Question 28
Let $x_1,$ $x_2,$ $\ldots,$ $x_n$ be a sequence of integers such that
(i) $-1 \le x_i \le 2,$ for $i = 1,$ $2,$ $3,$ $\ldots,$ $n$;
(ii) $x_1 + x_2 + \cdots + x_n = 19;$ 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 $x_1^3 + x_2^3 + \cdots + x_n^3,$ respectively. What is $\dfrac{M}{m}?$
(i) $-1 \le x_i \le 2,$ for $i = 1,$ $2,$ $3,$ $\ldots,$ $n$;
(ii) $x_1 + x_2 + \cdots + x_n = 19;$ 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 $x_1^3 + x_2^3 + \cdots + x_n^3,$ respectively. What is $\dfrac{M}{m}?$