ILL 1984 Problem 49

Let $n > 1$ and $x_i \in \mathbb{R}$ for $i = 1,\cdots, n$ . Set $$S_k = x_1^k+ x^k_2+\cdots+ x^k_n$$ for $k \ge 1$ . If $S_1 = S_2 =\cdots= S_{n+1}$ , show that $x_i \in \{0, 1\}$ for every $i = 1, 2,\cdots, n.$