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HMMT 二月 2019 · 冲刺赛 · 第 10 题

HMMT February 2019 — Guts Round — Problem 10

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [ 5 ] Let √ √ √ √ √ √ √ √ √ √ √ √ A = (1 + 2 2 + 3 3 + 6 6)(2 + 6 2 + 3 + 3 6)(3 + 2 + 6 3 + 2 6)(6 + 3 2 + 2 3 + 6) , √ √ √ √ √ √ √ √ √ √ √ √ B = (1 + 3 2 + 2 3 + 6 6)(2 + 2 + 6 3 + 3 6)(3 + 6 2 + 3 + 2 6)(6 + 2 2 + 3 3 + 6) . Compute the value of A/B .
解析
  1. [ 5 ] Let √ √ √ √ √ √ √ √ √ √ √ √ A = (1 + 2 2 + 3 3 + 6 6)(2 + 6 2 + 3 + 3 6)(3 + 2 + 6 3 + 2 6)(6 + 3 2 + 2 3 + 6) , √ √ √ √ √ √ √ √ √ √ √ √ B = (1 + 3 2 + 2 3 + 6 6)(2 + 2 + 6 3 + 3 6)(3 + 6 2 + 3 + 2 6)(6 + 2 2 + 3 3 + 6) . Compute the value of A/B . Proposed by: Yuan Yao Answer: 1 Note that √ √ √ √ √ √ √ √ A = ((1 + 2 2)(1 + 3 3))((2 + 3)(1 + 3 2))((3 + 2)(1 + 2 3))((3 + 3)(2 + 2)) , √ √ √ √ √ √ √ √ B = ((1 + 3 2)(1 + 2 3))((2 + 2)(1 + 3 3))((3 + 3)(1 + 2 2))((2 + 3)(3 + 2)) . It is not difficult to check that they have the exact same set of factors, so A = B and thus the ratio is