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 . M . O = 2001. What is the largest possible value of the sum I + M + O?

This has been explained on the website I found this on but I don't understand it too much, so could someone please elaborate on this, thanks.

Think of a rectangle. The maximal area is achieved when it is a square.

Similarly for a rectangular prism, the maximum volume is when it is a cube.

so, in this case, the sum is just 3∛2001

To find the largest possible value of the sum I + M + O, we need to find the largest values for I, M, and O that satisfy the given condition.

We know that I, M, and O are positive integers and their product is 2001. So, we need to find the factors of 2001.

To find the factors, we can start with the smallest possible factor, which is 1, and keep incrementing it until we reach the square root of 2001 (approximately 44.7). If a number evenly divides 2001, it is a factor.

By doing this, we find the following factors of 2001: 1, 3, 23, 69, 29, 87, 667, and 2001.

Now, we need to find the largest possible combination of these factors such that I, M, and O are distinct and positive.

Since I, M, and O are distinct, we can't include any duplicate factors in our combination. Therefore, we can exclude the factors that occur more than once, like 1 and 3.

Considering the remaining factors (23, 69, 29, 87, 667, and 2001), we need to find three factors whose sum is the largest possible.

Sorting these factors in descending order, we have 2001, 667, 87, 69, 29, and 23.

The sum of the three largest factors is 2001 + 667 + 87 = 2755.

Therefore, the largest possible value of the sum I + M + O, given the conditions, is 2755.