返回题库

HMMT 二月 2003 · 冲刺赛 · 第 9 题

HMMT February 2003 — Guts Round — Problem 9

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

  1. [6] For x a real number, let f ( x ) = 0 if x < 1 and f ( x ) = 2 x − 2 if x ≥ 1. How manysolutions are there to the equationf ( f ( f ( f ( x )))) = x ?
    HARVARD-MIT MATHEMATICS TOURNAMENT, MARCH 15, 2003 — GUTS ROUND

英文原题

[6] For x a real number, let f ( x ) = 0 if x < 1 and f ( x ) = 2 x − 2 if x ≥ 1. How many
solutions are there to the equation
f ( f ( f ( f ( x )))) = x ?
1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
HARVARD-MIT MATHEMATICS TOURNAMENT, MARCH 15, 2003 — GUTS ROUND

解析

英文解析

  1. For x a real number, let f ( x ) = 0 if x < 1 and f ( x ) = 2 x − 2 if x ≥ 1. How manysolutions are there to the equationf ( f ( f ( f ( x )))) = x ?
    Solution: 2
    Certainly 0 , 2 are fixed points of f and therefore solutions. On the other hand, therecan be no solutions for x < 0, since f is nonnegative-valued; for 0 < x < 2, we have
    0 ≤ f ( x ) < x < 2 (and f (0) = 0), so iteration only produces values below x , and forx > 2, f ( x ) > x , and iteration produces higher values. So there are no other solutions.