\(\Sigma\dfrac{1}{a^4\left(b+c\right)^2}=\Sigma\dfrac{a^2b^2c^2}{a^4\left(b+c\right)^2}=\Sigma\dfrac{\left(bc\right)^2}{\left(ab+ac\right)^2}\)
Đặt : \(ab=x;bc=y;ac=z\)
\(\Rightarrow\Sigma\dfrac{\left(bc\right)^2}{\left(ac+ab\right)^2}=\Sigma\dfrac{x^2}{\left(y+z\right)^2}=\Sigma\left(\dfrac{x}{y+z}\right)^2\)
Đặt \(\dfrac{x}{y+z}=n\); \(\dfrac{y}{z+x}=n\); \(\dfrac{z}{x+y}=k\)
\(\Rightarrow\Sigma\left(\dfrac{x}{y+z}\right)^2=m^2+n^2+k^2\)
Theo BĐT Nezbit
\(\Rightarrow n+m+k\ge\dfrac{3}{2}\)
Áp dụng BĐT Bunhiacopxki: \(m^2+n^2+k^2\ge\dfrac{\left(m+n+k\right)^2}{3}\ge\dfrac{\left(\dfrac{3}{2}\right)^2}{3}=\dfrac{3}{4}\)
=> ĐPCM
