Problem:
Suppose that is a finite set of positive integers. If the greatest integer in is removed from , then the average value (arithmetic mean) of the integers remaining is 32 . If the least integer is is also removed, then the average value of the integers remaining is 35 . If the greatest integer is then returned to the set, the average value of the integers rises to 40 . The greatest integer in the original set is 72 greater than the least integer in . What is the average value of all the integers in the set ?
Answer Choices:
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Solution:
Let the lowest value be and the highest , and let the sum be and the amount of numbers . We have
Clearing denominators gives
We use to turn the first equation into . Since we substitute it into the equation which gives . Turning the second into using , we see and . Therefore, the average is .
Let be the greatest integer, be the smallest, be the sum of the numbers in excluding and , and be the number of elements in . Then,
First, when the greatest integer is removed,
When the smallest integer is also removed,
When the greatest integer is added back,
We are given that . After you substitute , you have 3 equations with 3 unknowns and .
This can be easily solved to yield . Therefore, the average value of all integers in the set is
The problems on this page are the property of the MAA's American Mathematics Competitions