Question 65
For every integer $k \geqslant 0$, let $a_{k}$ denote the first digit of the number $2^{k}$, written in base 10. For example, $a_{5}=3$ is the first digit of $2^{5}=32$.
Let $n \geqslant 1$ be an integer. Prove that, among the digits from 1 to 9, there is one that is equal to at most $n / 17$ of the $n$ digits $a_{0}, a_{1}, a_{2}, \ldots, a_{n-1}$.
Let $n \geqslant 1$ be an integer. Prove that, among the digits from 1 to 9, there is one that is equal to at most $n / 17$ of the $n$ digits $a_{0}, a_{1}, a_{2}, \ldots, a_{n-1}$.