\[\begin{array}{l}
M = \frac{a}{{a + b}} + \frac{b}{{b + c}} + \frac{c}{{a + c}}\,\,\,\,\,\,(a;b;c \in {Z^ + })\\
Ta\,\,co:\\
\frac{a}{{a + b}} > \frac{a}{{a + b + c}};\\
\frac{b}{{b + c}} > \frac{b}{{a + b + c}};\\
\frac{c}{{a + c}} > \frac{c}{{a + b + c}}\\
= > M = \frac{a}{{a + b}} + \frac{b}{{b + c}} + \frac{c}{{a + c}}\, > \frac{a}{{a + b + c}} + \frac{b}{{a + b + c}} + \frac{c}{{a + b + c}} = 1\,\,(1)\\
Lai\,\,co:\\
a < a + b = > \frac{a}{{a + b}} < \frac{{a + c}}{{a + b + c}};\,\\
b < b + c = > \frac{b}{{b + c}} < \frac{{b + a}}{{a + b + c}};\,\\
c < a + c = > \frac{c}{{a + c}} < \frac{{c + b}}{{a + b + c}}\\
= > M = \frac{a}{{a + b}} + \frac{b}{{b + c}} + \frac{c}{{a + c}} < \frac{{a + c}}{{a + b + c}} + \frac{{b + a}}{{a + b + c}} + \frac{{c + b}}{{a + b + c}} = 2\,\,(2)\\
Tu\,\,(1)\,\,va\,\,(2)\,\,suy\,\,ra\,\,1 < M < 2\\
= > M \notin Z
\end{array}\]
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