2020 IMO Shortlist 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 $n \geq 0$. (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 $n \geq 1$ ? (United Kingdom)

对于任意奇素数$p$和任意整数$n$，令$d_{p}(n) \in\{0,1, \ldots, p-1\}$表示$n$除以$p$时的余数。如果$a_{0}$是与$p$互质的正整数，且$n \geq 0$的$a_{n+1}=a_{n}+d_{p}\left(a_{n}\right)$，我们称$\left(a_{0}, a_{1}, a_{2}, \ldots\right)$是$p$序列。 (a) 是否存在无穷多个素数 $p$，其中存在 $p$ 序列 $\left(a_{0}, a_{1}, a_{2}, \ldots\right)$ 和 $\left(b_{0}, b_{1}, b_{2}, \ldots\right)$，使得 $a_{n}>b_{n}$ 对于无穷多个 $n$，以及$b_{n}>a_{n}$ 表示无限多个 $n$ ？ (b) 是否存在无限多个素数$p$，其中存在$p$序列$\left(a_{0}, a_{1}, a_{2}, \ldots\right)$和$\left(b_{0}, b_{1}, b_{2}, \ldots\right)$，使得$a_{0}<b_{0}$，但$a_{n}>b_{n}$所有 $n \geq 1$ ？ (英国)