Question 7
(FRA 5) Let real numbers $x_1,x_2,\ldots,x_n$ satisfy $0\lt x_1\lt x_2\lt \cdots\lt x_n\lt 1$ and set $x_0=0$, $x_{n+1}=1$. Suppose that these numbers satisfy the following system of equations:
$$\sum_{\substack{j=0\\j\ne i}}^{n+1}\frac1{x_i-x_j}=0\quad(i=1,2,\ldots,n).\tag{1}$$
Prove that $x_{n+1-i}=1-x_i$ for $i=1,2,\ldots,n$.
$$\sum_{\substack{j=0\\j\ne i}}^{n+1}\frac1{x_i-x_j}=0\quad(i=1,2,\ldots,n).\tag{1}$$
Prove that $x_{n+1-i}=1-x_i$ for $i=1,2,\ldots,n$.