Problem:
A set of distinct positive integers has the following property: for every integer in , the arithmetic mean of the set of values obtained by deleting from is an integer. Given that belongs to and that is the largest element of , what is the greatest number of elements that can have?
Solution:
Let be the members of , and let
It is given that is an integer for any integer between and , inclusive. Note that, for any integers and between and , inclusive,
which must be an integer. Also, , and when , this implies that each element of is more than a multiple of . It follows that , implying that . Since is a divisor of , conclude that or or or , so is at most . A thirty-element set with the requested property is obtained by setting for and .
The problems on this page are the property of the MAA's American Mathematics Competitions