2002 AMC 12A Problems/Problem 15
- The following problem is from both the 2002 AMC 12A #15 and 2002 AMC 10A #21, so both problems redirect to this page.
Contents
Problem
The mean, median, unique mode, and range of a collection of eight integers are all equal to 8. The largest integer that can be an element of this collection is
Solution 1
As the unique mode is , there are at least two s.
As the range is and one of the numbers is , the largest one can be at most .
If the largest one is , then the smallest one is , and thus the mean is strictly larger than , which is a contradiction.
If we have 2 8's we can add find the numbers 4, 6, 7, 8, 8, 9, 10, 12. This is a possible solution but has not reached the maximum.
If we have 4 8's we can find the numbers 6, 6, 6, 8, 8, 8, 8, 14.
We can also see that they satisfy the need for the mode, median, and range to be 8. This means that the answer will be . ~By QWERTYUIOPASDFGHJKLZXCVBNM
Solution 2
We could express this collection as integers through , with being the smallest and being the largest.
Since the mode is , we know that and must also be . If they were not, the other numbers, which are lesser and greater than and respectively, would not be able to satisfy the condition that is the mode.
There are terms and the mean is . This tells us that the sum of all the numbers is .
We want to maximize the value of , so we set and to as well.
Knowing that we want to minimize numbers and that the range is , we set , , and equal to .
{, , , , , , , } {, , , , , , , }
Letting the sum of all the numbers be , we find that , which simplifies to . Solving, we get . ~By SK80, mod_x and huniiya for minor edits
Solution 3
Let's say that we choose numbers to be 8, and the largest number to be . There will therefore be numbers with the value of .
This gives the following expression . Solving for gives . The bank of choices we have for is represented by answer choices (A), (B), (C), (D), and (E).
We find that the value for gives creating the set , which satisfies all conditions.
This gives .
Solution 4
Since the mean of the numbers is and there are numbers, the sum of all numbers is . Since the range is also , let the smallest value in the set . Therefore the largest number in the set is . The mode is the number which appears most often, so we can let the th to th largest integers . To maximize the largest value, we will let the nd and rd smallest integers also equal .
In order this will be:
Now we can set the expression equal to (because of what we determined in the st step):
Simplyifying,
Therefore, since is the smallest number in the set, the largest number in the set will be , which is .
~goofytaipan
Video Solution by OmegaLearn
https://youtu.be/xqo0PgH-h8Y?t=848
~ pi_is_3.14
Video Solution
https://www.youtube.com/watch?v=Sic_pM_cmrA ~David
See Also
2002 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 14 |
Followed by Problem 16 |
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 |
2002 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 20 |
Followed by Problem 22 | |
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 |
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