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HMMT 二月 2011 · 冲刺赛 · 第 7 题

HMMT February 2011 — Guts Round — Problem 7

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [ 6 ] For any positive real numbers a and b , define a ◦ b = a + b + 2 ab . Find all positive real numbers 2 x such that x ◦ 9 x = 121.
解析
  1. [ 6 ] For any positive real numbers a and b , define a ◦ b = a + b + 2 ab . Find all positive real numbers 2 x such that x ◦ 9 x = 121. √ 31 − 3 53 Answer: 2 √ √ √ 2 2 2 Since a ◦ b = ( a + b ) , we have x ◦ 9 x = ( x + 3 x ) . Moreover, since x is positive, we have √ √ √ √ − 3+ 53 31 − 3 53 x + 3 x = 11, and the only possible solution is that x = , so x = . 2 2