Find all real polynomials $\mathrm{P}$ such that $\mathrm{P}(0)=0$ and $\mathrm{P}\left(\mathrm{X}^{2}+1\right)=\mathrm{P}(\mathrm{X})^{2}+1$.


Let $\left(u_{n}\right)$ be the real sequence such that $u_{0}=0$ and $u_{n+1}=u_{n}^{2}+1$. An immediate induction shows that $P\left(u_{n}\right)=u_{n}$. The sequence $\left(u_{n}\right)$ being strictly increasing, the polynomial $P(X)-X$ has an infinity of roots, which proves that $P(X)=X$, which is indeed a solution to the problem.