Ta có
$\dfrac{a}{2b} . \dfrac{b}{2c} . \dfrac{c}{2d} . \dfrac{d}{2a} = \left( \dfrac{a}{2b} \right)^4$
$\Leftrightarrow \dfrac{1}{2.2.2.2} = \left( \dfrac{a}{2b} \right)^4$
$\Leftrightarrow \dfrac{1}{2} = \dfrac{a}{2b}$
$\Leftrightarrow 1 = \dfrac{a}{b}$
$\Leftrightarrow a = b$
CMTT ta suy ra $a = b = c = d$.
Vậy ta tính
$A = 2013a - 2012 . \dfrac{b}{c+d} + 2013b - 2012 . \dfrac{c}{a+d} + 2013c - 2012.\dfrac{d}{a+b} + 2013d - 2012 . \dfrac{a}{b+c}$
$= 2013 a + 2013b + 2013c + 2013d - 2012 \left( \dfrac{b}{c+d} + \dfrac{c}{a+d} + \dfrac{d}{a+b} + \dfrac{a}{b+c}\right)$
Do $a = b = c = d$ nên
$\dfrac{b}{c+d} = \dfrac{a}{b+c} = \dfrac{c}{a+d} = \dfrac{d}{a+b} = \dfrac{1}{2}$
Thay vào ta có
$A = 2013.4 - 2012.\dfrac{1}{2} . 4$
$= 2013.4 - 1006.4$
$= 4(2013-1006)$
$= 4.1007 = 4028$
Vậy $A = 4028$.
