<s>[INST] Prove that $\frac1{a+b+1}+\frac1{b+c+1}+\frac1{c+a+1}\le1$ If $abc=1$ then
$$\frac1{a+b+1}+\frac1{b+c+1}+\frac1{c+a+1}\le1$$
I have tried AM-GM and C-S and can't seem to find a solution. What is the best way to prove it?
 [/INST] Also we can use the following reasoning.
For positives $a$, $b$ and $c$ let $a=x^3$, $b=y^3$ and $c=z^3$.
Hence, $\sum\limits_{cyc}\frac{1}{a+b+1}=\sum\limits_{cyc}\frac{1}{x^3+y^3+xyz}\leq\sum\limits_{cyc}\frac{1}{x^2y+y^2x+xyz}=\sum\limits_{cyc}\frac{z}{x+y+z}=1$,
but I think the first way is better.
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