HMMT 十一月 2010 · 冲刺赛 · 第 22 题

HMMT November 2010 — Guts Round — Problem 22

[ 12 ] Let g ( x ) = (1 + x + x + · · · ) for all values of x for which the right hand side converges. Let 1 3 g ( x ) = g ( g ( x )) for all integers n ≥ 2. What is the largest integer r such that g ( x ) is defined for n 1 n − 1 r some real number x ?

解析

[ 12 ] Let g ( x ) = (1 + x + x + · · · ) for all values of x for which the right hand side converges. Let 1 3 g ( x ) = g ( g ( x )) for all integers n ≥ 2. What is the largest integer r such that g ( x ) is defined for n 1 n − 1 r some 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