2000 AMC 12 Problem 12

Attempt Problem 12 of the 2000 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2000 AMC 12 solutions, or check the answer key.

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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}

62

72

92

102

112

AMC + AM + MC + CA

= (A + 1)(M + 1)(C + 1)

- (A + M + C) - 1

The three factors

A+1, M+1, C+1

sum to

15,

so their product is largest when they are equal

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.

Problem 12 in other years