[tex]\it x = \dfrac{\overline{a,b(c)} + \overline{b,c(a)} + \overline{a,b(c)}}{a+b+c} \Rightarrow 10x = \dfrac{\overline{ab,(c)} + \overline{bc,(a)} + \overline{ab,(c)}} {a+b+c} =
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= \dfrac{\overline{ab} +0,(c) +\overline{bc} +0,(a) +\overline{ca} +0,(b)}{a+b+c}=\\ \\ \\ = \dfrac{10a+b +\dfrac{c}{9} +10b+c +\dfrac{a}{9} +10c+a +\dfrac{b}{9}}{a+b+c}=[/tex]
[tex]\it =\dfrac{ 11a+11b+11c+\dfrac{a+b+c}{9}}{a+b+c} = \dfrac{ 11(a+b+c)+\dfrac{a+b+c}{9}}{a+b+c}=\\ \\ \\=\dfrac{(a+b+c)(11+\dfrac{1}{9})}{a+b+c} = \dfrac{100}{9}
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10x= \dfrac{100}{9} \Rightarrow x = \dfrac{10}{9}
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9x= 9\cdot\dfrac{10}{9} =10 \in \mathbb{N}[/tex]
