AHSME 1999 Problem 28

Let $x_1, x_2, \ldots , x_n$ be a sequence of integers such that (i) $-1 \le x_i \le 2$ for $i = 1,2, \ldots n$ (ii) $x_1 + \cdots + x_n = 19$; and (iii) $x_1^2 + x_2^2 + \cdots + x_n^2 = 99$. Let $m$ and $M$ be the minimal and maximal possible values of $x_1^3 + \cdots + x_n^3$, respectively. Then $\frac Mm =$ $\mathrm{(A) \ }3 \mathrm{(B) \ }4 \mathrm{(C) \ }5 \mathrm{(D) \ }6 \mathrm{(E) \ }7$