Áp dụng tính chất dãy tỉ số bằng nhau ta có
$\begin{align}
& \frac{2012a+b+c+d}{a}=\frac{2012b+a+c+d}{b}=\frac{2012c+a+b+d}{c}=\frac{2012d+a+b+c}{d} \\
& =\frac{2015\left( a+b+c+d \right)}{a+b+c+d}=2015 \\
\end{align}$
Suy ra $\left\{ \begin{align}
& 2012a+b+c+d=2015a \\
& 2012b+a+c+d=2015b \\
& 2012c+a+b+d=2015c \\
& 2012d+a+b+c=2015d \\
\end{align} \right.$$\Rightarrow \left\{ \begin{align}
& a+b+c+d=4a \\
& a+b+c+d=4b \\
& a+b+c+d=4c \\
& a+b+c+d=4d \\
\end{align} \right.$$\Rightarrow a=b=c=d$
Khi đó
$\begin{align}
& M=\frac{a+b}{c+d}+\frac{b+c}{d+a}+\frac{c+d}{a+b}+\frac{d+a}{b+c} \\
& =\frac{a+a}{a+a}+\frac{b+b}{b+b}+\frac{c+c}{c+c}+\frac{d+d}{d+d} \\
& =1+1+1+1=4 \\
\end{align}$