2000 AMC 10 Problems/Problem 23
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
As occurs three times and each of the three other values just once, regardless of what we choose the mode will always be .
The sum of all numbers is , therefore the mean is .
The six known values, in sorted order, are . From this sequence we conclude: If , the median will be . If , the median will be . Finally, if , the median will be .
We will now examine each of these three cases separately.
In the case , both the median and the mode are 2, therefore we can not get any non-constant arithmetic progression.
In the case we have , because . Therefore our three values in order are . We want this to be an arithmetic progression. From the first two terms the difference must be . Therefore the third term must be .
Solving we get the only solution for this case: .
The case remains. Once again, we have , therefore the order is . The only solution is when , i. e., .
The sum of all solutions is therefore .
See Also
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 |