1980 USAMO Problems/Problem 5
Problem
If are reals such that , show that
Solution
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.
See Also
1980 USAMO (Problems • Resources) | ||
Preceded by Problem 4 |
Followed by Last Question | |
1 • 2 • 3 • 4 • 5 | ||
All USAMO Problems and Solutions |
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