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2013 AMC 10B Problem 14

Problem 14 of 25IntermediateAlgebra

Define a♣b=a2b−ab2. a\clubsuit b=a^2b-ab^2 . Which of the following describes the set of points (x,y) (x, y) for which x♣y=y♣x? x\clubsuit y=y\clubsuit x ?

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Solution

The condition is x♣y=y♣xx\clubsuit y=y\clubsuit x, so x2y−xy2=y2x−yx2x^2y-xy^2=y^2x-yx^2. Combining like terms gives 2x2y−2xy2=2xy(x−y)=02x^2y-2xy^2=2xy(x-y)=0. Thus x=0x=0, y=0y=0, or x=yx=y. These are three lines. Thus, the correct answer is E .

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Concepts: custom operation · factoring

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