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HMMT 二月 2011 · 冲刺赛 · 第 15 题

HMMT February 2011 — Guts Round — Problem 15

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [ 10 ] Find all irrational numbers x such that x − 17 x and x + 4 x are both rational numbers.
解析
  1. [ 10 ] Find all irrational numbers x such that x − 17 x and x + 4 x are both rational numbers. √ Answer: − 2 ± 5 Guts Round √ 2 2 2 From x + 4 x ∈ Q , we deduce that ( x + 2) = x + 4 x + 4 is also rational, and hence x = − 2 ± y , √ √ 3 where y is rational. Then x − 17 x = (26 − 6 y ) ± ( y − 5) y , which forces y to be 5. Hence x = − 2 ± 5. It is easy to check that both values satisfy the problem conditions.