Given $a,b,c$ are positive satisfy $ab+bc+ca\ge3$. FInd the maximum value of $$\frac{1}{1+a^2+b^2}+\frac{1}{1+b^2+c^2}+\frac{1}{1+c^2+a^2}$$
By C-S $(a^2+b^2+1)(1+1+c^2)\geq (a+b+c)^2$
$\Leftrightarrow \frac{1}{a^2+b^2+1}\leq \frac{2+c^2}{(a+b+c)^2}$
$\frac{1}{a^2+b^2+1}+\frac{1}{b^2+c^2+1}+\frac{1}{c ^2+a^2+1}\leq \frac{6+a^2+b^2+c^2}{(a+b+c)^2}$
$\Leftrightarrow ab+bc+ca \geq 3\Leftrightarrow (a+b+c)^2 \geq 6+a^2+b^2+c^2$
Hence Max=1. Right or Wrong. If wrong can fix for me ?