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HMMT 二月 2016 · 冲刺赛 · 第 13 题

HMMT February 2016 — Guts Round — Problem 13

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [ 9 ] A right triangle has side lengths a , b , and 2016 in some order, where a and b are positive integers.
    Determine the smallest possible perimeter of the triangle.
解析

英文解析

  1. [ 9 ] A right triangle has side lengths a , b , and 2016 in some order, where a and b are positive integers.
    Determine the smallest possible perimeter of the triangle.
    Proposed by: Evan Chen
    √
    Answer: 48 + 2016
    2 2
    There are no integer solutions to a + b = 2016 due to the presence of the prime 7 on the right-handside (by Fermat’ s Christmas Theorem). Assuming a < b , the minimal solution ( a, b ) = (3 , 45) whichgives the answer above.