Problem

Source: Iran 2024 3rd round number theory exam p3

Tags: number theory



The prime number $p$ and a positive integer $k$ are given. Assume that $P(x)\in \mathbb Z[X]$ is a polynomial with coefficients in the set $\{0,1,\cdots,p-1\}$ with least degree which satisfies the following property: There exists a permutaion of numbers $1,2,\cdots,p-1$ around a circle such that for any $k$ consecutive numbers $a_1,a_2,\cdots,a_k$ one has $$ p | P(a_1)+P(a_2)+\cdots+ P(a_k). $$Prove that $P(x)$ is of the form $ax^d+b$. Proposed by Yahya Motevassel