2009 USAMO Problems/Problem 4
Contents
[hide]Problem
For let , , ..., be positive real numbers such that
Prove that .
Solution 1
Assume without loss of generality that . Now we seek to prove that .
By the Cauchy-Schwarz Inequality, Since , clearly , dividing yields:
as desired.
Solution 2
Assume without loss of generality that . Using the Cauchy–Bunyakovsky–Schwarz inequality and the inequality given, (Note that since as given!) This implies that as desired.
~Deng Tianle, username: Leole
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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