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HMMT 十一月 2016 · 团队赛 · 第 2 题

HMMT November 2016 — Team Round — Problem 2

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [ 3 ] What is the smallest possible perimeter of a triangle whose side lengths are all squares of distinctpositive integers? 5
解析

英文解析

  1. [ 3 ] What is the smallest possible perimeter of a triangle whose side lengths are all squares of distinctpositive 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