Ta có:
$\dfrac{ab}{a+b}=\dfrac{bc}{b+c}=\dfrac{ca}{a+c}$
$\Leftrightarrow \dfrac{a+b}{ab}=\dfrac{b+c}{bc}=\dfrac{a+c}{ac}$
$\Leftrightarrow \dfrac{1}{a}+\dfrac{1}{b}=\dfrac{1}{b}+\dfrac{1}{c}=\dfrac{1}{a}+\dfrac{1}{c}$
$\Leftrightarrow \dfrac{1}{a}=\dfrac{1}{b}=\dfrac{1}{c}$
$\Leftrightarrow a=b=c$
Suy ra: $A=\dfrac{2039}{3}$