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MCIV Inequalities 1 Problem 10 (and partially 12)
MCIV Inequalities 1 Problem 10 (and partially 12)
This problem seems like it can be done with something like Muirhead's or simple bashing, but I can't find a way to do it. In fact, I've tried solving number 12 and seeing if I can use it to solve 10 but I can't do either so I'm stuck. Is there a way I am missing? I've avoided using anything messy (like squaring or cubing the already cubic equations).
Thanks in advance!
Re: MCIV Inequalities 1 Problem 10 (and partially 12)
For Problem 10, rearrange the inequality to
$$a^3b^3 + b^3c^3 + c^3a^3 - 3a^2b^2c^2\geq a^3+b^3+c^3-3abc,$$
and factor both sides. The fact that $a\geq 1$, $b\geq1$ and $c\geq1$ is important.
Problem 12 is a typical proof. Remember $x$, $y$ and $z$ are symmetric, and you may assume $x\geq y\geq z$ without loss of generality.
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