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How can I prove that if $a,b,c,d>0$ then

$$a^3+b^3+c^3+d^3\geq abc+abd+acd+bcd?$$

I think there is some simple proof but I can't remember... is this a special case of some general inequality?

I need it to prove that if $a^3+2c=3ab$ then all of 4 positive roots of following polynomial is the same.

$x^4+ax^3+bx^2+cx+d$

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  • $\begingroup$ Mathematical formulae look better in $\LaTeX$. Here is a quick tutorial. $\endgroup$
    – Τίμων
    Apr 11, 2016 at 21:15
  • $\begingroup$ Are you sure the LHS is correct? $\endgroup$
    – peter.petrov
    Apr 11, 2016 at 21:17
  • $\begingroup$ I think LHS should be $a^3+b^3+c^3+d^3$... $\endgroup$
    – user327929
    Apr 11, 2016 at 21:19
  • $\begingroup$ If what's up there is correct, then the LHS simplifies down to $3a^3+b^3$, which means I could easily choose $c,d$ so that this inequality is false. $\endgroup$
    – Simply Beautiful Art
    Apr 11, 2016 at 21:21
  • $\begingroup$ I assumed that BBB was correct about the LHS and edited accordingly; if that is wrong, you can fix it. See meta.math.stackexchange.com/questions/5020/… for explanations of the markup I used. $\endgroup$
    – David K
    Apr 11, 2016 at 21:26

1 Answer 1

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hint: Use AM-GM inequality $4$ times:

$$x^3+y^3+z^3 \geq 3xyz$$

with $(x,y,z) = (a,b,c), (a,c,d), (a,b,d), (b,c,d)$, and add up.

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  • $\begingroup$ OH MY MY... TNX $\endgroup$
    – MR_BD
    Apr 11, 2016 at 21:28

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