KöMaL 2018 A727

For any finite sequence $(x_1,\ldots,x_n)$, denote by $N(x_1,\ldots,x_n)$ the number of ordered index pairs $(i,j)$ for which $1\le i<j\le n$ and $x_i=x_j$. Let $p$ be an odd prime, $1\le n<p$, and let $a_1,a_2,\ldots,a_n$ and $b_1,b_2,\ldots,b_n$ be arbitrary residue classes modulo $p$. Prove that there exists a permutation $\pi$ of the indices $1,2,\ldots,n$ for which $N\big(a_1+b_{\pi(1)},a_2+b_{\pi(2)},\ldots,a_n+b_{\pi(n)}\big) \le \min \big(N(a_1,a_2,\ldots,a_n), N(b_1,b_2,\ldots,b_n) \big).$