<s>[INST] Prove $a^3+b^3+c^3\geq a^2b+b^2c+c^2a$ 
if $a,b,c$ are positive real numbers,Prove:$$a^3+b^3+c^3\geq a^2b+b^2c+c^2a$$

Things I have done so far: I know the fact that $$a^3+b^3+c^3\geq\frac{1}{2}[ab(a+b)+bc(b+c)+ca(c+a)]$$
However i tried to reach the problem inequality,but I was not succesful.
Source of problem: school exam.
 [/INST] By aM-GM:
$$\frac{2a^3+b^3}3\ge(a^6b^3)^{\frac13}=a^2b$$
$$\frac{2b^3+c^3}3\ge(b^6c^3)^{\frac13}=b^2c$$
$$\frac{2c^3+a^3}3\ge(c^6a^3)^{\frac13}=c^2a$$
Add them.
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