Difference between revisions of "2003 JBMO Problems/Problem 4"
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− | <math>\Sigma \frac {\frac {p}{q+2r}*\Sigma p | + | <math>\Sigma \frac {\frac {p}{q+2r}*\Sigma p(q+2r)}{\Sigma p(q+2r)} \geq \frac {(\Sigma p)^2}{\Sigma p(q+2r)} = \frac {(\Sigma p)^2}{3\Sigma pq}</math> |
Revision as of 05:27, 8 June 2019
Problem
Let . Prove that
Solution
Since and
, we have that
and
are always positive.
Hence, and
must also be positive.
From the inequality , we obtain that
and, analogously,
. Similarly,
and
.
Now,
Substituting and
, we now need to prove
.
We have
By Cauchy-Schwarz,
Since , we have
.
Thus,
So,