Consider quadrilateral ABCD whose diagonals are perpendicular and meet at point E.

Minimize the perimeter of ABCD where AB, BC, CD, DA, EA, EB, EC, ED are all different integers.
(Or prove no such quadrilateral exists.)

(In reply to First half of solution by Tristan)

I had imagined finding a way of making all the ratios perfectly fit, since they only dealt with factors 2,3, and 5, but I quickly found that it's easier just to use a pair of ratios like this:

15 *(4/3)
20 *(12/5)
48 *(3/4)
36 *(5/12)

The sides of that quadrilateral are 25, 52, 60, and 39, summing to 176.

I haven't checked yet to see if other pairs of ratios work better, so it's possible that this is not the minimum.

I think it would be quite elegant if someone found a solution that used four different ratios, though it would probably have a higher perimeter.

Posted by Tristan on 2005-08-29 17:17:54