Problem 1-8 Meticulous Manipulations

Problem Posts

[Difficulty 2] [PDF]

QUESTION
(2012 PMO Area Stage) If $x+y+xy=1$, where $x$, $y$ are nonzero real numbers, find the value of $$xy+\frac{1}{xy}-\frac{y}{x}-\frac{x}{y}$$

Laconic Solution Sketch
Manipulate.



Discussion
The name of the game is $xy$, probably the nastiest thing over here. Expand the desired expression: $$xy+\frac{1}{xy}-\frac{y}{x}-\frac{x}{y} = \frac{x^2y^2+1-y^2-x^2}{xy}$$ Ugly, isn't it? But $x^2+y^2$ gives us a tip-off to square both sides of the given equation:
$$\begin{eqnarray*} x+y+xy & = & 1\\ x+y & = & 1-xy\\ x^{2}+y^{2}+2xy & = & 1-2xy+x^{2}y^{2}\\ 4xy & = & 1+x^{2}y^{2}-y^{2}-x^{2}\\ 4 & = & \frac{1+x^{2}y^{2}-y^{2}-x^{2}}{xy} \end{eqnarray*}$$ and we are finished.

Solution
The above equation array suffices.

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Problem 1-5 also features a nasty manipulation.