ELMO SL 2010 A5

Given a prime $p$ , let $d(a,b)$ be the number of integers $c$ such that $1 \leq c < p$ , and the remainders when $ac$ and $bc$ are divided by $p$ are both at most $\frac{p}{3}$ . Determine the maximum value of $$\sqrt{\sum_{a=1}^{p-1}\sum_{b=1}^{p-1}d(a,b)(x_a + 1)(x_b + 1)} - \sqrt{\sum_{a=1}^{p-1}\sum_{b=1}^{p-1}d(a,b)x_ax_b}$$ over all $(p-1)$ -tuples $(x_1,x_2,\ldots,x_{p-1})$ of real numbers.

Brian Hamrick.