Hollow rectangle at the edge!

A rectangular frame of a uniform density is placed on a table so that its edges are parallel to the table's edges, and only one corner of the frame is on the table. Let \(A\) be the area contained by the frame that is not over the table, and let \(B\) be the area contained by the frame that is over the table.

What is the maximum ratio of area \(\frac{A}{B}\) for which the frame will not fall off the table?

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