HMMT 十一月 2016 · 团队赛 · 第 2 题

HMMT November 2016 — Team Round — Problem 2

[ 3 ] What is the smallest possible perimeter of a triangle whose side lengths are all squares of distinct positive integers? 5

解析

[ 3 ] What is the smallest possible perimeter of a triangle whose side lengths are all squares of distinct positive integers? Proposed by: Eshaan Nichani Answer: 77 2 2 2 There exist a triangle with side lengths 4 , 5 , 6 , which has perimeter 77. If the sides have lengths 2 2 2 2 2 2 2 2 a , b , c with 0 < a < b < c , then a + b > c by the triangle inequality. Therefore ( b − 1) + b ≥ 2 2 2 2 2 2 2 2 2 a + b > c ≥ ( b + 1) . Solving this inequality gives b > 4. If b ≥ 6, then a + b + c ≥ 6 + 7 > 77. If b = 5, then c ≥ 7 is impossible, while c = 6 forces a = 4, which gives a perimeter of 77 . 5