We are given a polynomial $P$ with integer coefficients such that $P(0) \neq 0$ and $P(P(0))=0$. We denote $m$ as the number of integer roots of $P$.

Find all possible values of $m$.
Note that $P(0)$ is a root of $P$, so $m \geqslant 1$. Moreover, suppose that $m$ has at least 4 distinct roots. We can assume that $P(0)$ is one of them, and denote $a_{1}, a_{2}$ and $a_{3}$ the other three. We can then factorize $P$ and write

$$
P=\left(X-a_{1}\right)\left(X-a_{2}\right)\left(X-a_{3}\right)(X-P(0)) Q
$$

where $Q$ has integer coefficients. By taking $X=0$, we get

$$
P(0)=a_{1} a_{2} a_{3} P(0) Q(0)
$$

Since $P(0) \neq 0$, we have $a_{1} a_{2} a_{3}=1$. The numbers $a_{1}, a_{2}$ and $a_{3}$ therefore divide 1, so they are $\pm 1$, so two of them are equal, which is absurd. Therefore, we have $m \leqslant 3$.

Finally, the values $m=1, m=2$ and $m=3$ can all be achieved, as shown by the following examples:

$$
P(X)=1-X, P(X)=(X-1)(X-2) \text { and } P(X)=\left(X^{2}-1\right)(X-2) \text {. }
$$