VN-TST 2005 Problem 6

$n$ is called diamond 2005 if $n=\overline{...ab999...99999cd...}$ , e.g. $2005 \times 9$ . Let $\{a_n\}:a_n< C\cdot n,\{a_n\}$ is increasing. Prove that $\{a_n\}$ contain infinite diamond 2005.

Compare with this problem.