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2008 AMC 10B Problem 5

Problem 5 of 25EasierAlgebra

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?

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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\,\$\,(y-x)^2 =((x−y)2−(x−y)2)2=\left((x-y)^2-(x-y)^2\right)^2 =02=0.=0^2=0. Thus, the correct answer is A.

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Concepts: custom operation · symmetry (algebra)

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