PAMO 2022 Problem 5

Let $r$ be a positive integer. Find the smallest positive integer $m$ satisfying the condition: For all sets $A_1, A_2, \dots, A_r$ with $A_i \cap A_j = \emptyset$ , for all $i \neq j$ , and $\bigcup_{k = 1}^{r} A_k = \{ 1, 2, \dots, m \}$ , there exists $a, b \in A_k$ for some $k$ such that $1 \leq \frac{b}{a} \leq 1 + \frac{1}{2022}$ .