Bài 5:
TH1: a+b+c+d≠0
\(\dfrac{2019a+b+c+d}{a}=\dfrac{a+2019b+c+d}{b}=\dfrac{a+b+2019c+d}{c}=\dfrac{a+b+c+2019d}{d}\)
\(\Rightarrow\dfrac{2019a+b+c+d}{a}-2018=\dfrac{a+2019b+c+d}{b}-2018=\dfrac{a+b+2019c+d}{c}-2018=\dfrac{a+b+c+2019d}{d}-2018\)
\(\Rightarrow\dfrac{a+b+c+d}{a}=\dfrac{a+b+c+d}{b}=\dfrac{a+b+c+d}{c}=\dfrac{a+b+c+d}{d}\)
\(\Rightarrow a=b=c=d\)
\(M=\dfrac{a+b}{c+d}+\dfrac{b+c}{d+a}+\dfrac{c+d}{a+b}+\dfrac{d+a}{b+c}\)
\(\Rightarrow M=1+1+1+1\)
\(\Rightarrow M=4\)
TH1: a+b+c+d=0
\(\Rightarrow\left\{{}\begin{matrix}a+b=-\left(c+d\right)\\a+c=-\left(b+d\right)\\b+c=-\left(a+d\right)\\d+a=-\left(b+c\right)\end{matrix}\right.\)
\(M=\dfrac{a+b}{c+d}+\dfrac{b+c}{d+a}+\dfrac{c+d}{a+b}+\dfrac{d+a}{b+c}\)
\(\Rightarrow M=-1-1-1-1\)
\(\Rightarrow M=-4\)
