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Sequence where 2011 divides S_{2011a} - S_{a}

Source: Spanish Mathematical Olympiad, National Stage, 2011

August 9, 2011
algebrapolynomialinductionfloor functionmodular arithmeticnumber theorynumber theory unsolved

Problem Statement

The sequence S0,S1,S2,…S_0,S_1,S_2,\ldots is defined by[*]Sn=1S_n=1 for 0≤n≤20110\le n\le 2011, and [*]Sn+2012=Sn+2011+SnS_{n+2012}=S_{n+2011}+S_n for n≥0n\ge 0.Prove that S2011a−SaS_{2011a}-S_a is a multiple of 20112011 for all nonnegative integers aa.
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