A problem on the largest divisor $d$ of $N$ with $d\leq \sqrt{N}$

For a given number $N$, we consider the problem of computing two integers $1\leq r,f < N$ such that the set $$\mathcal{X}(N,r,f) = \{(a+b)-(f+\frac{Nr+1}{f}): ab=Nr\}$$ consists only of positive integers. Computing a solution to the problem is equivalent to finding a pair $(r,f)$ satisfying