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

