If $a,b,c ∈ \mathbb R^+$ prove $(\frac{a}{b+c} + \frac{1}{2})(\frac{b}{c+a} + \frac{1}{2})(\frac{c}{a+b} + \frac{1}{2}) \ge 1$

Question

$a,b,c ∈ \mathbb R^+$ $$(\frac{a}{b+c} + \frac{1}{2})(\frac{b}{c+a} + \frac{1}{2})(\frac{c}{a+b} + \frac{1}{2}) \ge 1$$

For this one I have little to no idea how to solve it. I tried some AM-GM and also some other manipulations but didn't give me anything that would equal or sum to 1. Tittu Lemma seems like a good option to solve this one , but using $\sqrt{a}$ in order to apply it , still doesn't give any good result. Or I am probably giving up too early.

Any help is welcomed.

Answer 1

Here's the solution

I hope my writing won't be an issue

Answer 2

Because by AM-GM $$\prod_{cyc}\left(\frac{a}{b+c}+\frac{1}{2}\right)=\frac{1}{8}\prod_{cyc}\frac{a+b+a+c}{b+c}\geq\frac{1}{8}\prod_{cyc}\frac{2\sqrt{(a+b)(a+c)}}{b+c}=1.$$