Áp dụng tính chất dãy tỉ số bằng nhau ta có:

\(\frac{x-y-z}{x}=\frac{-x+y-z}{y}=\frac{-x-y+z}{z}=\frac{x-y-z-x+y-z-x-y+z}{x+y+z}\)\(=\frac{-\left(x+y+z\right)}{x+y+z}\)

Nếu   \(x+y+z=0\)thì   \(\hept{\begin{cases}x+y=-z\\y+z=-x\\z+x=-y\end{cases}}\)

\(A=\left(1+\frac{y}{x}\right)\left(1+\frac{z}{y}\right)\left(1+\frac{x}{z}\right)\)

\(=\frac{x+y}{x}.\frac{y+z}{y}.\frac{z+x}{z}\)

\(=\frac{-z}{x}.\frac{-x}{y}.\frac{-y}{z}=-1\)

Nếu  \(x+y+z\ne0\)thì   \(\frac{x-y-z}{x}=\frac{-x+y-z}{y}=\frac{-x-y+z}{z}=-1\)

suy ra:   \(\frac{x-y-z}{x}=-1\)            \(\Rightarrow\)       \(x-y-z=-x\)          \(\Rightarrow\)     \(y+z=2x\)

             \(\frac{-x+y-z}{y}=-1\)                     \(-x+y-z=-y\)                         \(x+z=2y\)

             \(\frac{-x-y+z}{z}=-1\)                    \(-x-y+z=-z\)                         \(x+y=2z\)

\(A=\left(1+\frac{y}{x}\right)\left(1+\frac{z}{y}\right)\left(1+\frac{x}{z}\right)\)

\(=\frac{x+y}{x}.\frac{y+z}{y}.\frac{x+z}{z}\)

\(=\frac{2z}{x}.\frac{2x}{y}.\frac{2y}{z}=8\)

Ta có: \(\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)^2=\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}+2\left(\frac{1}{xy}+\frac{1}{xz}+\frac{1}{yz}\right)\)

\(\left(\sqrt{3}\right)^2=P+\frac{2\left(z+y+x\right)}{xyz}\) 

Mà x+y+z=xyz

=> P+2=3=>P=1

Vậy P=1

x + y + z = 0 \(\Leftrightarrow\)\(\hept{\begin{cases}y+z=-x\\x+y=-z\\x+z=-y\end{cases}}\)\(\Leftrightarrow\)\(\hept{\begin{cases}(y+z)^2=(-x)^2\\(x+y)^2=(-z)^2\\(x+z)^2=(-y)^2\end{cases}}\)\(\Leftrightarrow\)\(\hept{\begin{cases}y^2+2yz+z^2=x^2\\x^2+2xy+y^2=z^2\\x^2+2xz+z^2=y^2\end{cases}}\)\(\Leftrightarrow\)\(\hept{\begin{cases}y^2+z^2-x^2=-2yz\\x^2+y^2-z^2=-2xy\\x^2+z^2-y^2=-2xz\end{cases}}\)

Thay vào P ta được:

P=\(\frac{1}{-2yz}\)\(+\)\(\frac{1}{-2xy}\)\(+\)\(\frac{1}{-2xz}\)\(=\)\(\frac{-x}{2xyz}\)\(+\)\(\frac{-z}{2xyz}\)\(+\)\(\frac{-y}{2xyz}\)\(=\)\(\frac{-(x+y+z)}{2xyz}\)\(=\)0 \((x+y+z=0)\)

Vậy với \(x+y+z=0\)và \(xyz\ne0\)thì \(P=0\)

\(x+y+z=xyz\Rightarrow\frac{1}{xy}+\frac{1}{xz}+\frac{1}{yz}=1\)
\(\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)^2=1,63205^2\Rightarrow C+2\left(\frac{1}{xy}+\frac{1}{xz}+\frac{1}{yz}\right)=1.63205^2\)
\(\Rightarrow C+2=1.63205^2\Rightarrow C=1.63205^2-2\)

xin lỗi mình mới học lớp 6

đối vs bài bn thì mk ko giải được rùi

\(A=\left(1-\frac{z}{x}\right)\left(1-\frac{x}{y}\right)\left(1+\frac{y}{z}\right)\)

\(A=\frac{x-z}{x}\cdot\frac{y-x}{y}\cdot\frac{y+z}{z}\)

Do \(x-y-z=0\)

\(\Rightarrow x-z=y;y-x=-z;y+z=x\)

Khi đó \(A=\frac{y}{x}\cdot\frac{-z}{y}\cdot\frac{x}{z}=-1\)

Vậy A=-1

\(\frac{1}{xy+x+1}+\frac{y}{yz+y+1}+\frac{1}{xyz+yz+y}\)

\(=\frac{1}{xy+x+1}+\frac{y}{yz+y+1}+\frac{1}{1+yz+y}\)

\(=\frac{1}{xy+x+1}+\frac{y+1}{yz+y+1}\)

\(=\frac{yz}{xy\cdot yz+xyz+yz}+\frac{y+1}{yz+y+1}\)

\(=\frac{yz}{yz+y+1}+\frac{y+1}{yz+y+1}\)

\(=\frac{yz+y+1}{yz+y+1}\)

\(=1\)