Ta có :
\(\frac{2a+b+c+d}{a}=\frac{a+2b+c+d}{b}=\frac{a+b+2c+d}{c}=\frac{a+b+c+2d}{d}\)
\(\Leftrightarrow\)\(\frac{2a+b+c+d}{a}-1=\frac{a+2b+c+d}{b}-1=\frac{a+b+2c+d}{c}-1=\frac{a+b+c+2d}{d}-1\)
\(\Leftrightarrow\)\(\frac{a+b+c+d}{a}=\frac{a+b+c+d}{b}=\frac{a+b+c+d}{c}=\frac{a+b+c+d}{d}\)
Áp dụng tính chất dãy tỉ số bằng nhau ta có :
\(\frac{a+b+c+d}{a}=\frac{a+b+c+d}{b}=\frac{a+b+c+d}{c}=\frac{a+b+c+d}{d}=\frac{4\left(a+b+c+d\right)}{a+b+c+d}=4\)
+) Nếu \(a+b+c+d=0\)
\(\Rightarrow\)\(a+b=-\left(c+d\right)\)
\(\Rightarrow\)\(b+c=-\left(d+a\right)\)
\(\Rightarrow\)\(c+d=-\left(a+b\right)\)
\(\Rightarrow\)\(d+a=-\left(b+c\right)\)
Suy ra :
\(M=\frac{a+b}{c+d}+\frac{b+c}{d+a}+\frac{c+d}{a+b}+\frac{d+a}{b+c}+2017\)
\(M=\frac{-\left(c+d\right)}{d+d}+\frac{-\left(d+a\right)}{d+a}+\frac{-\left(a+b\right)}{a+b}+\frac{-\left(b+c\right)}{b+c}+2017\)
\(M=\left(-1\right)+\left(-1\right)+\left(-1\right)+\left(-1\right)+2017=-4+2017=2013\)
+) Nếu \(a+b+c+d\ne0\)
Do đó :
\(\frac{a+b+c+d}{a}=4\)\(\Rightarrow\)\(a+b+c+d=4a\)\(\left(1\right)\)
\(\frac{a+b+c+d}{b}=4\)\(\Rightarrow\)\(a+b+c+d=4b\)\(\left(2\right)\)
\(\frac{a+b+c+d}{c}=4\)\(\Rightarrow\)\(a+b+c+d=4c\)\(\left(3\right)\)
\(\frac{a+b+c+d}{d}=4\)\(\Rightarrow\)\(a+b+c+d=4d\)\(\left(4\right)\)
Từ (1), (2), (3) và (4) suy ra : \(4a=4b=4c=4d\)
\(\Leftrightarrow\)\(a=b=c=d\)
Suy ra :
\(M=\frac{a+b}{c+d}+\frac{b+c}{d+a}+\frac{c+d}{a+b}+\frac{d+a}{b+c}+2017\)
\(M=\frac{a+a}{a+a}+\frac{b+b}{b+b}+\frac{c+c}{c+c}+\frac{d+d}{d+d}+2017\)
\(M=1+1+1+1+2017=4+2017=2021\)
Vậy \(M=2013\) hoặc \(M=2021\)
Chúc bạn học tốt ~
