Đáp án:
\[A = - 3991\]
Giải thích các bước giải:
Áp dụng tính chất dãy tỉ số bằng nhau ta có:
\(\begin{array}{l}
\dfrac{{a + b - 7}}{{4c}} = \dfrac{{b + c + 3}}{{4a}} = \dfrac{{a + c + 4}}{{4b}} = \\
= \dfrac{{\left( {a + b - 7} \right) + \left( {b + c + 3} \right) + \left( {a + c + 4} \right)}}{{4c + 4a + 4b}}\\
= \dfrac{{2.\left( {a + b + c} \right)}}{{4.\left( {a + b + c} \right)}} = \dfrac{1}{2}\\
\Rightarrow \dfrac{{a + b + c}}{2} = \dfrac{{a + b - 7}}{{4c}} = \dfrac{{b + c + 3}}{{4a}} = \dfrac{{a + c + 4}}{{4b}} = \dfrac{1}{2}\\
\Leftrightarrow \left\{ \begin{array}{l}
\dfrac{{a + b + c}}{2} = \dfrac{1}{2}\\
\dfrac{{a + b - 7}}{{4c}} = \dfrac{1}{2}\\
\dfrac{{b + c + 3}}{{4a}} = \dfrac{1}{2}\\
\dfrac{{a + c + 4}}{{4b}} = \dfrac{1}{2}
\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}
a + b + c = 1\\
a + b - 7 = 2c\\
b + c + 3 = 2a\\
a + c + 4 = 2b
\end{array} \right.\\
a + b + c = 1 \Leftrightarrow \left\{ \begin{array}{l}
b + c = 1 - a\\
c + a = 1 - b\\
a + b = 1 - c
\end{array} \right.\\
\left\{ \begin{array}{l}
a + b - 7 = 2c\\
b + c + 3 = 2a\\
a + c + 4 = 2b
\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}
1 - c - 7 = 2c\\
1 - a + 3 = 2a\\
1 - b + 4 = 2b
\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}
3c = - 6\\
3a = 4\\
3b = 5
\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}
c = - 2\\
a = \dfrac{4}{3}\\
b = \dfrac{5}{3}
\end{array} \right.
\end{array}\)
Vậy \(A = 20a + 11b + 2018c = 20.\dfrac{4}{3} + 11.\dfrac{5}{3} + 2018.\left( { - 2} \right) = - 3991\)
