<s>[INST] Proof that $\frac{a}{a+3b+3c}+\frac{b}{b+3a+3c}+\frac{c}{c+3a+3b} \ge \frac{3}{7}$ for all $a,b,c > 0$ So I am trtying to proof that $\frac{a}{a+3b+3c}+\frac{b}{b+3a+3c}+\frac{c}{c+3a+3b} \ge \frac{3}{7}$ for all  $a,b,c > 0$. First I tried with Cauchy–Schwarz inequality but got nowhere.
Now I am trying to find a convex function, so I can use jensen's inequality, but I can't come up with one which works.. Has anyone an idea?
 [/INST] We need to prove that 
$$\sum\limits_{cyc}\left(\frac{a}{a+3b+3c}-\frac{1}{7}\right)\geq0$$ or
$$\sum\limits_{cyc}\frac{2a-b-c}{a+3b+3c}\geq0$$ or
$$\sum\limits_{cyc}\frac{a-b-(c-a)}{a+3b+3c}\geq0$$ or
$$\sum_{cyc}(a-b)\left(\frac{1}{a+3b+3c}-\frac{1}{b+3c+3a}\right)\geq0$$ or
$$\sum\limits_{cyc}\frac{(a-b)^2}{(a+3b+3c)(b+3c+3a)}\geq0.$$
Done!
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