Skip to main content

2004 AMC 12B Problem 2

Problem 2 of 25EasierProblem-Solving Techniques

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.
AoPS wiki

Tagged: optimization · casework

More practice