2000 AMC 10 Problem 20

Question

Let

A,

M,

and

C

be nonnegative integers such that

A + M + C = 10.

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}

49

59

69

79

89

Add

1

to each variable and expand

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

The expression equals

(A+1)(M+1)(C+1) - 11,

so maximize a product of three positive integers summing to

13

Notice that

\begin{gathered} A \cdot M \cdot C + AM + MC + CA \\ = (A+1)(M+1)(C+1) \\ {}- (A + M + C) - 1 \\ = (A+1)(M+1)(C+1) - 11. \end{gathered}

We maximize a product of three positive integers summing to

13.

The most balanced split is

4, 4, 5,

giving

4 \cdot 4 \cdot 5 = 80.

The maximum is

80 - 11 = 69.

Thus, the correct answer is C.

Problem 20 in other years