2000 AMC 10 Problem 20

Attempt Problem 20 of the 2000 AMC 10 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 10 solutions, or check the answer key.

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20.

Let $A,$ $M,$ and $C$ be nonnegative integers such that $A + M + C = 10.$ What is the maximum value of $\begin{aligned} &A \cdot M \cdot C + A \cdot M \\ &\quad {}+ M \cdot C + C \cdot A? \end{aligned}$

$49$

$59$

$69$

$79$

$89$

Answer: C

Concepts:factoring optimization

Difficulty rating: 1820

Solution:

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.

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