Ta có: \(\frac{x}{y+z+t}=\frac{y}{z+t+x}=\frac{z}{x+t+y}=\frac{t}{x+y+z}\)

Thêm 1 vào mỗi phân số ta được:

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

\(\Rightarrow\frac{x+y+z+t}{y+z+t}=\frac{x+y+z+t}{z+t+x}=\frac{x+y+z+t}{x+t+y}=\frac{x+y+z+t}{x+y+z}\)

- Nếu x + y + z + t \(\ne\) 0 thì x = y = z = t

\(\Rightarrow P=\frac{x+y}{z+t}+\frac{y+z}{t+x}+\frac{z+t}{x+y}+\frac{t+x}{y+z}=\frac{x+x}{x+x}+\frac{x+x}{x+x}+\frac{x+x}{x+x}+\frac{x+x}{x+x}=1+1+1+1=4\)

- Nếu x + y + z + t = 0 thì x + y = -(z + t)

                                         y + z = -(t + x)

                                         z + t = -(x + y)

                                         t + x = -(y + z)

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

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1, ta co \(\frac{x}{5}=\frac{y}{6}=\frac{x}{20}=\frac{y}{24}\)

\(\frac{y}{8}=\frac{z}{7}=\frac{y}{24}=\frac{z}{21}\)

=>\(\frac{x}{20}=\frac{y}{24}=\frac{z}{21}=\frac{x+y-z}{20+24-21}=\frac{69}{23}=3\)

=>\(x=3\cdot20=60\)

    \(y=3\cdot24=72\)

    \(z=3\cdot21=63\)

3. ta co \(\frac{x}{15}=\frac{y}{7}=\frac{z}{3}=\frac{t}{1}=\frac{x+y-z+t}{15-7+3-1}=\frac{10}{10}=1\)

=> \(x=1\cdot15=15\)

     \(y=1\cdot7=7\)

     \(z=1\cdot3=3\)

     \(t=1\cdot1=1\)

Áp dụng tính chất của dãy tỉ số bằng nhau, ta được:

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

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

TH1: x+y+z+t<>0

=>\(\frac{x}{y+z+t}=\frac{y}{x+z+t}=\frac{z}{x+y+t}=\frac{t}{x+y+z}=\frac13\)

=>y+z+t=3x; x+z+t=3y; x+y+t=3z; x+y+z=3t

=>x=y=z=t

Ta có: \(P = \frac{x+y}{z+t} + \frac{y+z}{t+x} + \frac{z+t}{x+y} + \frac{t+x}{y+z}\)

\(=\frac{x+x}{x+x}+\frac{x+x}{x+x}+\frac{x+x}{x+x}+\frac{x+x}{x+x}\)

=1+1+1+1

=4

TH2: x+y+z+t=0

=>x+y=-(z+t); y+z=-(t+x)

Ta có: \(P = \frac{x+y}{z+t} + \frac{y+z}{t+x} + \frac{z+t}{x+y} + \frac{t+x}{y+z}\)

=-1-1-1-1

=-4