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2008 AMC 10B Problem 5

Below is the professionally curated solution for Problem 5 of the 2008 AMC 10B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2008 AMC 10B solutions, or check the answer key.

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Concepts:custom operationsymmetry (algebra)

Difficulty rating: 880

5.

For real numbers aa and b,b, define a $ b=(a−b)2.a\,\$\,b=(a-b)^2. What is (x−y)2 $ (y−x)2?(x-y)^2\,\$\,(y-x)^2?

00

x2+y2x^2+y^2

2x22x^2

2y22y^2

4xy4xy

Solution:

Since (y−x)2=(x−y)2,(y-x)^2=(x-y)^2, the two inputs are identical.

Therefore (x−y)2 $ (y−x)2=((x−y)2−(x−y)2)2=02=0.(x-y)^2\,\$\,(y-x)^2=\left((x-y)^2-(x-y)^2\right)^2=0^2=0.

Thus, the correct answer is A.

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