OSN 2011 Problem 6

Let a sequence of integers $a_0, a_1, a_2, \cdots, a_{2010}$ such that $a_0 = 1$ and $2011$ divides $a_{k-1}a_k - k$ for all $k = 1, 2, \cdots, 2010$ . Prove that $2011$ divides $a_{2010} + 1$ .