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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