\(A=\left(1+\frac{x}{y}\right)\left(1+\frac{y}{z}\right)\left(1+\frac{z}{x}\right)\)\(=\left(\frac{x+y}{y}\right)\left(\frac{y+z}{z}\right)\left(\frac{z+x}{x}\right)\)
Xét 2 TH
+> Nếu \(x+y+z=0\)
=> \(\hept{\begin{cases}x+y=-z\\y+z=-x\\x+z=-y\end{cases}}\)
=> \(A=\left(-\frac{z}{y}\right)\left(-\frac{x}{z}\right)\left(-\frac{y}{x}\right)=-1\)
+> Nếu \(x+y+z\ne0\)
\(\frac{x+y+2013z}{z}=\frac{y+z+2013x}{x}=\frac{x+z+2013y}{y}\)
=> \(\frac{x+y}{z}+2013=\frac{y+z}{x}+2013=\frac{z+x}{y}+2013\)
=>\(\frac{x+y}{z}=\frac{y+z}{x}=\frac{z+x}{y}\)\(=\frac{x+y+y+z+z+x}{x+y+z}=2\)
=> \(\hept{\begin{cases}x+y=2z\\y+z=2x\\z+x=2y\end{cases}}\)
=> A = 2.2.2=8
Ta có :
\(A=\frac{x+y+2013z}{z}=\frac{y+z+2013x}{x}=\frac{x+z+2013}{y}\)
\(\Leftrightarrow A=\frac{x+y}{z}+2013=\frac{y+z}{x}+2013=\frac{x+z}{y}+2013=2015\)( Chỗ này áp dụng Tc của dãy tỉ số bằng nhau là ra )
\(\Leftrightarrow\frac{x+y}{z}=\frac{y+z}{x}=\frac{x+z}{y}=2\)
\(\Rightarrow\hept{\begin{cases}x+y=2z\\y+z=2x\\x+z=2y\end{cases}}\)
Thay vào ta có :
\(\left(1+\frac{x}{y}\right)\left(1+\frac{y}{z}\right)\left(1+\frac{z}{x}\right)\)
\(=\left(\frac{x+y}{y}\right)\left(\frac{y+z}{z}\right)\left(\frac{x+z}{x}\right)\)
\(=\frac{2z.2x.2y}{xyz}=\frac{8xyz}{xyz}=8\)
Vậy ...........
