Mock AIME 6 2006-2007 Problems/Problem 12

Problem

Let $x_k$ be the largest positive rational solution $x$ to the equation $(2007-x)(x+2007^{-k})^k=1$ for all integers $k\ge 2$ . For each $k$ , let $x_k=\frac{a_k}{b_k}$ , where $a_k$ and $b_k$ are relatively prime positive integers. If $S=\sum_{k=2}^{2007} (2007b_k-a_k),$ what is the remainder when $S$ is divided by $1000$ ?

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Mock AIME 6 2006-2007 Problems/Problem 12

Problem

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