Pinko wrote: Find all polynomials $P\in \mathbb{R}[x]$, for which $P(P(x))=\lfloor P^2 (x)\rfloor$ is true for $\forall x\in \mathbb{Z}$. $P(P(x))$ is continuous and integer, so constant. So $P^2(x)$ is bounded and, since polynomial, constant. It remains to solve $c=\lfloor c^2\rfloor$ which implies $c\in\mathbb Z$ and so $c=c^2$ Hence the two solutions $\boxed{\text{S1 : }P(x)=0\quad\forall x}$ and $\boxed{\text{S2 : }P(x)=1\quad\forall x}$