2006 IMO Shortlist Problems/A4
Prove the inequality for positive real numbers .
Note that Suppose that . Note that is an increasing function of both and . It follows that if , then i.e., is an increasing function of .
Since is also an increasing function of , it follows from Chebyshev's Inequality that or Now, for fixed , both and are increasing functions of . It follows again from Chebyshev's Inequality that or which in sum becomes If we denote , then in summary, we thus have as desired.
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