\(\dfrac{x}{y+z+t}=\dfrac{y}{x+t+z}=\dfrac{z}{t+y+x}\)
\(\Rightarrow\dfrac{x}{y+z+t}+1=\dfrac{y}{x+t+z}+1=\dfrac{z}{t+y+x}+1\)
\(\Rightarrow\dfrac{x+y+z+t}{y+z+t}=\dfrac{x+y+z+t}{x+t+z}=\dfrac{x+y+z+t}{t+y+x}\)
\(\Rightarrow\left[{}\begin{matrix}x=y=z=t\\x+y+z+t=0\end{matrix}\right.\)
\(\circledast\) Khi \(x=y=z=t\) thì
\(P=\dfrac{x+y}{z+t}+\dfrac{y+z}{x+t}+\dfrac{z+t}{x+y}+\dfrac{t+x}{y+z}=1+1+1+1=4\)
\(\circledast\) Khi \(x+y+z+t=0\) thì:\(\left\{{}\begin{matrix}x+y=-\left(z+t\right)\\y+z=-\left(x+t\right)\\z+t=-\left(x+y\right)\\t+x=-\left(y+z\right)\end{matrix}\right.\)
\(P=\dfrac{x+y}{z+t}+\dfrac{y+z}{x+t}+\dfrac{z+t}{x+y}+\dfrac{t+x}{y+z}=\left(-1\right)+\left(-1\right)+\left(-1\right)+\left(-1\right)=-4\)