Suppose $x,y\geq 0$ and $b,c,d\geq 1$ are integers. Prove or find a counterexample to the following inequality $$ \frac{1}{1+\frac{b+d}{\sqrt{x^{2}+c^{2}}}} + \frac{1}{1+\frac{c+d}{\sqrt{y^{2}+b^{2}}}}\geq \frac{1}{1+\frac{d}{\sqrt{(x+b)^{2}+(y+c)^{2}}}}. $$

closed as too localized by Felipe Voloch, Vladimir Dotsenko, Chris Godsil, Suvrit, Daniel Moskovich Aug 1 '12 at 1:00
This question is unlikely to help any future visitors; it is only relevant to a small geographic area, a specific moment in time, or an extraordinarily narrow situation that is not generally applicable to the worldwide audience of the internet. For help making this question more broadly applicable, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.
OK, I'll take kodlu's word as a proof this time. :). Let $u=\sqrt{x^2+c^2}$, $v=\sqrt{y^2+b^2}$, $w=\sqrt{(x+b)^2+(y+c)^2}$. We need to show that $$ \frac{u}{u+b+d}+\frac{v}{v+c+d}\ge \frac{w}{w+d} $$ Note that $w\le u+v$ (triangle inequality on the plane) and that $t\mapsto \frac{t}{t+d}$ is increasing in $t$, so it will suffice to show that $$ \frac{u}{u+b+d}+\frac{v}{v+c+d}\ge \frac{u+v}{u+v+d}. $$ However, $b\le v$ and $c\le u$, so the LHS is at least $$ \frac{u}{u+v+d}+\frac{v}{v+u+d}= \frac{u+v}{u+v+d}. $$ As I said, I wouldn't be surprised to see it on some decent high school math. contest. 

