 = 15$ . This is achievable since the three sides $\{3, 5, 7\}$ form the sides of a triangle.

To show it is best possible, note that $\{3, 5, 7\}$ are the three smallest odd primes; so any smaller triangle would need to have a side of length $2$ . However, the two longer lengths would then be different odd primes; this would cause them to differ by at least $2$ , which is impossible.

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