SIMON-MARAIS 2022 A1

Let $ABCD$ be a unit square, and let $P$ be a point inside triangle $ABC$. Let $E$ and $F$ be the points on $AB$ and $AD$, respectively, such that $AEPF$ is a rectangle. Let $x$ and $y$ denote the lengths of $AE$ and $AF$, respectively.

Determine a two-variable polynomial $f(x,y)$ with the property that the area of $AEPF$ is greater than the area of each of the quadrilaterals $EBCP$ and $FPCD$ if and only if $f(x,y) > 0$.