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NT: Smallest degree of polynomial

Source: Kyiv City MO 2022 Round 2, Problem 10.4

January 30, 2022
number theorypolynomialalgebra

Problem Statement

Prime p>2p>2 and a polynomial QQ with integer coefficients are such that there are no integers 1≤i<j≤p−11 \le i < j \le p-1 for which (Q(j)−Q(i))(jQ(j)−iQ(i))(Q(j)-Q(i))(jQ(j)-iQ(i)) is divisible by pp. What is the smallest possible degree of QQ?
(Proposed by Anton Trygub)
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