Difference between revisions of "2011 USAMO Problems/Problem 1"
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We note that the product of the three (positive) terms is 1/8, therefore by AM-GM their mean is at least 1/2, and thus their sum is at least 3/2 and we are done. | We note that the product of the three (positive) terms is 1/8, therefore by AM-GM their mean is at least 1/2, and thus their sum is at least 3/2 and we are done. | ||
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+ | ==See also== | ||
+ | {{USAMO newbox|year=2011|beforetext=|before=First Problem|num-a=2}} |
Revision as of 10:36, 29 April 2011
Problem
Let , , be positive real numbers such that . Prove that
Solution
Since it is natural to consider a change of variables: with the inverse mapping given by: With this change of variables, the constraint becomes while the left side of the inequality we need to prove is now
Therefore it remains to prove that
We note that the product of the three (positive) terms is 1/8, therefore by AM-GM their mean is at least 1/2, and thus their sum is at least 3/2 and we are done.
See also
2011 USAMO (Problems • Resources) | ||
First Problem | Followed by Problem 2 | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAMO Problems and Solutions |