2000 AMC 12 Problem 12

Below is the professionally curated solution for Problem 12 of the 2000 AMC 12, 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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Concepts:factoring AM-GM Inequality optimization

Difficulty rating: 1650

12.

Let $A,$ $M,$ and $C$ be nonnegative integers such that $A + M + C = 12.$ What is the maximum value of $A \cdot M \cdot C + A \cdot M + M \cdot C + C \cdot A?$

$62$

$72$

$92$

$102$

$112$

Solution:

Observe that $AMC + AM + MC + CA = (A + 1)(M + 1)(C + 1) - (A + M + C) - 1.$

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.

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