2022 AMC 8 Problem 2

Attempt Problem 2 of the 2022 AMC 8 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2022 AMC 8 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

2.

Consider these two operations: a ⧫ b=a2−b2a⋆b=(a−b)2\begin{align*} a\,\blacklozenge\,b &= a^2 - b^2 \\ a \star b &= (a - b)^2 \end{align*}
Compute the value:
(5 ⧫ 3)⋆6 (5\,\blacklozenge\,3) \star 6
−20-20
44
1616
100100
220220
Answer: D
Concepts:custom operationorder of operations
Small hint:
Evaluate 5 ⧫ 35\,\blacklozenge\,3 first
Big hint:
Substitute the value from the first operation into ⋆6\star 6
Solution:
Using the definitions, (5 ⧫ 3)⋆6=(52−32)⋆6=16⋆6=(16−6)2=100\begin{align*} (5\,\blacklozenge\,3) \star 6 &= (5^2-3^2) \star 6 \\ &=16 \star 6 \\ &=(16-6)^2\\ &= 100\end{align*}
Thus, the correct answer is D.
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