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.
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