Question

Let $A,$ $M,$ and $C$ be nonnegative integers such that $A + M + C = 12.$ What is the maximum value of $\begin{aligned} &A \cdot M \cdot C + A \cdot M \\ &\quad {}+ M \cdot C + C \cdot A? \end{aligned}$

$62$

$72$

$92$

$102$

$112$

Answer: E

Concepts:factoring AM-GM Inequality optimization

Difficulty rating: 1650

Solution:

Observe that $\begin{aligned} &AMC + AM + MC + CA \\ &\quad = (A + 1)(M + 1)(C + 1) \\ &\quad {}- (A + M + C) - 1. \end{aligned}$

Since $A + M + C = 12,$ this equals $(A + 1)(M + 1)(C + 1) - 13.$ The three factors sum to $15,$ so their product is maximized when each equals $5,$ giving $5^3 = 125.$

The maximum value is $125 - 13 = 112.$

Thus, the correct answer is E.

2000 AMC 12 Problem 12

Answer: E

Difficulty rating: 1650

Solution:

Problem 12 in Other Years