The First Anniversary Problem

Algebra Level 4

Let \(n\) be a positive integer and \(x_1, x_2, \ldots, x_n\) be integers satisfying the following conditions:

  1. \(-1 \le x_i \le 2\) for all \(i = 1, 2, \ldots, n\)
  2. \(\displaystyle \sum_{i=1}^n x_i = 19\)
  3. \(\displaystyle \sum_{i=1}^n x_i^2 = 99\)

Out of all possible \(n, x_1, x_2, \ldots, x_n\) satisfying the above conditions, find the sum of the minimum and the maximum values of \(\displaystyle \sum_{i=1}^n x_i^3\).


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