HMMT 十一月 2010 · 冲刺赛 · 第 22 题

英文解析

22. [ 12 ] Let g ( x ) = (1 + x + x + · · · ) for all values of x for which the right hand side converges. Letg ( x ) = g ( g ( x )) for all integers n ≥ 2. What is the largest integer r such that g ( x ) is defined for 31
    n 1 n − 1 rsome real number x ?
    Answer: 5 Notice that the series is geometric with ratio x , so it converges if − 1 < x < 1. Also
    1 1
    notice that where g ( x ) is defined, it is equal to . The image of g ( x ) is then the interval ( , ∞ ).
    1 1
    3(1 − x ) 6
    1 2
    The image of g ( x ) is simply the values of g ( x ) for x in ( , 1), which is the interval ( , ∞ ). Similarly,
    2 1
    6 5
    5 3 4
    the image of g ( x ) is ( , ∞ ), the image of g ( x ) is ( , ∞ ), and the image of g ( x ) is ( , ∞ ). As this
    3 4 5
    9 4 3
    does not intersect the interval ( − 1 , 1), g ( x ) is not defined for any x , so the answer is 5. 6