Problem

Source: Iran MO 3rd round 2016 mid-terms - Number Theory P1

Tags: number theory, algebra, polynomial, Iran



Let $F$ be a subset of the set of positive integers with at least two elements and $P(x)$ be a polynomial with integer coefficients such that for any two distinct elements of $F$ like $a$ and $b$, the following two conditions hold $a+b \in F$, and $\gcd(P(a),P(b))=1$. Prove that $P(x)$ is a constant polynomial.