2006 IMO Shortlist Problems/A4
Problem
Prove the inequality for positive real numbers .
Solution
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.
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.