number theory - Prove that powers of any fixed prime $p$ contain arbitrarily many consecutive equal digits.

Prove that powers of any fixed prime $p$ contain arbitrarily many consecutive equal digits.

It is an intuitive re-statement of Baltic Way 2012 (I think there are shortlists in Baltic Way every year and this is a part of the 2012 shortlist):

Prove that, for every prime $p$ and positive integer $a$, there exists a positive integer $n$ such that $p^n$ contains $a$ consecutive equal digits.

It is a tough one and I haven't found a solution on the Internet.