AUSTRIAN-POLISH 1979 Problem 6

A positive integer $n$ and a real number $a$ are given. Find all $n$ -tuples $(x_1, ... ,x_n)$ of real numbers that satisfy the system of equations $$\sum_{i=1}^{n} x_i^k= a^k \,\,\, for \,\,\, k = 1,2, ... ,n$$