Let $P \in \mathbb{R}[X]$ be a monic polynomial of degree 2020 such that $P(n)=n$ for all $n \in \llbracket 0,2019 \rrbracket$. Calculate $P(2020)$.

Hint: Introduce the polynomial $Q=P-X$.
We know that $Q=P-X$ is a unitary polynomial of degree 2020 and vanishes at $k$ for all $k \in \llbracket 0,2019 \rrbracket$. Thus, $Q=X(X-1) \cdot(X-2019)$. We can evaluate at 2020 the identity $P=X(X-1) \cdot(X-$ $2019)+X$:

$$
P(2020)=2020!+2020
$$