Question 57
Show how to find a permutation $\sigma \in S_k$ such that $p \gt 2$, $p \in \mathbb{R}$, $a_1, \ldots, a_k$ are distinct elements of $\mathbb{Z}_p$, and $b_1, \ldots, b_k$ are elements of $\mathbb{Z}_p$ (some may be equal), while $a_1 + b_{\sigma(1)}, \ldots, a_k + b_{\sigma(k)}$ is distinct elements of $\mathbb{Z}_p$.