Problem:
Camila writes down five positive integers. The unique mode of these integers is greater than their median, and the median is greater than their arithmetic mean. What is the least possible value for the mode?
Answer Choices:
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Solution:
Assume for simplicity that the five integers are written in nondecreasing order. Let be the unique mode of these integers. Because the mode is 2 greater than the third number, the three greatest integers must be , and , in that order. Let and be the two least integers. Because the arithmetic mean of the five integers equals ,
which can be rewritten as . Because the mode is unique, and are distinct positive integers. This equation requires to be even, so the least possible value of occurs when , giving . This minimum mode is achieved with the integers , 11 , and 11 .
The problems on this page are the property of the MAA's American Mathematics Competitions