Question

Cho 2a+b+c+d =a+b+c+2d =a+b+2c+d =a+b+c+2d

a b c d

Tinh M=a+b + b+c + c+d + d+a

c+d d+a a+b b+c

Answer 1

Theo bài ra \(\dfrac{2a+b+c+d}{a}=\dfrac{a+2b+c+d}{b}=\dfrac{a+b+2c+d}{c}=\dfrac{a+b+c+2d}{d}\)

\(\Rightarrow\)

\(\dfrac{2a+b+c+d}{a}-1=\dfrac{a+2b+c+d}{b}-1=\dfrac{a+b+2c+d}{c}-1=\dfrac{a+b+c+2d}{d}-1\)\(\Rightarrow\)\(\dfrac{a+b+c+d}{a}=\dfrac{a+b+c+d}{b}=\dfrac{a+b+c+d}{c}=\dfrac{a+b+c+d}{d}\)+)Nếu a+b+c+d=0

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

\(\Rightarrow\)M=1+1+1+1=4

+)Nếu a+b+c+d=0

\(\Rightarrow\)\(\left\{{}\begin{matrix}a+b=-\left(c+d\right)\\b+c=-\left(a+d\right)\\c+d=-\left(a+b\right)\\d+a=-\left(b+c\right)\end{matrix}\right.\)

\(\Rightarrow\)M=(-1)+(-1)+(-1)+(-1)=-4

Vậy M=4 hoặc M=-4