Find all the values that the expresion $\sum_{cyc} \frac{a}{a+b+d}$ takes.

Question

Let $a,b,c,d \in \mathbb{R}^+$. Find all the values that the expresion $\frac{a}{a+b+d} + \frac{b}{a+b+c} + \frac{c}{b+c+d} + \frac{d}{a+c+d}$ takes. $$\\$$ I had this problem in a test. I found that $1 = \sum_{cyc} \frac{a}{a+b+c+d} < \sum_{cyc} \frac{a}{a+b+d} < \frac{a}{a+b}+\frac{b}{b+a}+\frac{c}{c+d}+\frac{d}{d+c}=2$. I tried to find a $ \ge \le$ instead of a $><$.However I think that the fact that $1<S<2$ can help, but I don't know how to continue.

Answer 1

For $d=c\rightarrow0^+$ we have $$\sum_{cyc}\frac{a}{a+b+d}\rightarrow\frac{a}{a+b}+\frac{b}{a+b}=1.$$ For $b=d\rightarrow0^+$ we obtain: $$\sum_{cyc}\frac{a}{a+b+d}\rightarrow\frac{a}{a}+\frac{c}{c}=2.$$ Now, use continuity.