Problem:
Let A,M, and C be nonnegative integers such that A+M+C=10. What is the maximum value of Aâ‹…Mâ‹…C+Aâ‹…M+Mâ‹…C+Câ‹…A?
Answer Choices:
A. 49
B. 59
C. 69
D. 79
E. 89
Solution:
Note that
AMC+AM+MC+CA=(A+1)(M+1)(C+1)−(A+M+C)−1=pqr−11,
where p,q, and r are positive integers whose sum is 13. A case-by-case analysis shows that pqt is largest when two of the numbers p,q,r are 4 and the third is 5. Thus the answer is 4⋅4⋅5−11=69.
Answer: C​.
The problems on this page are the property of the MAA's American Mathematics Competitions