Find all polynomials $P \in \mathbb{R}[X]$ such that $P(2 x)=P(x)$ for $x \in \mathbb{R}$.
It is about showing that $P$ is constant.

There are several ways to solve this exercise. We write $P(X)=\sum_{k=0}^{n} a_{k} X^{k}$. By rigidity, we get for all $k$ between 0 and $n, 2^{k} a_{k}=a_{k}$. Therefore, $a_{k}=0$ for $k \neq 0$ and $P$ is constant. Conversely, a constant $P$ works.