Determine all polynomials such that $P\left(x^{2}\right)=x P(x)$ for $x \in \mathbb{R}$
Let's first look at the degree. For this, we assume $P \neq 0$ (The case $P=0$ works).

We have $2 \cdot \operatorname{deg} P=1+\operatorname{deg} P$ so $\operatorname{deg} P=1$.

Furthermore, by evaluating at $x=0$, we have $P(0)=0$, so $P(X)=\lambda X$ with $\lambda \in \mathbb{R}^{*}$ which works. Thus, the solutions are the polynomials