Ta có: \(\frac{2012a+b+c+d}{a}-2011=\frac{a+2012b+c+d}{b}-2011=\frac{a+b+2012c+d}{c}-2011\)
\(=\frac{a+b+c+2012d}{d}-2011\)
\(\Rightarrow\frac{a+b+c+d}{a}=\frac{a+b+c+d}{b}=\frac{a+b+c+d}{c}=\frac{a+b+c+d}{d}\)
+) Xét \(a+b+c+d=0\)
\(\Rightarrow a+b=-\left(c+d\right);b+c=-\left(a+d\right);c+d=-\left(a+b\right);a+d=-\left(b+c\right)\)
\(M=\frac{a+b}{c+d}+\frac{b+c}{d+a}+\frac{c+d}{a+b}+\frac{d+a}{b+c}\)
\(=\frac{a+b}{-\left(a+b\right)}+\frac{b+c}{-\left(b+c\right)}+\frac{c+d}{-\left(c+d\right)}+\frac{d+a}{-\left(d+a\right)}\)
\(=\left(-1\right)+\left(-1\right)+\left(-1\right)+\left(-1\right)=-4\)
+) Xét \(a+b+c+d\) khác 0 \(\Rightarrow a=b=c=d\)
\(M=\frac{a+b}{c+d}+\frac{b+c}{d+a}+\frac{c+d}{a+b}+\frac{d+a}{b+c}=1+1+1+1=4\)
Vậy...
