IMC 2023 · Problem 4

Day 130th IMC · Blagoevgrad, Bulgaria

Statement

Let $p$ be a prime number and let $k$ be a positive integer. Suppose that the numbers $a_i = i^k + i$ for $i = 0, 1, \ldots, p-1$ form a complete residue system modulo $p$ . What is the set of possible remainders of $a_2$ upon division by $p$ ?

Official solution

Hidden so you can work on the problem first.

Proposed by Tigran Hakobyan, Yerevan State University, Armenia.