Lời giải

Ta có: a² – 2b = c² – 2a

⇔ a² – c² = 2b – 2a

⇔ (a – c)(a + c) = 2(b – a)

\( \Leftrightarrow a + c = \frac{{2\left( {b - a} \right)}}{{a - c}}\)

\( \Leftrightarrow a + c + 2 = \frac{{2\left( {b - a} \right)}}{{a - c}} + 2\)

\( \Leftrightarrow a + c + 2 = \frac{{2\left( {b - a} \right)}}{{a - c}} + 2\)

\[ \Leftrightarrow a + c + 2 = \frac{{2\left( {b - a} \right) + 2\left( {a - c} \right)}}{{a - c}}\]

\[ \Leftrightarrow a + c + 2 = \frac{{2b - 2a + 2a - 2c}}{{a - c}}\]

\[ \Leftrightarrow a + c + 2 = \frac{{2\left( {b - c} \right)}}{{a - c}}\]

Chứng minh tương tự ta có:

\(a + b + 2 = \frac{{2\left( {a - c} \right)}}{{a - b}}\) và \(b + c + 2 = \frac{{2\left( {b - a} \right)}}{{b - c}}\)

Suy ra A = (a + b + 2)(b + c + 2)(c + a + 2)

               \[ = \frac{{2\left( {a - c} \right)}}{{a - b}}.\frac{{2\left( {b - a} \right)}}{{b - c}}.\frac{{2\left( {b - c} \right)}}{{a - c}} = - 8\]

Vậy A = – 8.