1980 USAMO Problems/Problem 5
If are reals such that , show that
Rewrite the given inequality so that is isolated on the right side. Set the left side to be . Now a routine computation shows
which shows that is convex (concave up) in all three variables. Thus the maxima can only occur at the endpoints, i.e. if and only if . Checking all eight cases shows that the value of the expression cannot exceed 1.
|1980 USAMO (Problems • Resources)|
|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.