1976 USAMO Problems/Problem 4
If the sum of the lengths of the six edges of a trirectangular tetrahedron (i.e., ) is , determine its maximum volume.
Let the side lengths of , , and be , , and , respectively. Therefore . Let the volume of the tetrahedron be . Therefore .
Note that implies , which means , which implies , which means , which implies . Equality holds only when . Therefore
is true from AM-GM, with equality only when . So . This means that , or , or , with equality only when . Therefore the maximum volume is .
Note that so Proceed as before.
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
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