Vì $abc=2019$ nên ta có :
$M = \dfrac{2019a}{2019+2019a+ab} + \dfrac{b}{2019+b+bc}+\dfrac{c}{1+c+ca} $
$ = \dfrac{a^2bc}{abc+a^2bc+ab}+\dfrac{b}{abc+b+bc}+\dfrac{c}{1+c+ca} $
$=\dfrac{a^2bc}{ab.(c+ac+1)}+\dfrac{b}{b.(ac+1+c)}+\dfrac{c}{1+c+ca} $
$=\dfrac{ac}{c+ca+1}+\dfrac{1}{c+ac+1}+\dfrac{c}{c+ca+1} $
$=1$