Erdős problem 443

Let $m,n\geq 1$. What is $$\# \{ k(m-k) : 1\leq k\leq m/2\} \cap \{ l(n-l) : 1\leq l\leq n/2\}?$$ Is it $\leq (mn)^{o(1)}$ for all sufficiently large $m,n$?