Y by Sedro
Given distinct real numbers
satisfying the equation:
![\[
(x^2y - y^2z)(1 - zx) = (xy^2 - zx^2)(1 - yz).
\]](//latex.artofproblemsolving.com/d/1/c/d1c11cb351b59ae5e79edc1e1997834b5402638a.png)
Prove that:
![\[
x + y + z = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}.
\]](//latex.artofproblemsolving.com/c/1/2/c122da8411c836a38fb2ea639338c3fd9749a619.png)

![\[
(x^2y - y^2z)(1 - zx) = (xy^2 - zx^2)(1 - yz).
\]](http://latex.artofproblemsolving.com/d/1/c/d1c11cb351b59ae5e79edc1e1997834b5402638a.png)
Prove that:
![\[
x + y + z = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}.
\]](http://latex.artofproblemsolving.com/c/1/2/c122da8411c836a38fb2ea639338c3fd9749a619.png)
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