HMMT 十一月 2008 · 冲刺赛 · 第 24 题

HMMT November 2008 — Guts Round — Problem 24

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

解析

[ 12 ] Let f ( x ) = . Let f ( x ) = f ( f ( x )), with f ( x ) = f ( x ). What is f (2008)? 1 − x 4 − 1 2 1 x − 1 3 Answer: Notice that, if x 6 = 0 , 1, then f ( x ) = = , which means that f ( x ) = 1 2007 x 1 − 1 − x 1 n 2007 2008 = x . So f is periodic with period n = 3, which means that f ( x ) = x so f (2008) = x − 1 1 − x − 1 f (2008) = . 2007 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . st 1 HARVARD-MIT NOVEMBER TOURNAMENT, 8 SATURDAY 2008 — GUTS ROUND