\(\left(a+b+c\right)\left(a^2+b^2+c^2-ab-bc-ca\right)=0\)
\(\Leftrightarrow\orbr{\begin{cases}a+b+c=0\\a^2+b^2+c^2-ab-bc-ca=0\end{cases}}\)
TH1: Với a+b+c=0\(\Rightarrow\hept{\begin{cases}a+b=-c\\b+c=-a\\c+a=-b\end{cases}}\)
Ta có:\(S=\left(1+\frac{a}{b}\right)\left(1+\frac{b}{c}\right)\left(1+\frac{c}{a}\right)\)
\(=\frac{a+b}{b}.\frac{b+c}{c}.\frac{a+c}{a}\)
\(=\frac{-c}{b}.\frac{-a}{c}.\frac{-b}{a}\)
\(=-1\)
TH2: \(a^2+b^2+c^2-ab-bc-ca=0\)
\(\Rightarrow2a^2+2b^2+2c^2-2ab-2bc-2ca=0\)
\(\Leftrightarrow\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2=0\left(1\right)\)
Vì \(\hept{\begin{cases}\left(a-b\right)^2\ge0;\forall a,b,c\\\left(b-c\right)^2\ge0;\forall a,b,c\\\left(c-a\right)^2\ge0;\forall a,b,c\end{cases}}\)\(\Rightarrow\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2\ge0;\forall a,b,c\left(2\right)\)
Từ (1) và (2)\(\Rightarrow\hept{\begin{cases}\left(a-b\right)^2=0\\\left(b-c\right)^2=0\\\left(c-a\right)^2=0\end{cases}}\Leftrightarrow\hept{\begin{cases}a=b\\b=c\\c=a\end{cases}\Leftrightarrow}a=b=c\)
Ta có: \(S=\left(1+\frac{a}{b}\right)\left(1+\frac{b}{c}\right)\left(1+\frac{c}{a}\right)\)
\(=2.2.2=8\)
Vậy .... ( ko bít ghi kiểu gì luôn -.- )
