Let $P$ be a polynomial of degree 2022 such that $P(1)=1, P(2)=2, \ldots P(2022)=2022$ and $P(0)=1$

Determine $P(-1)$
The polynomial strongly resembles the polynomial $X$. More precisely, $P(X)-X$ has roots $1,2, \ldots, 2022$, so there exists $Q$ such that $P(X)-X=Q(X) \cdot(X-1) \cdot(X-2) \cdot \ldots \cdot(X-2022)$. By the degree, there exists $\lambda$ such that $P(X)=X+\lambda(X-1) \ldots(X-2022)$.

Evaluating at 0, we have $1=\lambda 2022$!, hence $\lambda=\frac{1}{2022!}$.

We thus obtain $P(-1)=-1+2023=2022$.