Đáp án:
`x/(y+z+t)=y/(x+z+t)=z/(x+y+t)=t/(x+y+z)`
$\text{Áp dụng tính chất dãy tỉ số bằng nhau ta có:}$
`x/(y+z+t)=y/(x+z+t)=z/(x+y+t)=t/(x+y+z)=(x+y+z+t)/(y+z+t+x+z+t+x+y+t+x+y+z)`
`=(x+y+z+t)/(3.(x+y+z+t))`
$\text{Xét}$ `(x+y+z+t)/(3.(x+y+z+t))=0`
`=>` $\left\{\begin{matrix}y+z=-(z+t)& \\y+z=-(t+x)&\\z+t=-(x+y)\\t+x=-(y+z)& \end{matrix}\right.$
`=> P= (x+y)/(z+t)+(y+z)/(t+x)+(z+t)/(x+y)+(t+x)/(y+z)`
`=(-1)+(-1)+(-1)+(-1)=-4`
$\text{Xét}$ `(x+y+z+t)/(3.(x+y+z+t))\ne0`
`=>(x+y+z+t)/(3.(x+y+z+t))=1/3`
`=>` $\left\{\begin{matrix}y+z+t=3x& \\x+z+t=3y&\\x+y+t=3z\\x+y+z=3t& \end{matrix}\right.$
`=>` $\left\{\begin{matrix}y+z+t+x=4x& \\x+y+z+t=4y&\\x+y+z+t=4z\\x+y+z+t=4t& \end{matrix}\right.$
`=> 4x=4y=4z=4t`
`=> x=y=z=t`
`P= (x+y)/(z+t)+(y+z)/(t+x)+(z+t)/(x+y)+(t+x)/(y+z)`
`=1+1+1+1=4`
$\text{Vậy P=4}$
