I first started simplifying the expression:$$(a-b)^3+(b-c)^3+(c-a)^3=3(a-b)(b-c)(c-a)$$I went one more step further and wrote the expression $(a-b)(b-c)(c-a)$ as:$$ab^2-a^2b+b^2c-bc^2+c^2a-ac^2$$Now, what can I else do? I know that $(a,b,c)=(1,2,3)$ is a solution to this, but I do not want to use this values directly. I need to use the values of the three expression to evaluate this expression of the question. Is it really possible? I tried some of more steps I could:$$ab^2-a^2b+b^2c-bc^2+c^2a-ac^2$$$$=a^2b+ab^2+3abc+b^2c+bc^2+ac^2+a^2c-3abc-2(a^2b+bc^2+ac^2)$$$$=ab(a+b+c)+bc(a+b+c)+ca(a+b+c)$$But I am helpless again. Need some guidelines. Is it really solvable?
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