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