Giải thích các bước giải:
Ta có:
\[\begin{array}{l}
\left\{ \begin{array}{l}
x = by + cz\\
y = ax + cz\\
z = ax + by
\end{array} \right. \Rightarrow x + y + z = by + cz + ax + cz + ax + by\\
\Rightarrow x + y + z = 2\left( {ax + by + cz} \right)\\
\Rightarrow x\left( {1 - 2a} \right) + y\left( {1 - 2b} \right) + z\left( {1 - 2c} \right) = 0,\,\,\,\,\,\forall x,y,z\\
\Rightarrow \left\{ \begin{array}{l}
1 - 2a = 0\\
1 - 2b = 0\\
1 - 2c = 0
\end{array} \right. \Leftrightarrow a = b = c = \frac{1}{2}\\
\Rightarrow \frac{1}{{1 + a}} + \frac{1}{{1 + b}} + \frac{1}{{1 + c}} = \frac{1}{{1 + \frac{1}{2}}} + \frac{1}{{1 + \frac{1}{2}}} + \frac{1}{{1 + \frac{1}{2}}} = 3.\frac{1}{{\frac{3}{2}}} = 3.\frac{2}{3} = 2
\end{array}\]
