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HMMT 二月 2011 · ALGGEO 赛 · 第 9 题

HMMT February 2011 — ALGGEO Round — Problem 9

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. For all real numbers x , let 1 f ( x ) = √ . 2011 2011 1 − x 2011 Evaluate ( f ( f ( . . . ( f (2011)) . . . ))) , where f is applied 2010 times.
解析
  1. For all real numbers x , let 1 √ f ( x ) = . 2011 2011 1 − x 2011 Evaluate ( f ( f ( . . . ( f (2011)) . . . ))) , where f is applied 2010 times. √ 2011 2011 1 − x 2011 Answer: 2011 Direct calculation shows that f ( f ( x )) = and f ( f ( f ( x ))) = x . − x 2011 Hence ( f ( f ( . . . ( f ( x )) . . . ))) = x , where f is applied 2010 times. So ( f ( f ( . . . ( f (2011)) . . . ))) = 2011 2011 .