Determine if there exists a set $S$ of $5783$ different real numbers with the following property: For every $a,b\in S$ (not necessarily distinct) there are $c\neq d$ in $S$ so that $a\cdot b=c+d$.
Source: 2023 Israel National Olympiad P6
Tags: algebra, Sets
Determine if there exists a set $S$ of $5783$ different real numbers with the following property: For every $a,b\in S$ (not necessarily distinct) there are $c\neq d$ in $S$ so that $a\cdot b=c+d$.