2000 AMC 10 Problem 1

Below is the professionally curated solution for Problem 1 of the 2000 AMC 10, 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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Concepts:prime factorization optimization

Difficulty rating: 960

In the year 2001, the United States will host the International Mathematical Olympiad. Let $I,$ $M,$ and $O$ be distinct positive integers such that the product $I \cdot M \cdot O = 2001.$ What is the largest possible value of the sum $I + M + O?$

$23$

$55$

$99$

$111$

$671$

Solution:

Factoring gives $2001 = 3 \cdot 23 \cdot 29.$

To maximize the sum with the product fixed, spread the factors as much as possible: take $I = 1$ and combine the two largest primes, $M = 3$ and $O = 23 \cdot 29 = 667.$

The sum is $1 + 3 + 667 = 671.$

Thus, the correct answer is E.

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