2024 AMC 12B Problems/Problem 11
Contents
[hide]Problem
Let . What is the mean of ?
Solution 1
Add up with , with , and with . Notice by the Pythagorean identity. Since we can pair up with and keep going until with , we get Hence the mean is
~kafuu_chino
Solution 2
We can add a term into the list, and the total sum of the terms won't be affected since . Once is added into the list, the average of the terms is clearly . Hence the total sum of the terms is . To get the average of the original , we merely divide by to get . Hence the mean is
~tsun26
Solution 3 (Inductive Reasoning)
If we use radians to rewrite the question, we have: . Notice that have no specialty beyond any other integers, so we can use some inductive processes.
If we change to :
If we change to :
By intuition, although not rigorous at all, we can guess out the solution if we change into , we get . Thus, if we plug in , we get
~Prof. Joker
Solution 4
~Kathan
Solution 4
Note that . We want to determine .
Graphing , we can pair and so on. We are left with .
Our answer is
~vinyx
Video Solution 1 by SpreadTheMathLove
https://www.youtube.com/watch?v=gJq7DhLNnZ4&t=0s
See also
2024 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 10 |
Followed by Problem 12 |
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 |
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