IMO Shortlist 2008 problem A7
Dodao/la:
arhiva2. travnja 2012. Prove that for any four positive real numbers
![a](/media/m/6/d/2/6d2832265560bb67cf117009608524f6.png)
,
![b](/media/m/e/e/c/eec0d7323095a1f2101fc1a74d069df6.png)
,
![c](/media/m/e/a/3/ea344283b6fa26e4a02989dd1fb52a51.png)
,
![d](/media/m/f/7/d/f7d3dcc684965febe6006946a72e0cd3.png)
the inequality
holds. Determine all cases of equality.
Author: Darij Grinberg (Problem Proposal), Christian Reiher (Solution), Germany
%V0
Prove that for any four positive real numbers $a$, $b$, $c$, $d$ the inequality
$$\frac {(a - b)(a - c)}{a + b + c} + \frac {(b - c)(b - d)}{b + c + d} + \frac {(c - d)(c - a)}{c + d + a} + \frac {(d - a)(d - b)}{d + a + b} \geqslant 0$$
holds. Determine all cases of equality.
Author: Darij Grinberg (Problem Proposal), Christian Reiher (Solution), Germany
Izvor: Međunarodna matematička olimpijada, shortlist 2008