2006 IMO Shortlist Problems/A4
Revision as of 11:12, 29 December 2007 by Boy Soprano II (talk | contribs) (whoops, submitted it before I was done the first time)
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.