IMO Shortlist 2011 problem A7
Dodao/la:
arhiva23. lipnja 2013. Let
![a,b](/media/m/7/d/8/7d8bdace47e602448e6040957d8cf923.png)
and
![c](/media/m/e/a/3/ea344283b6fa26e4a02989dd1fb52a51.png)
be positive real numbers satisfying
![\min(a+b,b+c,c+a) > \sqrt{2}](/media/m/c/9/6/c96a424bf119518a826de113d013859d.png)
and
![a^2+b^2+c^2=3.](/media/m/0/6/f/06f5df8703e773b3a6cc8b8fd0a7e567.png)
Prove that
![\frac{a}{(b+c-a)^2} + \frac{b}{(c+a-b)^2} + \frac{c}{(a+b-c)^2} \geq \frac{3}{(abc)^2}.](/media/m/e/8/1/e81e75f134b138e75941068df0a70d40.png)
Proposed by Titu Andrescu, Saudi Arabia
%V0
Let $a,b$ and $c$ be positive real numbers satisfying $\min(a+b,b+c,c+a) > \sqrt{2}$ and $a^2+b^2+c^2=3.$ Prove that
$$\frac{a}{(b+c-a)^2} + \frac{b}{(c+a-b)^2} + \frac{c}{(a+b-c)^2} \geq \frac{3}{(abc)^2}.$$
Proposed by Titu Andrescu, Saudi Arabia
Izvor: Međunarodna matematička olimpijada, shortlist 2011