\(a^3+b^3+c^3=3abc\)
<=> \(a^3+b^3+c^3-3abc=0\)
<=> \(\left(a+b+c\right)\left(a^2+b^2+c^2-ab-bc-ca\right)=0\)
<=> \(\orbr{\begin{cases}a+b+c=0\\a^2+b^2+c^2-ab-bc-ca=0\end{cases}}\)
Xét: \(a^2+b^2+c^2-ab-bc-ca=0\)
<=> \(2a^{ 2}+2b^2+2c^2-2ab-2bc-2ca=0\)
<=> \(\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2=0\)
<=> \(\hept{\begin{cases}a-b=0\\b-c=0\\c-a=0\end{cases}}\) <=> \(\hept{\begin{cases}a=b\\b=c\\c=a\end{cases}}\)<=> \(a=b=c\)
=> đpcm