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HMMT 十一月 2008 · 冲刺赛 · 第 24 题

HMMT November 2008 — Guts Round — Problem 24

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

  1. [ 12 ] Let f ( x ) = . Let f ( x ) = f ( f ( x )), with f ( x ) = f ( x ). What is f (2008)?
    1 − x
    1 HARVARD-MIT NOVEMBER TOURNAMENT, 8 NOVEMBER 2008 — GUTS ROUNDst

英文原题

[ 12 ] Let f ( x ) = 1
1 − x . Let f k +1 ( x ) = f ( f k ( x )), with f 1 ( x ) = f ( x ). What is f 2008 (2008)?
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 st HARVARD-MIT NOVEMBER TOURNAMENT, 8 NOVEMBER 2008 — GUTS ROUND

解析

英文解析

  1. [ 12 ] Let f ( x ) = . Let f ( x ) = f ( f ( x )), with f ( x ) = f ( x ). What is f (2008)?
    1 − x − 1 2 1 x − 1 34
    Answer: Notice that, if x 6 = 0 , 1, then f ( x ) = = , which means that f ( x ) =
    2007 x 1
    1 −
    1 − xn 2007 20081 = x . So f is periodic with period n = 3, which means that f ( x ) = x so f (2008) =
    x − 1
    1 −
    − 1 xf (2008) = .
    2007
    1 HARVARD-MIT NOVEMBER TOURNAMENT, 8 SATURDAY 2008 — GUTS ROUNDst