Question

Cho dãy tỉ số bằng nhau: \(\frac{2a+b+c+d}{a}=\frac{a+2b+c+d}{b}=\frac{a+b+2c+d}{c}=\frac{a+b+c+2d}{d}\)

Tính giá trji của biểu thức \(M=\frac{a+b}{c+d}+\frac{b+c}{d+a}+\frac{c+d}{a+b}+\frac{d+a}{b+c}\)

Answer 1

Ta co : \(\frac{2a+b+c+d}{a-1}=\frac{a+2b+c+d}{b-1}=\frac{a+b+2c+d}{c-1}=\frac{a+b+c+2d}{d-1}\)

\(\Rightarrow\frac{a+b+c+d}{a}=\frac{a+b+c+d}{b}=\frac{a+b+c+d}{c}=\frac{a+b+c+d}{d}\)

Xet 2 truong hop 

TH1: 

\(a+b+c+d=0\)\(\Rightarrow a+b=-\left(c+d\right);b+c=-\left(a+d\right);c+d=-\left(a+d\right)\)

Khi do \(M=\left(-1\right).4=-4\)

TH2:

\(a+b+c+d\ne0\)

\(\Rightarrow a=b=c=d\)

Khi do \(M=1.4=4\)

Vay : M=4 hoac M=-4

**** nhe