Problem 3. Let's call a three-digit number interesting if at least one of its digits is divisible by 3. What is the maximum number of consecutive interesting numbers that can exist? (Provide an example and prove that it is impossible to have more consecutive numbers.)


Answer: 122.

Solution. The numbers $289,290, \ldots, 299,300, \ldots, 399,400, \ldots, 409,410$ are interesting (recall that 0 is divisible by 3), and there are 122 of them in total. Let's prove that a larger number is not possible.

Suppose we managed to find a larger number of consecutive interesting numbers; choose 123 consecutive ones from them.

Let's call a hundred consecutive numbers, all of which have the same hundreds digit that is divisible by 3, an interesting hundred. Note that before any interesting hundred, there are only 11 interesting numbers ending in $89,90, \ldots, 99$, and the 12th number ends in 88 and is not interesting. Similarly, after an interesting hundred, there are also only 11 interesting numbers ending in $00, \ldots, 09,10$, and the 12th number ends in 11 and is also not interesting.

If our sequence of 123 numbers intersects with some interesting hundred, then it contains at least 12 numbers either before or after this hundred. Consequently, at least one number in it is not interesting.

If our sequence of 123 numbers does not intersect with an interesting hundred, then it contains at least one number ending in 55 (as well as any other combination of digits). But this number is not interesting, as none of its digits are divisible by 3.

## Criteria

0 6. Incorrect answer and incorrect (or missing) example.

16 . Correct answer provided.

1 b. Correct example provided, but the count of numbers in it is incorrect.

2 6. Correct example and answer provided, but no justification that a larger number of numbers is impossible.

4 b. Correct answer, correct example, and justification provided.