\(\frac{x}{y+z+t}=\frac{y}{z+t+x}=\frac{z}{t+x+y}=\frac{t}{x+y+z}\)
\(=\frac{x+y+z+t}{y+z+t+z+t+x+t+x+y+x+y+z}=\frac{x+y+z+t}{3x+3y+3z+3t}\)
\(=\frac{x+y+z+t}{3\left(x+y+z+t\right)}=\frac{1}{3}\)
\(\Rightarrow x=y=z=t\)
\(=\frac{x+y}{z+t}+\frac{y+z}{x+t}+\frac{z+t}{x+y}+\frac{x+t}{x+z}=\frac{x+x}{x+x}+\frac{y+y}{y+y}+\frac{z+z}{z+z}+\frac{t+t}{t+t}=4\)