Let $P(X)$ be a monic polynomial of degree 2017 such that $P(1)=1, P(2)=2, \ldots, P(2017)=$ 2017. What is the value of $P(2018)$?
The conditions of the statement prevent us from directly accessing the roots of $P$. However, the polynomial $Q(X)=P(X)-X$ is monic, of degree 2017, and we know 2017 of its roots: the integers from 1 to 2017. Therefore, we can explicitly determine this polynomial: $Q(X)=(X-1)(X-2) \cdots(X-2017)$. From this, we deduce that $P(X)=Q(X)+X=$ $(X-1)(X-2) \cdots(X-2017)+X$ and that $P(2018)=2017!+2018$.