Giải thích các bước giải:
Áp dụng tính chất dãy tỉ số bằng nhau ta có:
\[\begin{array}{l}
\frac{x}{{2x + y + z}} = \frac{y}{{x + 2y + z}} = \frac{z}{{x + y + 2z}} = \frac{{x + y + z}}{{\left( {2x + y + z} \right) + \left( {x + 2y + z} \right) + \left( {x + y + 2z} \right)}} = \frac{{x + y + z}}{{4\left( {x + y + z} \right)}} = \frac{1}{4}\\
\Rightarrow \left\{ \begin{array}{l}
\frac{x}{{2x + y + z}} = \frac{1}{4}\\
\frac{y}{{x + 2y + z}} = \frac{1}{4}\\
\frac{z}{{x + y + 2z}} = \frac{1}{4}
\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}
2x = y + z\\
2y = z + x\\
2z = x + y
\end{array} \right.\\
\Rightarrow P = \frac{{x + y}}{{3z}} + \frac{{y + z}}{{4x}} + \frac{{z + x}}{{6y}} = \frac{{2z}}{{3z}} + \frac{{2x}}{{4x}} + \frac{{2y}}{{6y}} = \frac{2}{3} + \frac{1}{2} + \frac{1}{3} = \frac{3}{2}
\end{array}\]