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2011 divides a_{k-1}a_k - k for all k = 1,2,...,2010

Source: Indonesian Mathematics Olympiad 2011, Day 2, Problem 6

September 14, 2011
modular arithmeticnumber theory proposednumber theory

Problem Statement

Let a sequence of integers a0,a1,a2,⋯ ,a2010a_0, a_1, a_2, \cdots, a_{2010} such that a0=1a_0 = 1 and 20112011 divides ak−1ak−ka_{k-1}a_k - k for all k=1,2,⋯ ,2010k = 1, 2, \cdots, 2010. Prove that 20112011 divides a2010+1a_{2010} + 1.
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