2009 USAMO Problems/Problem 4
For let , , ..., be positive real numbers such that
Prove that .
Assume without loss of generality that . Now we seek to prove that .
By the Cauchy-Schwarz Inequality, Since , clearly , dividing yields:
Alternative Solution (by Deng Tianle, username: Leole) 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.
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