IMO Shortlist 2006 problem A4
Dodao/la:
arhiva2. travnja 2012. Prove the inequality:
for positive reals
![a_{1}](/media/m/0/6/5/0653090dabb5d1972cd7a7dfcd31abc1.png)
,
![a_{2}](/media/m/5/5/6/5565dac5c7f1dadb0e60c273c1d11c80.png)
, ...,
![a_{n}](/media/m/e/1/b/e1bf963ddae5d084fba54d8a7aa04acc.png)
.
%V0
Prove the inequality:
$\displaystyle \sum_{i < j}{\frac {a_{i}a_{j}}{a_{i} + a_{j}}}\leq \frac {n}{2(a_{1} + a_{2} + ... + a_{n})}\cdot \sum_{i < j}{a_{i}a_{j}}$
for positive reals $a_{1}$, $a_{2}$, ..., $a_{n}$.
Izvor: Međunarodna matematička olimpijada, shortlist 2006