2021 AMC 12B Problems/Problem 12
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 . If the least integer in is also removed, then the average value of the integers remaining is . If the great integer is then returned to the set, the average value of the integers rises to The greatest integer in the original set is greater than the least integer in . What is the average value of all the integers in the set
Solution
Let be the greatest integer, be the smallest, be the sum of the numbers in S excluding and , and be the number of elements in S.
Then,
Firstly, 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 , , .
average value of all integers in the set , D)
~ SoySoy4444
See Also
2021 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 11 |
Followed by Problem 13 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.