Problem:
What is the median of the following list of 4040 numbers?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
We can see that which is less than . Therefore, there are of the numbers greater than . Also, there are numbers that are less than or equal to .
Since there are duplicates/extras, it will shift up our median's placement down . Had the list of numbers been , the median of the whole set would be .
Thus, our answer is .
OR
As we are trying to find the median of a -term set, we must find the average of the th and st terms.
Since is slightly greater than , we know that the perfect squares through are less than , and the rest are greater. Thus, from the number to the number , there are terms. Since is less than and less than , we will only need to consider the perfect square terms going down from the th term, , after going down terms. Since the th and st terms are only and terms away from the th term, we can simply subtract from and from to get the two terms, which are and . Averaging the two, we get .
The problems on this page are the property of the MAA's American Mathematics Competitions