Question 198
Three strictly increasing sequences
$$
a_{1}, a_{2}, a_{3}, \ldots, \quad b_{1}, b_{2}, b_{3}, \ldots, \quad c_{1}, c_{2}, c_{3}, \ldots
$$
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) $a_{n+1}\gt b_{n}$;
(iii) the number $c_{n+1} c_{n}-(n+1) c_{n+1}-n c_{n}$ is even.
Find $a_{2010}, b_{2010}$, and $c_{2010}$.
$$
a_{1}, a_{2}, a_{3}, \ldots, \quad b_{1}, b_{2}, b_{3}, \ldots, \quad c_{1}, c_{2}, c_{3}, \ldots
$$
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) $a_{n+1}\gt b_{n}$;
(iii) the number $c_{n+1} c_{n}-(n+1) c_{n+1}-n c_{n}$ is even.
Find $a_{2010}, b_{2010}$, and $c_{2010}$.