2012 AMC 8 Problems/Problem 22
Contents
Problem
Let be a set of nine distinct integers. Six of the elements are 2, 3, 4, 6, 9, and 14. What is the number of possible values of the median of
?
Solution 1
First, we find that the minimum value of the median of will be
.
We then experiment with sequences of numbers to determine other possible medians.
Median:
Sequence:
Median:
Sequence:
Median:
Sequence:
Median:
Sequence:
Median:
Sequence:
Median:
Sequence:
Median:
Sequence:
Any number greater than also cannot be a median of set
.
Therefore, the answer is
Solution 2
Let the values of the missing integers be . We will find the bound of the possible medians.
The smallest possible median will happen when we order the set as . The median is
.
The largest possible median will happen when we order the set as . The median is
Therefore, the median must be between and
inclusive, yielding
possible medians,
.
~superagh
See Also
2012 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 21 |
Followed by Problem 23 | |
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 AJHSME/AMC 8 Problems and Solutions |
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