Ta có [tex](a-b)^{2}+(b-c)^{2}+(c-a)^{2}=6abc[/tex]
=[tex]2(a^{2}+b^{2}+c^{2}-ab-bc-ca)=6abc[/tex]
=>[tex]a^{2}+b^{2}+c^{2}-ab-bc-ca=3abc[/tex]
Nhân 2 vế với a+b+c,ta có
[tex](a+b+c)(a^{2}+b^{2}+c^{2}-ab-bc-ca)=3abc(a+b+c)[/tex]
=>[tex]a^{3}+b^{3}+c^{3}-3abc=3abc(a+b+c)[/tex]
=>[tex]a^{3}+b^{3}+c^{3}=3abc(a+b+c)+3abc=>a^{3}+b^{3}+c^{3}=3abc(a+b+c+1)[/tex]