Difference between revisions of "2004 USAMO Problems/Problem 5"
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− | + | This follows from [[Hölder's Inequality]] | |
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− | + | when we take <math>m_{1,1} = a^3</math>, <math>m_{2,2} = b^3</math>, <math>m_{3,3} = c^3</math>, and <math>m_{x,y} = 1 </math> when <math> x \neq y </math>. We have equality if and only if <math>a = b = c = 1 </math>. | |
− | + | ''It is also possible to solve this inequality by expanding terms and applying brute force, either before or after proving that <math>x^5 - x^2 + 3 \ge x^3 + 2 </math>.'' | |
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Revision as of 23:11, 1 July 2008
Problem 5
(Titu Andreescu) Let , , and be positive real numbers. Prove that
.
Solutions
We first note that for positive , . We may prove this in the following ways:
- Since and must be both lesser than, both equal to, or both greater than 1, by the rearrangement inequality, .
- Since and have the same sign, , with equality when .
- By weighted AM-GM, and . Adding these gives the desired inequality. Equivalently, the desired inequality is a case of Muirhead's Inequality.
It thus becomes sufficient to prove that
.
This follows from Hölder's Inequality
when we take , , , and when . We have equality if and only if .
It is also possible to solve this inequality by expanding terms and applying brute force, either before or after proving that .
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.