Middle European Mathematical Olympiad 2010 Problem T-1

Three strictly increasing sequences $$a_1, a_2, a_3, \ldots, \qquad b_1, b_2, b_3, \ldots, \qquad c_1, c_2, c_3, \ldots$$ of positive integers are given. Every positive integer belongs to exactly one of the three sequences. For every positive integer $n$, the following conditions hold:

(i) $c_{a_n} = b_n + 1$;

(ii) $a_{n+1} > b_n$;

(iii) the number $c_{n+1}c_n - (n+1)c_{n+1} - nc_n$ is even.

Find $a_{2010}$, $b_{2010}$, and $c_{2010}$.