CN-TST 2023 Problem 20

Let $a,b,d$ be integers such that $\left|a\right| \geqslant 2$ , $d \geqslant 0$ and $b \geqslant \left( \left|a\right| + 1\right)^{d + 1}$ . For a real coefficient polynomial $f$ of degree $d$ and integer $n$ , let $r_n$ denote the residue of $\left[ f(n) \cdot a^n \right]$ mod $b$ . If $\left \{ r_n \right \}$ is eventually periodic, prove that all the coefficients of $f$ are rational.