USAMO 2018 Problem 4

Let $p$ be a prime, and let $a_1, \dots, a_p$ be integers. Show that there exists an integer $k$ such that the numbers $$a_1 + k, a_2 + 2k, \dots, a_p + pk$$ produce at least $\tfrac{1}{2} p$ distinct remainders upon division by $p$ .

Proposed by Ankan Bhattacharya