HMMT 二月 2006 · 冲刺赛 · 第 10 题

HMMT February 2006 — Guts Round — Problem 10

[7] Let f ( x ) = x − 2 x . How many distinct real numbers c satisfy f ( f ( f ( f ( c )))) = 3? 2 n +7 n +136

解析

Let f ( x ) = x − 2 x . How many distinct real numbers c satisfy f ( f ( f ( f ( c )))) = 3? Answer: 9 − 1 − 1 − 1 − 1 Solution: We see the size of the set f ( f ( f ( f (3)))) . Note that f ( x ) = 2 ( x − 1) − 1 = 3 has two solutions: x = 3 and x = − 1, and that the fixed points f ( x ) = x are x = 3 and x = 0. Therefore, the number of real solutions is equal to the number of distinct real numbers c such that c = 3, c = − 1, f ( c ) = − 1 or f ( f ( c )) = − 1, or f ( f ( f ( c ))) = − 1. The equation f ( x ) = − 1 has exactly one root x = 1. Thus, the last three equations are equivalent to c = 1 , f ( c ) = 1, and f ( f ( c )) = 1. f ( c ) = 1 has √ two solutions, c = 1 ± 2, and for each of these two values c there are two preimages. It follows that the answer is 1 + 1 + 1 + 2 + 4 = 9. 2 n +7 n +136