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MCIV Inequalities 1 Problem 10 (and partially 12)

 
 
YuDL的头像
MCIV Inequalities 1 Problem 10 (and partially 12)
YuDL - 2019年07月5日 Friday 08:41
 

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!

 
ProfessorAreteem的头像
Re: MCIV Inequalities 1 Problem 10 (and partially 12)
ProfessorAreteem - 2019年07月7日 Sunday 18:08
 

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.