Difference between revisions of "2019 AMC 10B Problems/Problem 13"
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==Video Solution== | ==Video Solution== | ||
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+ | ~IceMatrix | ||
==See Also== | ==See Also== |
Revision as of 21:37, 7 September 2020
- The following problem is from both the 2019 AMC 10B #13 and 2019 AMC 12B #7, so both problems redirect to this page.
Contents
[hide]Problem
What is the sum of all real numbers for which the median of the numbers and is equal to the mean of those five numbers?
Solution
The mean is .
There are three possibilities for the median: it is either , , or .
Let's start with .
has solution , and the sequence is , which does have median , so this is a valid solution.
Now let the median be .
gives , so the sequence is , which has median , so this is not valid.
Finally we let the median be .
, and the sequence is , which has median . This case is therefore again not valid.
Hence the only possible value of is
Video Solution
https://youtu.be/mXvetCMMzpU?t=448
~IceMatrix
See Also
2019 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
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 |
2019 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 6 |
Followed by Problem 8 |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.