Problem:
A list of positive integers has a unique mode, which occurs exactly times. What is the least number of distinct values that can occur in the list?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
The list has entries that are not equal to the mode. Because the mode is unique, each of these entries can occur at most times. There must be at least distinct values in the list that are different from the mode, because if there were fewer than this many such values, then the size of the list would be at most . (The ceiling function notation represents the least integer greater than or equal to .) Therefore the least possible number of distinct values that can occur in the list is 225 . One list satisfying the conditions of the problem contains instances of each of the numbers through , instances of the number , and one instance of .
The problems on this page are the property of the MAA's American Mathematics Competitions