Skip to main content

2004 AMC 12B Problem 2

Problem 2 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?

Answer choices

Show solution

Solution

To maximize, set d=0.d = 0. With a,b,ca, b, c taking 1,2,3,1, 2, 3, the term c⋅abc \cdot a^b is largest when c=1c = 1 and ab=32=9.a^b = 3^2 = 9. This gives 1⋅9−0=9,1 \cdot 9 - 0 = 9, which beats 23=82^3 = 8 and the other assignments. Thus, the correct answer is D.

More practice

Concepts: optimization · casework

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