# Joel's Problem 4: Maximum of maximums

Geometry Level pending

Consider a rectangle R with vertices A, B, C, D in clockwise order. Among all points P on segment CD, let the maximum value of $$PA \cdot PB$$ be $$f (R)$$. Among all rectangles Q of area $$\sqrt {2}+1$$, let $$M$$ be the maximum value of $$f (Q)$$. However, only some rectangles can have a point on segment CD such that $$PA \cdot PB=M$$. Let $$S$$ be the set of all such rectangles. Find the maximum possible value of $$\frac {AB}{BC}$$ over all rectangles in $$S$$.

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