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What values can the expression $(x-y)(y-z)(z-x)$ take if it is known that $\sqrt{x-y+z}=\sqrt{x}-\sqrt{y}+\sqrt{z}$?
Let $\sqrt{x}=a, \sqrt{y}=b, \sqrt{z}=c, \sqrt{x-y+z}=d$, we get the system $a-b+c=d$ $a^{2}-b^{2}+c^{2}=d^{2}$, or

$a-b=d-c$

$a^{2}-b^{2}=d^{2}-c^{2}$.

If $a=b$, then $x=y$. Otherwise, dividing the second equation by the first, we get $a+b=d+c$.

Subtracting the first equation from this, we get $b=c$, that is, $y=z$.

In any case, $(x-y)(y-z)(z-x)=0$.

Answer

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