Question 6
A positive integer is called a “2005-diamond number” if its decimal representation contains 2005 consecutive digits all equal to 9. Let $(a_n)_{n\ge1}$ be a strictly increasing sequence of positive integers satisfying $a_n\lt nC$ for all $n\ge1$, where $C$ is some fixed positive real constant. Prove that the sequence contains infinitely many 2005-diamond numbers.
The supplied PDF does not state a time limit. This edition uses the user-approved 240 minutes per session, 480 minutes total for both sessions; this is an editorial convention. The supplied source has no question point values. Each problem uses 1 editorial point for import compatibility; this is not an official scoring scale. Essay grading is manual; the rubric is editorial. Import multipliers: correct 1, incorrect 0, blank 0.
The supplied PDF does not state a time limit. This edition uses the user-approved 240 minutes per session, 480 minutes total for both sessions; this is an editorial convention. The supplied source has no question point values. Each problem uses 1 editorial point for import compatibility; this is not an official scoring scale. Essay grading is manual; the rubric is editorial. Import multipliers: correct 1, incorrect 0, blank 0.