1993 AIME Problem 14

A rectangle that is inscribed in a larger rectangle (with one vertex on each side) is called *unstuck* if it is possible to rotate (however slightly) the smaller rectangle about its center within the confines of the larger. Of all the rectangles that can be inscribed unstuck in a 6 by 8 rectangle, the smallest perimeter has the form \sqrt{N}\,, for a positive integer N\,. Find N\,.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.