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HMMT 二月 2017 · 冲刺赛 · 第 25 题

HMMT February 2017 — Guts Round — Problem 25

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [ 15 ] Find all real numbers x satisfying the equation x − 8 = 16 x + 1.
解析
  1. [ 15 ] Find all real numbers x satisfying the equation x − 8 = 16 x + 1. Proposed by: Alexander Katz √ Answer: − 2 , 1 ± 5 √ √ 3 3 3 x − 8 − 1 Let f ( x ) = . Then f ( x ) = 8 x + 8 = 2 x + 1, and so the given equation is equivalent to 8 − 1 f ( x ) = f ( x ). This implies f ( f ( x )) = x . However, as f is monotonically increasing, this implies that 3 x − 8 3 2 f ( x ) = x . As a result, we have = x = ⇒ x − 8 x − 8 = 0 = ⇒ ( x + 2)( x − 2 x − 4) = 0, and so 8 √ x = − 2 , 1 ± 5.