2007 AIME I Problem 11

For each positive integer p, let b(p) denote the unique positive integer k such that |k-\sqrt{p}| < \frac{1}{2}. For example, b(6) = 2 and b(23) = 5. If S = \sum_{p=1}^{2007} b(p), find the remainder when S is divided by 1000.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.