2012 AMC 12A Problem 24

Let \{a_k\}_{k=1}^{2011} be the sequence of real numbers defined by a_1=0.201, a_2=(0.2011)^{a_1}, a_3=(0.20101)^{a_2}, a_4=(0.201011)^{a_3}, and in general,

a_k=\begin{cases} (0.\underline{20101\cdots 0101}_{\,k+2\text{ digits}})^{a_{k-1}} & \text{if }k\text{ is odd,}\\ (0.\underline{20101\cdots 01011}_{\,k+2\text{ digits}})^{a_{k-1}}& \text{if }k\text{ is even.} \end{cases}

Rearranging the numbers in the sequence \{a_k\}_{k=1}^{2011} in decreasing order produces a new sequence \{b_k\}_{k=1}^{2011}. What is the sum of all integers k, 1\le k \le 2011, such that a_k=b_k?

\textbf{(A)}\ 671\qquad\textbf{(B)}\ 1006\qquad\textbf{(C)}\ 1341\qquad\textbf{(D)}\ 2011\qquad\textbf{(E)}\ 2012

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.