Problem:
Let be a list of positive integers - not necessarily distinct-in which the number appears. The average (arithmetic mean) of the numbers in is . However, if is removed, the average of the remaining numbers drops to . What is the largest number that can appear in
Solution:
Let denote the number of positive integers in , and let denote their sum. Then, on the basis of the information given,
Solving these equations simultaneously yields and . Now, to maximize the largest element of , one must minimize the others. To attain this, must contain eleven 's, a and a , since .
The problems on this page are the property of the MAA's American Mathematics Competitions