HMMT 十一月 2014 · 团队赛 · 第 3 题

HMMT November 2014 — Team Round — Problem 3

[ 5 ] The side lengths of a triangle are distinct positive integers. One of the side lengths is a multiple of 42, and another is a multiple of 72. What is the minimum possible length of the third side? Enumeration

解析

[ 5 ] The side lengths of a triangle are distinct positive integers. One of the side lengths is a multiple of 42, and another is a multiple of 72. What is the minimum possible length of the third side? Answer: 7 Suppose that two of the side lengths are 42 a and 72 b , for some positive integers a and b . Let c be the third side length. We know that 42 a is not equal to 72 b , since the side lengths are distinct. Also, 6 | 42 a − 72 b . Therefore, by the triangle inequality, we get c > | 42 a − 72 b | ≥ 6 and thus c ≥ 7 . Hence, the minimum length of the third side is 7 and equality is obtained when a = 7 and b = 4.