1990 USAMO 第 2 题

A sequence of functions $\, \{f_n(x) \} \,$ is defined recursively as follows:

$$$$
\begin{aligned} f_1(x) &= \sqrt {x^2 + 48}, \quad \text{and} \\ f_{n + 1}(x) &= \sqrt {x^2 + 6f_n(x)} \quad \text{for } n \geq 1. \end{aligned}
$$$$

(Recall that $\sqrt {\makebox[5mm]{}}$ is understood to represent the positive square root.) For each positive integer $n$, find all real solutions of the equation $\, f_n(x) = 2x \,$.

函数序列 $\, \{f_n(x) \} \,$ 递归定义如下：

$$

$$

\begin{对齐} f_1(x) &= \sqrt {x^2 + 48}, \quad \text{and} \\ f_{n + 1}(x) &= \sqrt {x^2 + 6f_n(x)} \quad \text{for } n \geq 1。 \end{对齐}

$$

$$

(回想一下 $\sqrt {\makebox[5mm]{}}$ 被理解为表示正平方根。)对于每个正整数 $n$，找到方程 $\, f_n(x) = 2x \,$ 的所有实数解。