Question

thách ai làm được bài này nè:

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

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

Answer 1

Ta có:

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

\(=\frac{2\left(x+y+z+t\right)}{x+y+z+t}\left(tcdtsbn\right)\)=2

\(\Rightarrow y+z+t=2x;z+t+x=2y;\)

\(t+x+y=2z;x+y+z=2t\)

Tu do de CM x=y=z=t

Khi do 

\(A=1+1+1+1=4\)

Answer 2

Xet \(x+y+z+t=0\)

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

Xet \(x+y+z+t\ne0\)

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

\(\Rightarrow x=y=z=t\ne0\)

\(\Rightarrow A=4\)