Let \(p\) be a prime number at least three and let \(k\) be a positive integer smaller than \(p\). Given \(a_1,\dots, a_k\in \mathbb{F}_p\) and distinct elements \(b_1,\dots, b_k\in \mathbb{F}_p\), prove that there exists a permutation \(\sigma\) of \([k]\) such that the values of \(a_i + b_{\sigma(i)}\) are distinct modulo \(p\).