MHB Can You Solve These Unique System of Equations?

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The discussion presents a problem of the week (POTW) involving a unique system of equations: x^3 + y^3 + z^3 = x + y + z and x^2 + y^2 + z^2 = xyz. Participants are encouraged to find all positive real solutions to these equations. The thread notes that no responses have been submitted yet, indicating a lack of engagement with the problem. A reminder is provided for users to refer to the guidelines for participation. The absence of solutions highlights the challenge posed by the equations.
anemone
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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 https://mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to https://mathhelpboards.com/forms.php?do=form&fid=2!
 
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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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