2022 AMC 8 Problems/Problem 2

Problem

Consider these two operations: $\begin{align*} a \, \blacklozenge \, b &= a^2 - b^2\\ a \, \bigstar \, b &= (a - b)^2 \end{align*}$ What is the value of $(5 \, \blacklozenge \, 3) \, \bigstar \, 6?$

$\textbf{(A) } {-}20 \qquad \textbf{(B) } 4 \qquad \textbf{(C) } 16 \qquad \textbf{(D) } 100 \qquad \textbf{(E) } 220$

Solution

We get that $(5 \, \blacklozenge \, 3) = 5^2 - 3^2 = 16$ , so $(5 \, \blacklozenge \, 3) \, \bigstar \, 6 = 16 \, \bigstar \, 6 = (16 - 6)^2 = 10^2 = \boxed{\textbf{(D) } 100}.$

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2022 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 1	Followed by Problem 3
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All AJHSME/AMC 8 Problems and Solutions

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2022 AMC 8 Problems/Problem 2

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