2009 USAMO Problems/Problem 4
Problem
For let
,
, ...,
be positive real numbers such that
![$(a_1+a_2+ ... +a_n)\left( {1 \over a_1} + {1 \over a_2} + ... +{1 \over a_n} \right) \le \left(n+ {1 \over 2} \right) ^2$](http://latex.artofproblemsolving.com/5/9/6/5963cac8c96d5f27b431ca50755a0e1b60ed69c1.png)
Prove that .
Solution
Assume without loss of generality that . Now we seek to prove that
.
By the Cauchy-Schwarz Inequality,
Since
, clearly
, dividing yields:
as desired.
Alternative Solution (by Deng Tianle, username: Leole)
Assume without loss of generality that .
Using the Cauchy–Bunyakovsky–Schwarz inequality and the inequality given,
This implies that
as desired.
See Also
2009 USAMO (Problems • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAMO Problems and Solutions |
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