Question 1
(FRA 1) Find all solutions of the following system of $n$ equations in $n$ variables:
$$\begin{aligned}x_1|x_1|-(x_1-a)|x_1-a|&=x_2|x_2|,\\x_2|x_2|-(x_2-a)|x_2-a|&=x_3|x_3|,\\&\cdots\\x_n|x_n|-(x_n-a)|x_n-a|&=x_1|x_1|,\end{aligned}$$
where $a$ is a given number.
$$\begin{aligned}x_1|x_1|-(x_1-a)|x_1-a|&=x_2|x_2|,\\x_2|x_2|-(x_2-a)|x_2-a|&=x_3|x_3|,\\&\cdots\\x_n|x_n|-(x_n-a)|x_n-a|&=x_1|x_1|,\end{aligned}$$
where $a$ is a given number.