\(a+b+c=\frac{9}{a}+\frac{9}{b}+\frac{9}{c}=9\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\)

\(\Rightarrow\left(a+b+c\right)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)=9\)

Ta có: \(\left(a+b+c\right)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)=1+1+1+\left(\frac{a}{b}+\frac{b}{a}\right)+\left(\frac{b}{c}+\frac{c}{b}\right)+\left(\frac{c}{a}+\frac{a}{c}\right)\ge3+2+2+2=9\)

Dấu "=" xảy ra khi a = b = c = 3 (do abc = 27)

....

Đặt \(P=\frac{a-b}{c}+\frac{b-c}{a}+\frac{c-a}{b}\) ; \(Q=\frac{c}{a-b}+\frac{a}{b-c}+\frac{b}{c-a}\)

Ta có : \(P=\frac{a-b}{c}+\frac{b-c}{a}+\frac{c-a}{b}=\frac{ab\left(a-b\right)+bc\left(b-c\right)+ac\left(c-a\right)}{abc}\)

Xét tử số của P  :  \(ab\left(a-b\right)+bc\left(b-c\right)+ac\left(c-a\right)=ab\left[-\left(b-c\right)-\left(c-a\right)\right]+bc\left(b-c\right)+ac\left(c-a\right)\)

\(=-ab\left(b-c\right)-ab\left(c-a\right)+bc\left(b-c\right)+ac\left(c-a\right)\)

\(=b\left(b-c\right)\left(c-a\right)+a\left(c-a\right)\left(c-b\right)=\left(b-c\right)\left(c-a\right)\left(b-a\right)\)

\(\Rightarrow P=\frac{\left(b-c\right)\left(c-a\right)\left(b-a\right)}{abc}\)

Lại có : \(Q=\frac{c}{a-b}+\frac{a}{b-c}+\frac{b}{c-a}\). Đặt \(a-b=x\); \(b-c=y\); \(c-a=z\)

Suy ra được : \(\hept{\begin{cases}x-y=a-b-b+c=a+c-2b=-3b\\y-z=b-c-c+a=a+b-2c=-3c\\z-x=c-a+b-a=b+c-2a=-3a\end{cases}\Rightarrow\hept{\begin{cases}b=-\frac{\left(x-y\right)}{3}\\c=-\frac{\left(y-z\right)}{3}\\a=-\frac{\left(z-x\right)}{3}\end{cases}}}\)

Ta có : \(Q=\frac{-\left(\frac{y-z}{3}\right)}{x}+\frac{-\left(\frac{z-x}{3}\right)}{y}+\frac{-\left(\frac{x-y}{3}\right)}{z}=-\frac{1}{3}.\left(\frac{y-z}{x}+\frac{z-x}{y}+\frac{x-y}{z}\right)\)

\(=-\frac{1}{3}\left(\frac{yz\left(y-z\right)+xz\left(z-x\right)+yx\left(x-y\right)}{xyz}\right)\)

Đến đây rút gọn tương tự với P được: \(Q=\frac{\left(x-z\right)\left(x-y\right)\left(z-y\right)}{3xyz}=\frac{\left(3a\right).\left(-3b\right).\left(3c\right)}{3\left(a-b\right)\left(b-c\right)\left(c-a\right)}\Rightarrow Q=\frac{-9abc}{\left(a-b\right)\left(b-c\right)\left(c-a\right)}\)

Vậy : \(PQ=\frac{\left(b-c\right)\left(c-a\right)\left(b-a\right)}{abc}.\frac{-9abc}{\left(a-b\right)\left(b-c\right)\left(c-a\right)}=9\)

Vậy ta có điều phải chứng minh.

\(\)

Đặt \(\left(\frac{a-b}{c},\frac{b-c}{a},\frac{c-a}{b}\right)\rightarrow\left(x,y,z\right)\)

Khi đó:\(\left(\frac{c}{a-b},\frac{a}{b-c},\frac{b}{c-a}\right)\rightarrow\left(\frac{1}{x},\frac{1}{y},\frac{1}{z}\right)\)

Ta có:

\(P\cdot Q=\left(x+y+z\right)\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)=3+\frac{y+z}{x}+\frac{z+x}{y}+\frac{x+y}{z}\)

Mặt khác:\(\frac{y+z}{x}=\left(\frac{b-c}{a}+\frac{c-a}{b}\right)\cdot\frac{c}{a-b}=\frac{b^2-bc+ac-a^2}{ab}\cdot\frac{c}{a-b}\)

\(=\frac{c\left(a-b\right)\left(c-a-b\right)}{ab\left(a-b\right)}=\frac{c\left(c-a-b\right)}{ab}=\frac{2c^2}{ab}\left(1\right)\)

Tương tự:\(\frac{x+z}{y}=\frac{2a^2}{bc}\left(2\right)\)

\(=\frac{x+y}{z}=\frac{2b^2}{ac}\left(3\right)\)

Từ ( 1 );( 2 );( 3 ) ta có:
\(P\cdot Q=3+\frac{2c^2}{ab}+\frac{2a^2}{bc}+\frac{2b^2}{ac}=3+\frac{2}{abc}\left(a^3+b^3+c^3\right)\)

