ISL 2020 N4

For any odd prime $p$ and any integer $n$, let $d_{p}(n) \in\{0,1, \ldots, p-1\}$ denote the remainder when $n$ is divided by $p$. We say that $\left(a_{0}, a_{1}, a_{2}, \ldots\right)$ is a $p$-sequence, if $a_{0}$ is a positive integer coprime to $p$, and $a_{n+1}=a_{n}+d_{p}\left(a_{n}\right)$ for $n \geqslant 0$. (a) Do there exist infinitely many primes $p$ for which there exist $p$-sequences $\left(a_{0}, a_{1}, a_{2}, \ldots\right)$ and $\left(b_{0}, b_{1}, b_{2}, \ldots\right)$ such that $a_{n}>b_{n}$ for infinitely many $n$, and $b_{n}>a_{n}$ for infinitely many $n$ ? (b) Do there exist infinitely many primes $p$ for which there exist $p$-sequences $\left(a_{0}, a_{1}, a_{2}, \ldots\right)$ and $\left(b_{0}, b_{1}, b_{2}, \ldots\right)$ such that $a_{0}<b_{0}$, but $a_{n}>b_{n}$ for all $n \geqslant 1$ ? (United Kingdom)