Irecently (yesterday) took the amc 10a, can anyone me with this problem? for a positive integer n and nonzero digits a, b and c, let be the n-digit integer each of whose digits is equal to a; let be the n-digit integer whose digits is equal to b; let be the 2n digit integer (not n digit) each of whose digits is equal to c. what is the greatest possible value of a + b + c for which there are at least two values of n such that ? *this was #25, and the questions have been posted to the art of problem solving website, so i'm guessing discussion is allowed.

First, we claim that is the digit number with all digits equal to one.

Note that is a one followed by zeroes, so subtracting one gives nines.  Divide that number by nine and you get ones, completing the proof.

Therefore, we have that , , and .

Let .  Then, we have:

.

Multiplying by gives:



Now, note that is not a valid input, since requires , so we safely divide by to get:







Because this is now a linear equation in , it has either zero, one, or infinitely many solutions.  Obviously, we need the latter to occur, which happens when and , since the coefficient of must cancel to zero and thus the RHS must equal zero as well.

Since , we must have .  Since must be a multiple of three, we plug in values.  If , we get and thus , which is impossible.  So doesn't work.

With , , and thus , so , so .  This gives as one possibility.

With , , and thus , so .  This obviously is worse.

We've gone through all the cases and the two possibilities are and , so our answer is .