KMO 2018 Problem 8

Let there be positive integers $a, c$ . Positive integer $b$ is a divisor of $ac-1$ . For a positive rational number $r$ which is less than $1$ , define the set $A(r)$ as follows. $$A(r) = \{m(r-ac)+nab| m, n \in \mathbb{Z} \}$$ Find all rational numbers $r$ which makes the minimum positive rational number in $A(r)$ greater than or equal to $\frac{ab}{a+b}$ .