PL-OM 1994 Problem 6

The distinct reals $x_1, x_2, ... , x_n$ ,( $n > 3$ ) satisfy $\sum_{i=1}^n x_i = 0$ , $\sum_{i=1}^n x_i ^2 = 1$ . Show that four of the numbers $a, b, c, d$ must satisfy: $$a + b + c + nabc \leq \sum_{i=1}^n x_i ^3 \leq a + b + d + nabd$$.