2004 AMC 10B Problem 5

Below is the professionally curated solution for Problem 5 of the 2004 AMC 10B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2004 AMC 10B solutions, or check the answer key.

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Concepts:exponent optimization casework

Difficulty rating: 1120

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?

$5$

$6$

$8$

$9$

$10$

Solution:

Setting $d = 0$ removes the subtraction, so we maximize $c \cdot a^b$ using $1, 2, 3.$

Taking $c = 1,$ $a = 3,$ $b = 2$ gives $3^2 = 9.$ The alternative $2^3 = 8$ is smaller, and any assignment with $c \gt 1$ forces a smaller power. The maximum is $9.$

Thus, the correct answer is D.

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