Problem:
Someone observed that . Find the largest positive integer for which can be expressed as the product of consecutive positive integers.
Solution:
If can be expressed as the product of consecutive integers, then there is a positive integer such that
We can express this last relation as
and expand to get
If , then the factors on the left of the previous equation all exceed and the equation cannot be true. On the other hand, and is an obvious solution to and shows that is the answer to the problem.
The problems on this page are the property of the MAA's American Mathematics Competitions