<s>[INST] If $a,b,c$ are positive real numbers, then $\frac{a}{ab+1}+\frac{b}{bc+1}+\frac{c}{ca+1}\geq \frac{3}{2}$ 
If $a$, $b$, and $c$ are positive real numbers such that $abc=1$, then prove that
$$\frac{a}{ab+1}+\frac{b}{bc+1}+\frac{c}{ca+1}\geq \frac{3}{2}$$

Progress
I think the relevant concept would be  the application of AM/GM inequality.
 [/INST] Writing $a=x/y$, $b=y/z$, and $c=z/x$, we simplify the LHS:
$$
\frac{xz}{xy+yz}+\frac{xy}{xz+yz}+\frac{yz}{xy+xz}=\frac{r}{s+t}+\frac{s}{r+t}+\frac{t}{s+r}
$$
where $r=xz$, $s=xy$, and $t=yz$. It remains to invoke Nesbitt's inequality. 

The linked wikipedia article contains a standard Cauchy-Schwarz proof:
$$
[(s+t)+(r+t)+(s+r)]\left[\frac{1}{s+t}+\frac{1}{r+t}+\frac{1}{s+r}\right]\geq9
$$
which after the simplification of the LHS implies
$$
\frac{r}{s+t}+\frac{s}{r+t}+\frac{t}{s+r}\geq\frac{3}{2}.
$$
</s>