Theo giả thiết, ta có: \(a^2b^2+b^2c^2+c^2a^2=a^2b^2c^2\Leftrightarrow\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}=1\)

Áp dụng BĐT AM - GM cho 5 số, ta được: \(\hept{\begin{cases}a.a.a.b.b\le\frac{a^5+a^5+a^5+b^5+b^5}{5}=\frac{3a^5+2b^5}{5}\\b.b.b.a.a\le\frac{b^5+b^5+b^5+a^5+a^5}{5}=\frac{3b^5+2a^5}{5}\end{cases}}\)

\(\Rightarrow\frac{5\left(a^5+b^5\right)}{5}\ge a^2b^2\left(a+b\right)\)hay \(a^5+b^5\ge a^2b^2\left(a+b\right)\)

\(\Rightarrow\frac{1}{\sqrt{a^5+b^5}}\le\frac{1}{ab\sqrt{a+b}}\)(1) .

Tương tự, ta có: \(\frac{1}{\sqrt{b^5+c^5}}\le\frac{1}{bc\sqrt{b+c}}\)(2); \(\frac{1}{\sqrt{c^5+a^5}}\le\frac{1}{ca\sqrt{c+a}}\)(3)

Cộng theo vế của 3 BĐT (1), (2), (3), ta được: \(VT=\Sigma_{cyc}\frac{1}{\sqrt{a^5+b^5}}\le\Sigma_{cyc}\frac{1}{ab\sqrt{a+b}}\)()

Xét \(\left(\Sigma_{cyc}\frac{1}{ab\sqrt{a+b}}\right)^2\le\left(\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\right)\left(\Sigma_{cyc}\frac{1}{b^2\left(a+b\right)}\right)\)\(=\Sigma_{cyc}\frac{1}{b^2\left(a+b\right)}\Rightarrow\Sigma_{cyc}\frac{1}{ab\sqrt{a+b}}\le\sqrt{\Sigma_{cyc}\frac{1}{b^2\left(a+b\right)}}\)(2)

Từ (1) và (2) suy ra \(\Sigma_{cyc}\frac{1}{\sqrt{a^5+b^5}}\le\sqrt{\Sigma_{cyc}\frac{1}{b^2\left(a+b\right)}}\)(đpcm)

Đẳng thức xảy ra khi \(a=b=c=\sqrt{3}\)

Áp dụng cosi ta có \(a.a.a.b.b\le\frac{3a^5+2b^5}{5};b.b.b.a.a\le\frac{3b^5+2a^5}{5}\)

=> \(a^5+b^5\ge a^2b^2\left(a+b\right)\)

Khi đó

\(VT\le\frac{1}{ab\sqrt{a+b}}+\frac{1}{bc\sqrt{b+c}}+\frac{1}{ac\sqrt{a+c}}\)

Áp dụng BĐT buniacoxki  ta có :

\((\frac{1}{ab\sqrt{a+b}}+\frac{1}{bc\sqrt{b+c}}+\frac{1}{ac\sqrt{a+c}})^2\le\left(\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\right)\left(\frac{1}{b^2\left(a+b\right)}+\frac{1}{c^2\left(b+c\right)}+...\right)\)

Mà 1/a^2+1/b^2+1/c^2=1(giả thiết)

=> \(VT\le VP\)(ĐPCM)

Dấu bằng xảy ra khi a=b=c=can(3)

hay quá 

Trần Phúc Khang Đúng ngay ý tưởng chế đề:3

\(VT=\sum\dfrac{a^2}{a+3abc+4\left(ab+ac\right)}\ge\dfrac{\left(a+b+c\right)^2}{a+b+c+9abc+8\left(ab+bc+ca\right)}\)

\(VT\ge\dfrac{\left(a+b+c\right)^2}{a+b+c+\dfrac{1}{3}\left(a+b+c\right)^3+\dfrac{8}{3}\left(a+b+c\right)^2}=\dfrac{1}{4}\)

Dấu "=" xảy ra khi và chỉ khi \(a=b=c=\dfrac{1}{3}\)

Đề bài bị sai con số bên vế phải

nguyem,ưdjxrckrk

Áp dụng cosi ta có 

\(\sqrt[4]{\frac{a}{b+c}}+\sqrt[4]{\frac{a}{b+c}}+\sqrt[4]{\frac{a}{b+c}}+\sqrt[4]{\frac{a}{b+c}}+\frac{b+c}{2a\sqrt[4]{2}}\ge5\sqrt[5]{\frac{1}{\sqrt[4]{2^5}}}=\frac{5}{\sqrt[4]{2}}\)

Khi đó

\(4P\ge\frac{15}{\sqrt[4]{2}}+\left(4-\frac{1}{2\sqrt[4]{2}}\right)\left(\frac{b+c}{a}+\frac{c+a}{b}+\frac{a+b}{c}\right)\)

Mà \(\frac{b+c}{a}+\frac{c+a}{b}+\frac{a+b}{c}=\left(\frac{a}{b}+\frac{b}{a}\right)+...\ge6\)

=> \(4P\ge\frac{15}{\sqrt[4]{2}}+\left(4-\frac{1}{2\sqrt[4]{2}}\right).6=24+\frac{12}{\sqrt[4]{2}}\)

=> \(P\ge6+\frac{3}{\sqrt[4]{2}}\)

dấu bằng xảy ra khi a=b=c