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2004 AMC 10B Problem 5

Problem 5 of 25EasierAlgebra

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

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Solution

Setting d=0d = 0 removes the subtraction, so we maximize c⋅abc \cdot a^b using 1,1, 2,2, 3.3. Taking c=1,c = 1, a=3,a = 3, b=2b = 2 gives 32=9.3^2 = 9. The alternative 23=82^3 = 8 is smaller, and any assignment with c>1c \gt 1 forces a smaller power. The maximum is 9.9. Thus, the correct answer is D.

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Concepts: exponent · optimization · casework

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.