2000 AMC 12 Problem 12

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}

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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.

Tagged: factoring · AM-GM Inequality · optimization