<s>[INST] $a,b,c>0$, prove: $\sum_{cyc}{\frac{a+\sqrt{bc}}{\sqrt{(a+b)(a+c)}+\sqrt{bc}}}\le2$ 
Let $a,b,c>0$. Prove that:  $$\!\!\frac{a+\sqrt{bc}}{\sqrt{(a+b)(a+c)}+\sqrt{bc}}+ \frac{b+\sqrt{ca}}{\sqrt{(b+c)(b+a)}+\sqrt{ca}}+ \frac{c+\sqrt{ab}}{\sqrt{(c+a)(c+b)}+\sqrt{ab}}\le2$$
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My approach using AM-GM: $\sqrt{(a+b)(a+c)}\ge2\sqrt[4]{a^2bc}$ ;  so we need to prove that: $$\sum_{cyc}{\frac{a+\sqrt{bc}}{2\sqrt[4]{a^2bc}+\sqrt{bc}}}\le2$$ Due to homogenious, I denote $abc=1$ which implies the new one variable inequality: $$\sum_{cyc}{\frac{a+\dfrac{1}{\sqrt{a}}}{2\sqrt[4]{a}+\dfrac{1}{\sqrt{a}}}}\le2$$
I am trying to find suitable term to finish my idea.
Is there any good way to full of my approach or other better idea? Thanks for help.
 [/INST] Here is SOS form I get. I used new algorithm for MatLab to kill it:
The OP is equivalent to:
$$\sum\frac{a^3\left(\sqrt{b}-\sqrt{c}\right)^4}{\left(\sqrt{(a+b)(a+c)}+\sqrt{bc}\right)(X_a+Y_a)}\ge0$$
Where :$X_a=a(2a+b+c)\sqrt{(a+b)(a+c)}$ and $Y_a=a\sqrt{bc}(a+b+c)+2a^2(a+b+c)-a^2\sqrt{bc}$
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