Ta có:\(a+b+c=0\)

\(\Rightarrow\left(a+b\right)^3=-c^3\)

\(\Rightarrow a^3+b^3+3ab\left(a+b\right)=-c^3\)

\(\Rightarrow a^3+b^3+c^3=3abc\)

Khi đó:\(P\cdot Q=3+\frac{2}{abc}\cdot3abc=9\)

Mách mk nốt 2 bài kia vs

Đặt \(M=\frac{b+c}{a}+\frac{2a+c}{b}+\frac{4\left(a+b\right)}{a+c}\)

\(=\frac{b}{a}+\frac{c}{a}+\frac{a}{b}+\frac{a+c}{b}+\frac{4a}{a+c}+\frac{4b}{a+c}\)

\(=\left(\frac{b}{a}+\frac{a}{b}\right)+\left(\frac{c+a}{a}+\frac{4a}{a+c}\right)+\left(\frac{a+c}{b}+\frac{4b}{a+c}\right)-1\)

Áp dụng BĐT AM-GM ta có:

\(M\ge2.\sqrt{\frac{b}{a}.\frac{a}{b}}+2.\sqrt{\frac{c+a}{a}.\frac{4a}{a+c}}+2.\sqrt{\frac{a+c}{b}.\frac{4b}{a+c}}-1=2+4+4-1=9\)

Dấu " = " xảy ra <=> a=b=c ( tự giải cụ thể nhé ).

Bài này hình như thừa điều kiện abc=1.

Nếu có chỗ nào sai sót xin chỉ giáo.

Thử :

Áp dụng BĐT Cosi ta đc : 

\(\frac{a}{c}+\frac{b}{a}+\frac{c}{b}\ge\frac{9}{a+b+c}\)

\(\frac{a}{c}+\frac{b}{a}+\frac{c}{b}\ge3\sqrt{\frac{a}{c}.\frac{b}{a}.\frac{c}{b}}=3\)

Dấu ''='' xảy ra khi \(\frac{9}{a+b+c}\Leftrightarrow\frac{9}{3+3+3}=1\)

\(\Leftrightarrow\orbr{\begin{cases}a=1\\b=1\end{cases};c=1}\)

Lần đầu lm cs vẻ sai phần trình bày 

No Name  làm thế này mới đúng

Áp dụng BĐT Cauchy-Schwarz ta có:

\(\frac{a}{c}+\frac{b}{c}+\frac{c}{a}\ge\frac{\left(a+b+c\right)^2}{ab+bc+ca}\)

Ta sẽ chứng minh

\(\frac{\left(a+b+c\right)^2}{ab+bc+ca}\ge\frac{9}{a+b+c}\Leftrightarrow\frac{3}{ab+bc+ca}+2\ge\frac{9}{a+b+c}\)

Đặt a+b+c=t thì ta cần chứng minh

\(\frac{6}{t^2-3}+2\ge\frac{9}{t}\Leftrightarrow\left(t+3\right)\left(t-3\right)^2\ge0\)

Dấu "=" xảy ra <=> a=b=c=1

Áp dụng bất đẳng thức Cauchy-Schwarz, ta được:

\(3\left(a^2+b^2+c^2\right)=\left(1^2+1^2+1^2\right)\left(a^2+b^2+c^2\right)\ge\left(a+b+c\right)^2\)

\(\Rightarrow\left(a+b+c\right)^2\le3.3=9\)hay \(a+b+c\le3\)(do \(a^2+b^2+c^2=3\))

Theo bất đẳng thức Mincopxki và bất đẳng thức Bunyakovsky dạng phân thức, ta được:

\(\sqrt{\frac{9}{\left(a+b\right)^2}+c^2}+\sqrt{\frac{9}{\left(b+c\right)^2}+a^2}+\sqrt{\frac{9}{\left(c+a\right)^2}+b^2}\)

\(\ge\sqrt{9\left(\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}\right)^2+\left(a+b+c\right)^2}\)

\(\ge\sqrt{9\left[\frac{9}{2\left(a+b+c\right)}\right]^2+\left(a+b+c\right)^2}\)

Đến đây, ta cần chứng minh rằng: \(\sqrt{9\left[\frac{9}{2\left(a+b+c\right)}\right]^2+\left(a+b+c\right)^2}\ge\frac{3\sqrt{13}}{2}\)(*)

Đặt \(t=a+b+c\Rightarrow0< t\le3\)

Khi đó, (*) trở thành \(\sqrt{9\left(\frac{9}{2t}\right)^2+t^2}\ge\frac{3\sqrt{13}}{2}\Leftrightarrow9\left(\frac{9}{2t}\right)^2+t^2\ge\frac{117}{4}\)

\(\Leftrightarrow\frac{\left(t-3\right)\left(2t-9\right)\left(t+3\right)\left(2t+9\right)}{4t^2}\ge0\)(đúng với mọi \(0< t\le3\))

Đẳng thức xảy ra khi a = b = c = 1