CAPS 2005 Problem 1

Let $n$ be a given positive integer. Solve the system $$x_1 + x_2^2 + x_3^3 + \cdots + x_n^n = n,$$ $$x_1 + 2x_2 + 3x_3 + \cdots + nx_n = \frac{n(n+1)}{2}$$ in the set of nonnegative real numbers.