I have a problem with solving these 3 symmetric inequations. I want to find positive values for these 6 variables such that the following inequations hold (or show that it is impossible). Please suggest me how to do, or any good tools for this:
$\frac{2a}{b+c}+1 \lt \frac{b'+c'}{a'}$ (1)
$\frac{2b}{c+a}+1 \lt \frac{c'+a'}{b'}$ (2)
$\frac{2c}{a+b}+1 \lt \frac{a'+b'}{c'}$ (3)
Thank you for your help.
Adding one to both sides and taking reciprocals gives the three inequalities: $$\begin{align*}\frac{b+c}{2(a+b+c)}&>\frac{a'}{a'+b'+c'} \\ \frac{c+a}{2(a+b+c)}&>\frac{b'}{a'+b'+c'} \\ \frac{a+b}{2(a+b+c)}&>\frac{c'}{a'+b'+c'}\end{align*}$$
Summing these up gives $1>1$, which is impossible. So no positive solutions exist.
Intuition: to combine the given inequalities, it would be optimal if the denominators were equal at least on one side. As it turns out, one can do better, makig the denominators equal on both sides simultaneously. That leads to the above solution.