<s>[INST] Find the value of $\frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y}$ If $$x+y+z=7$$ and $$\frac{1}{x+y}+\frac{1}{y+z}+\frac{1}{z+x}=\frac{7}{10}$$ Find the value of $$\frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y}$$
I tried but I got nothing
 [/INST] Ever heard of Ravi substitution? If we set $a=y+z,b=x+z,c=x+y$, we have:
$$a+b+c = 14,\qquad \frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{7}{10} $$
and we are asked to find:
$$ \sum_{cyc}\frac{b+c-a}{2a} = \sum_{cyc}\left(\frac{a+b+c}{2a}-1\right) = -3+\frac{1}{2}(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right).$$
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