1976 USAMO Problems/Problem 4
Contents
Problem
If the sum of the lengths of the six edges of a trirectangular tetrahedron (i.e., ) is , determine its maximum volume.
Solution
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 .
Solution 2
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.
See Also
1976 USAMO (Problems • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
1 • 2 • 3 • 4 • 5 | ||
All USAMO Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.