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Summation, Ceiling, and Floor Functions

Source: AIME I 2007 #7

March 15, 2007
functionceiling functionlogarithmsfloor functioncalculusintegrationLaTeX

Problem Statement

Let N=∑k=11000k(⌈log⁡2k⌉−⌊log⁡2k⌋).N= \sum_{k=1}^{1000}k(\lceil \log_{\sqrt{2}}k\rceil-\lfloor \log_{\sqrt{2}}k \rfloor). Find the remainder when N is divided by 1000. (Here ⌊x⌋\lfloor x \rfloor denotes the greatest integer that is less than or equal to x, and ⌈x⌉\lceil x \rceil denotes the least integer that is greater than or equal to x.)
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