<s>[INST] Prove or disprove the inequality if $a,b,c>0$, $a \geq b+c$. Prove or disprove the inequality
$$a^2b+a^2c+b^2a+b^2c+c^2a+c^2b \geq 7abc$$ if $$a,b,c>0, a \geq b+c.$$
I thought to use this evaluation:
$$a^2b+b^2c+c^2a \geq 3abc.$$
So we have:
$$a^2b+a^2c+b^2a+b^2c+c^2a+c^2b \geq 3abc+3abc=6abc,$$ which is obvious that $$6abc<7abc.$$
Is it right? I'm embarrassed that in my solution I did not have to use the condition that $$a \geq b+c.$$
Any hint would help a lot. thanks!
 [/INST] Since $$b+c\geq\frac{7bc-b^2-c^2}{2(b+c)},$$ we obtain:
$$\sum_{cyc}(a^2b+a^2c)-7abc=(b+c)a^2+(b^2+c^2-7bc)a+bc(b+c)\geq$$
$$\geq(b+c)^3+(b^2+c^2-7bc)(b+c)+bc(b+c)=2(b+c)(b-c)^2\geq0.$$
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