Question

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. Then $\dfrac{M}{m} = \, ?$

$3$

$4$

$5$

$6$

$7$

Solution:

Let $a, b, c$ be the numbers of $-1$ s, $1$ s, and $2$ s. Then $-a + b + 2c = 19$ and $a + b + 4c = 99,$ giving $a = 40 - c$ and $b = 59 - 3c$ with $0 \le c \le 19.$

The sum of cubes is $-a + b + 8c = 19 + 6c.$ The minimum is at $c = 0$ (value $19$ ) and the maximum at $c = 19$ (value $133$ ), so $\dfrac{M}{m} = \dfrac{133}{19} = 7.$

Thus, the correct answer is E.

1999 AMC 12 Problem 28

Solution: