Determine all real polynomials $P$ satisfying $P(2X) = 2P(X)$.
Suppose $P \neq 0$ is a solution of the equation, and let $c \neq 0$ be its leading coefficient. The leading coefficient of $2 P(X)$ is $2 c$, and that of $P(2 X)$ is $2^{n} c$. Thus, $2^{n} c = 2 c$, so $2^{n} = 2$, and therefore $n = 1$. Hence, $P$ is of the form $a X + b$ for real numbers $a$ and $b$.

By evaluating at 0, we get $P(0) = 2 P(0)$, so $P(0) = 0$, and thus $b = 0$. Therefore, in all cases, $P$ is of the form $a X$ for $a \in \mathbb{R}$.

Conversely, any polynomial of the form $a X$ for $a \in \mathbb{R}$ satisfies $P(2 X) = a \times 2 X = 2 a X = 2 P(X)$, and is indeed a solution. Therefore, the polynomial solutions are those of the form $a X$ for $a \in \mathbb{R}$.