KöMaL 2015 B4736

Let $n$ be a positive integer. Solve the simultaneous equations $\sum_{i=1}^{n} {|x_i|} = \sum_{i=1}^{n} \big|x_i^3\big| = \sum_{i=1}^{n} {\frac{2 {|x_i|}^3}{x_i^2+1}}.$