RMM 2019 Problem 6

Find all pairs of integers $(c, d)$, both greater than 1 , such that the following holds:

For any monic polynomial $Q$ of degree $d$ with integer coefficients and for any prime $p>c(2 c+1)$, there exists a set $S$ of at most $\left(\frac{2 c-1}{2 c+1}\right) p$ integers, such that

$$\bigcup_{s \in S}\{s, Q(s), Q(Q(s)), Q(Q(Q(s))), \ldots\}$$

is a complete residue system modulo $p$ (i.e., intersects with every residue class modulo $p$ ).

Each of the three problems is worth 7 points. Time allowed $4 \frac{1}{2}$ hours.