1999 AHSME Problem 28

Let x_1, x_2, \ldots , x_n be a sequence of integers such that

\text{(i)} -1 \le x_i \le 2 \text{for} i = 1,2, \ldots n

\text{(ii)} x_1 + \cdots + x_n = 19; \text{and}

\text{(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 \qquad \mathrm{(B) \ }4 \qquad \mathrm{(C) \ }5 \qquad \mathrm{(D) \ }6 \qquad \mathrm{(E) \ }7

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.