Find all nonconstant polynomials $P$ which satisfy $P(\{X\})=\{P(X)\}$, where $\{x\}$ is the fractional part of $x$.
I've tried to prove that the polynomial in question is linear, but I can't think of how to prove it, especially since we don't know anything about the constants
Hint
$P(\{X\})$ is periodic with a period of $1$ (since $\{X+1\}=\{X\}$), hence $\{P(X)\}$ is also periodic only if $P(X)$ is linear, because for non-linear $P(X)$ (WLOG we assume $\lim_{x\to\infty} P(X)=\infty$), by defining $$ I_k=\{x: k\le P(x)<k+1\}\quad,\quad k\in\Bbb Z $$we have $$\lim_{k\to \infty}|I_k|=0$$which means that $\{P(X)\}$ cannot be periodic and the statement is proved $\blacksquare$