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Summation Pattern Problem

Source: AIME I 2007 #11

March 15, 2007
algebrapolynomialAMCaxusus

Problem Statement

For each positive integer pp, let b(p)b(p) denote the unique positive integer kk such that ∣k−p∣<12|k-\sqrt{p}|<\frac{1}{2}. For example, b(6)=2b(6) = 2 and b(23)=5b(23)=5. If S=∑p=12007b(p)S = \textstyle\sum_{p=1}^{2007}b(p), find the remainder when S is divided by 1000.
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