Đặt x/y=y/z=z/t=t/x=k
=>x=yk; y=zk; z=tk; t=xk
=>x=yk; y=zk; z=xk^2; t=xk
=>x=yk; y=xk^3; z=xk^2' t=xk
=>x=xk^4; y=xk^3; z=xk^2; t=xk
=>xk^4-x=0
=>k=1 hoặc k=-1
\(M=\dfrac{2xk^4-xk^3}{xk^2+xk}+\dfrac{2xk^3-xk^2}{xk+xk^4}+\dfrac{2xk^2-xk}{xk^4+xk^3}+\dfrac{2xk-xk^4}{xk^3+xk^2}\)
\(=\dfrac{xk^3\left(2k-1\right)}{xk\left(k+1\right)}+\dfrac{xk^2\left(2k-1\right)}{xk\left(k+1\right)\left(k^2-k+1\right)}+\dfrac{xk\left(2k-1\right)}{xk^3\left(k+1\right)}+\dfrac{xk\left(2-k^3\right)}{xk^2\left(k+1\right)}\)
\(=k^2\left(2k-1\right)+\dfrac{k\left(2k-1\right)}{\left(k+1\right)\left(k^2-k+1\right)}+\dfrac{1}{k^2}\cdot\dfrac{2k-1}{k+1}+\dfrac{2-k^3}{k^2+k}\)
k=-1 thì loại
k=1 thì \(M=1^2\left(2\cdot1-1\right)+\dfrac{1\left(2\cdot1-1\right)}{\left(1^3+1\right)}+\dfrac{1}{1^2}\cdot\dfrac{1}{2}+\dfrac{2-1}{1+1}\)
\(=1+\dfrac{1}{2}+\dfrac{1}{2}+\dfrac{1}{2}=\dfrac{5}{2}\)
