2013 AMC 12A Problem 8

Problem 8 of 25EasierAlgebra

Given that

x

and

y

are distinct nonzero real numbers such that

x + \dfrac{2}{x} = y + \dfrac{2}{y},

what is

xy?

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Solution

Multiplying by

xy

gives

x^2 y + 2y = xy^2 + 2x,

\begin{gathered} x^2 y - xy^2 - 2x + 2y \\ = (x - y)(xy - 2) \\ = 0. \end{gathered}

Since

x \ne y,

it follows that

xy = 2.

Thus, the correct answer is D.

Concepts: algebraic manipulation · factoring

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.