Difference between revisions of "2000 AMC 12 Problems/Problem 14"
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+ | {{duplicate|[[2000 AMC 12 Problems|2000 AMC 12 #14]] and [[2000 AMC 10 Problems|2000 AMC 10 #23]]}} | ||
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== Problem == | == Problem == | ||
When the [[mean]], [[median]], and [[mode]] of the list | When the [[mean]], [[median]], and [[mode]] of the list | ||
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== See also == | == See also == | ||
{{AMC12 box|year=2000|num-b=13|num-a=15}} | {{AMC12 box|year=2000|num-b=13|num-a=15}} | ||
+ | {{AMC10 box|year=2000|num-b=22|num-a=24}} | ||
[[Category:Introductory Algebra Problems]] | [[Category:Introductory Algebra Problems]] |
Revision as of 00:06, 27 November 2011
- 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, which results in a contradiction.
- If the median is
(
), then the mean can either be
to form an arithmetic progression. Solving for
yields
respectively, of which only
works.
- If the median is
(
), then the mean can either be
to form an arithmetic progression. Solving for
yields
respectively, of which only
works.
The answer is .
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 |