Suppose $x_1, x_2, \dots, x_n$ satisfy the equations above. Then we have
$$\begin{aligned} 0 &= x_1 + x_2^2 + x_3^3 + \dots + x_n^n - n - (x_1 + 2x_2 + 3x_3 + \dots + nx_n - \frac{1}{2}n(n+1)) \\ &= (x_2^2 - 2x_2 + 2 - 1) + (x_3^3 - 3x_3 + 3 - 1) + \dots + (x_n^n - nx_n + n - 1). \end{aligned}$$
However, the expressions in the brackets are nonnegative. Indeed, for $k \ge 2$ and $x \ge 0$ we have, by the AM-GM inequality,
$$x^k + k - 1 = x^k + 1 + 1 + \dots + 1 \ge k \cdot \sqrt[k]{x^k} = kx$$
and the equality holds if and only if $x = 1$. Therefore we have $x_2 = x_3 = \dots = x_n = 1$ and, by the first equation, $x_1 = 1$.