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iceghost

Đặt $a-b=x \\ b-c=y \\ c-a=z$
Ta có : $x+y+z=(a-b)+(b-c)+(c-a)=0$
$\iff x+y=-z \\
\iff (x+y)^3=-z^3 \\
\iff x^3+3x^2y+3xy^2+y^3+z^3=0 \\
\iff x^3+y^3+z^3+3xy(x+y)=0 \\
\iff x^3+y^3+z^3+3xy(-z) =0 \\
\iff x^3+y^3+z^3=3xyz \\
\iff (a-b)^3+(b-c)^3+(c-a)^3=3(a-b)(b-c)(c-a)=3k \\
\implies \mathrm{dpcm}$
 
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