2013 AMC 10B Problem 14 — Solution | LIVE by Po-Shen Loh

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2013 AMC 10B Problem 14

Below is the professionally curated solution for Problem 14 of the 2013 AMC 10B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2013 AMC 10B solutions, or check the answer key.

All of the real AMC 8, AMC 10, AMC 12, and AIME problems in our complete solution collection are used with official legal permission of the Mathematical Association of America (MAA).

Concepts:custom operation factoring

Difficulty rating: 1140

14.

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

A finite set of points

One line

Two parallel lines

Two intersecting lines

Three lines

Solution:

The condition is $x\clubsuit y=y\clubsuit x$ , so $x^2y-xy^2=y^2x-yx^2$ .

Combining like terms gives $2x^2y-2xy^2=2xy(x-y)=0$ .

Thus $x=0$ , $y=0$ , or $x=y$ . These are three lines.

Thus, the correct answer is E .

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