2008 AMC 10B Problems/Problem 5

Problem

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

$\mathrm{(A)}\ 0\qquad\mathrm{(B)}\ x^2+y^2\qquad\mathrm{(C)}\ 2x^2\qquad\mathrm{(D)}\ 2y^2\qquad\mathrm{(E)}\ 4xy$

Solution

Since $(-a)^2 = a^2$ , it follows that $(x-y)^2 = (y-x)^2$ , and $(x-y)^2 * (y-x)^2 = [(x-y)^2 - (y-x)^2]^2 = [(x-y)^2 - (x-y)^2]^2 = 0\ \mathrm{(A)}.$

2008 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 4	Followed by Problem 6
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2008 AMC 10B Problems/Problem 5

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