1997 USAMO Problems/Problem 5
Problem
Prove that, for all positive real numbers
.
Solution 1
Because the inequality is homogenous (i.e. can be replaced with without changing the inequality other than by a factor of for some ), without loss of generality, let .
Lemma: Proof: Rearranging gives , which is a simple consequence of and
Thus, by :
Solution 2
Rearranging the AM-HM inequality, we get . Letting , , and , we get By AM-GM on , , and , we have So, . -Tigerzhang
This solution doesn’t work because , so
Solution 3
If we multiply each side by , we get that we must just prove that If we divide our LHS equation, we get that Make the astute observation that by Titu's Lemma, Therefore: If we expand it out, we get that Since our original equation is less than this, we get that
-KEVIN_LIU
See Also
1997 USAMO (Problems • Resources) | ||
Preceded by Problem 4 |
Followed by Problem 6 | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAMO Problems and Solutions |
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