返回题库

HMMT 二月 2011 · ALGCOMB 赛 · 第 9 题

HMMT February 2011 — ALGCOMB Round — Problem 9

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. For all real numbers x , letf ( x ) = √ .1
    2011
    2011
    1 − x
    2011
    Evaluate ( f ( f ( . . . ( f (2011)) . . . ))) , where f is applied 2010 times.
解析

英文解析

  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 .