Problem:
Let A,M, and C be nonnegative integers such that A+M+C=12. What is the maximum value of A⋅M⋅C+A⋅M+M⋅C+C⋅A ?
Answer Choices:
A. 62
B. 72
C. 92
D. 102
E. 112
Solution:
Note that
AMC+AM+MC+CA=(A+1)(M+1)(C+1)−(A+M+C)−1=pqr−13,
where p,q, and r are positive integers whose sum is 15 . A case-by-case analysis shows that pqr is largest when p=5,q=5, and r=5. Thus the answer is 5⋅5⋅5−13=112.
The problems on this page are the property of the MAA's American Mathematics Competitions