<s>[INST] Find the value of expression $^2+^2+^2$ given $++=1$ and $\frac 1x +\frac 1y +\frac 1z=0$ Find the value of expression $$^2+^2+^2$$ given $$++=1$$ and $$\frac 1x +\frac 1y +\frac 1z=0$$
I am sorry I don't really know how to tag this question. 
 [/INST] Assume that $x,y,z$ are the roots of a monic, cubic polynomial $p(t)$
$$ p(t) = (t-x)(t-y)(t-z) = t^3-e_1 t^2+e_2 t-e_3. $$
Since $x+y+z=1$ we have $e_1=1$. Similarly, from $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=0$ we get $e_2=0$.
$$x^2+y^2+z^2 = (x+y+z)^2-2(xy+xz+yz) = e_1^2-e_2$$
then leads to $x^2+y^2+z^2=\color{blue}{1}$.
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