2004 AMC 12B Problem 2

Attempt Problem 2 of the 2004 AMC 12B 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 2004 AMC 12B solutions, or check the answer key.

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

In the expression $c \cdot a^b - d,$ the values of $a, b, c,$ and $d$ are $0, 1, 2,$ and $3,$ although not necessarily in that order. What is the maximum possible value of the result?

$5$

$6$

$8$

$9$

$10$

Answer: D

Concepts:optimization casework

Difficulty rating: 1000

Solution:

To maximize, set $d = 0.$ With $a, b, c$ taking $1, 2, 3,$ the term $c \cdot a^b$ is largest when $c = 1$ and $a^b = 3^2 = 9.$ This gives $1 \cdot 9 - 0 = 9,$ which beats $2^3 = 8$ and the other assignments.

Thus, the correct answer is D.

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