gt $\Leftrightarrow \frac{1}{bc}+\frac{1}{ca}+\frac{1}{ab}=1$ (1)đặt $x=\frac{1}{a};y=\frac{1}{b};z=\frac{1}{c}$ ($x,y,z>0$ )
(1) $\Leftrightarrow xy+yz+zx=1$
VT of đpcm TT
$\frac{1}{\sqrt{1+\frac{1}{x^{2}}}}+\frac{1}{\sqrt{1+\frac{1}{y^{2}}}}+\frac{1}{\sqrt{1+\frac{1}{z^{2}}}}$
ta có $\frac{1}{\sqrt{1+\frac{1}{x^{2}}}}=\frac{1}{\sqrt{\frac{x^{2}+1}{x^{2}}}}=\frac{x}{\sqrt{(x+y)(x+z)}}$ ( do $xy+yz+zx=1$)
$\leq \frac{1}{2}(\frac{x}{x+y}+\frac{x}{x+z})$
TT $\Rightarrow VT\leq \frac{3}{2}$
dấu "=" $\Leftrightarrow a=b=c=\sqrt{3}$