Difference between revisions of "2000 AMC 12 Problems/Problem 14"
(soln) |
m (→See also: bad copy and paste :/ this is why we need javascript) |
||
Line 21: | Line 21: | ||
== See also == | == See also == | ||
− | {{AMC12 box|year=2000|num-b= | + | {{AMC12 box|year=2000|num-b=13|num-a=15}} |
[[Category:Introductory Algebra Problems]] | [[Category:Introductory Algebra Problems]] |
Revision as of 20:03, 4 January 2008
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
. 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 |