2000 AMC 12 Problems/Problem 14
(Redirected from 2000 AMC 10 Problems/Problem 23)
- The following problem is from both the 2000 AMC 12 #14 and 2000 AMC 10 #23, so both problems redirect to this page.
Problem
When the mean, median, and mode of the list
are arranged in increasing order, they form a non-constant arithmetic progression. What is the sum of all possible real values of ?
Solution
- The mean is .
- Arranged in increasing order, the list is , so the median is either or depending upon the value of .
- The mode is , since it appears three times.
We apply casework upon the median:
- If the median is (), then the arithmetic progression must be constant.
- If the median is (), because the mode is , the mean can either be to form an arithmetic progression. Solving for yields respectively, of which only works because it is larger than .
- If the median is (), we must have the arithmetic progression . Thus, we find that so .
The answer is .
Video Solution by OmegaLearn
https://youtu.be/IziHKOubUI8?t=933
~ pi_is_3.14
Video Solution
See also
2000 AMC 12 (Problems • Answer Key • Resources) | |
Preceded by Problem 13 |
Followed by Problem 15 |
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 |
2000 AMC 10 (Problems • Answer Key • Resources) | ||
Preceded by Problem 22 |
Followed by Problem 24 | |
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 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.