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Problem Of The Week #413 Apr 19th, 2020

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anemone

MHB POTW Director
Staff member
Feb 14, 2012
3,516
Here is this week's POTW:

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Find all positive real solutions to the following system of solution:

$x^3+y^3+z^3=x+y+z$

$x^2+y^2+z^2=xyz$

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Remember to read the POTW submission guidelines to find out how to submit your answers!
 
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anemone

MHB POTW Director
Staff member
Feb 14, 2012
3,516
I apologize as I just realized the reply to this POTW #413 has gone amiss.

I just checked and no one answered to this POTW. (Sadface) Nevertheless, you can refer to the suggested solution by other below:

We have $xyz=x^2+y^2+z^2>Y^2+z^2\ge 2yz$. Hence $x>2$ and $x^3>x$. Similarly, $y^3>y$ and $z^3>z$. Adding them up gives $x^3+y^3+z^3>x+y+z$ and this contradicts to what is given and hence, there are no solutions to the system.
 
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