(Polynomials 1) Find all $P \in \mathbb{R}[X]$ such that $P(0)=0$ and

$$
P\left(x^{2}+1\right)=P(x)^{2}+1
$$

for all real $x$.
We find $\mathrm{P}(1)=1, \mathrm{P}(2)=2, \mathrm{P}(5)=5$, etc. We then define the sequence $\left(a_{n}\right)$ by $a_{0}=0$ and $a_{n+1}=a_{n}^{2}+1$.

It is easily verified that it is strictly increasing, and we prove by induction on $\mathrm{n}$ that $P\left(a_{n}\right)=a_{n}$. Thus, the polynomial $Q(X)=P(X)-X$ has an infinite number of roots, so it is the zero polynomial. In other words, for any real $x, \mathrm{P}(\mathrm{x})=x$. Conversely, the identity is a solution to the problem.