2011 USAMO Problems/Problem 1
Problem
Let , , be positive real numbers such that . Prove that
Solutions
Solution 1
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.
Solution 2
Rearranging the condition yields that
Now note that
Summing this for all pairs of gives that
By AM-GM. Dividing by gives the desired inequality.
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See also
2011 USAMO (Problems • Resources) | ||
First Problem | Followed by Problem 2 | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAMO Problems and Solutions |
2011 USAJMO (Problems • Resources) | ||
Preceded by Problem 1 |
Followed by Problem 3 | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAJMO Problems and Solutions |