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Show that the sum is an integer divisible by 25

Source: Austrian Mathematical Olympiad 2001, Part 2, D1, P1

June 25, 2011
modular arithmeticfloor functionalgebra proposedalgebra

Problem Statement

Prove that 125∑k=02001[2k25]\frac{1}{25} \sum_{k=0}^{2001} \left[ \frac{2^k}{25}\right] is a positive integer.
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