Question

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 $I + M + O?$

$23$

$55$

$99$

$111$

$671$

Answer: E

Concepts:prime factorization optimization

Difficulty rating: 960

Solution:

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

If one factor is $1,$ the possible pairs for the other two factors are $(3,667),$ $(23,87),$ and $(29,69).$ Their corresponding sums with $1$ are $671,$ $111,$ and $99.$ (The pair $(1,2001)$ would repeat the factor $1.$ )

If no factor is $1,$ all three prime factors must be split among the three integers, giving only $3,23,29$ and a much smaller sum. Therefore, the largest possible sum is $1 + 3 + 667 = 671.$

Thus, the correct answer is E.

2000 AMC 10 Problem 1

Answer: E

Difficulty rating: 960

Solution:

Problem 1 in Other Years