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I need to prove several inequalities trivially. (i.e. without using AM-GM, re-arrangement etc). I just keep hitting a blank. Could anyone help?

$$x^{4}+y^{4}+z^{4}\geq x^{2}yz+xy^{2}z+xyz^{2}$$

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    $(x,y,z)=(1,1,1)$2012-04-08
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    @pedja: There are other solutions, e.g., $\langle x,y,z\rangle=\langle -1,1,1\rangle$.2012-04-08
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    Sorry I forgot to add that I have to prove this inequality holds true for all positive real numbers x,y,z.2012-04-08
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    Using Muirhead's inequality these can be easily proven.2012-12-21

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