2000 AMC 12 Problem 1

Attempt Problem 1 of the 2000 AMC 12 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 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

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

Factor

2001

into primes

To maximize the sum, make one factor as large as possible and the others small

Factoring gives

2001 = 3 \cdot 23 \cdot 29.

To maximize the sum of three distinct positive integers with this product, take

I = 1,

M = 3,

and

O = 23 \cdot 29 = 667.

The largest sum is

1 + 3 + 667 = 671.

Thus, the correct answer is E.

Problem 1 in other years