1984 IMO 第 1 题

(FRA 1) Find all solutions of the following system of $n$ equations in $n$ variables: $$\begin{aligned} & x_{1}\left|x_{1}\right|-\left(x_{1}-a\right)\left|x_{1}-a\right|=x_{2}\left|x_{2}\right| \\ & x_{2}\left|x_{2}\right|-\left(x_{2}-a\right)\left|x_{2}-a\right|=x_{3}\left|x_{3}\right| \\ & \cdots \\ & x_{n}\left|x_{n}\right|-\left(x_{n}-a\right)\left|x_{n}-a\right|=x_{1}\left|x_{1}\right| \end{aligned}$$ where $a$ is a given number.

(FRA 1) 找出以下 $n$ 个变量中 $n$ 方程组的所有解： $$\begin{aligned} & x_{1}\left|x_{1}\right|-\left(x_{1}-a\right)\left|x_{1}-a\right|=x_{2}\left|x_{2}\right| \\ & x_{2}\left|x_{2}\right|-\left(x_{2}-a\right)\left|x_{2}-a\right|=x_{3}\left|x_{3}\right| \\ & \cdots \\ & x_{n}\left|x_{n}\right|-\left(x_{n}-a\right)\left|x_{n}-a\right|=x_{1}\left|x_{1}\right| \end{aligned}$$ 其中 $a$ 是给定数字。