Problem

Source: Balkan MO 2016, Problem 3

Tags: polynomial, number theory



Find all monic polynomials $f$ with integer coefficients satisfying the following condition: there exists a positive integer $N$ such that $p$ divides $2(f(p)!)+1$ for every prime $p>N$ for which $f(p)$ is a positive integer. Note: A monic polynomial has a leading coefficient equal to 1. (Greece - Panagiotis Lolas and Silouanos Brazitikos)