Difference between revisions of "2002 AMC 12A Problems/Problem 15"
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Let the 8 numbers be <math>a, b, c, d, e, f, g, h</math>, arranged in increasing order. Since the range of the eight numbers is 8, <math>h=a+8</math>. | Let the 8 numbers be <math>a, b, c, d, e, f, g, h</math>, arranged in increasing order. Since the range of the eight numbers is 8, <math>h=a+8</math>. | ||
− | I claim that <math>d</math>,<math>e</math> must both be <math>8</math>. Since the median is 8, the mean of <math>d</math> and <math>e</math> must be 8. Let's assume that <math>d</math> and <math>e</math> aren't <math>8</math>. The mode of the collection is <math>8</math>, and if <math>d</math> and <math>e</math> aren't, then <math>8</math> must be between <math>d</math> and <math>e</math> (i.e. not in the collection). This is a contradiction, so <math>d</math> and <math>e</math> have to be 8. | + | I claim that <math>d</math>, <math>e</math> must both be <math>8</math>. Since the median is 8, the mean of <math>d</math> and <math>e</math> must be 8. Let's assume that <math>d</math> and <math>e</math> aren't <math>8</math>. The mode of the collection is <math>8</math>, and if <math>d</math> and <math>e</math> aren't, then <math>8</math> must be between <math>d</math> and <math>e</math> (i.e. not in the collection). This is a contradiction, so <math>d</math> and <math>e</math> have to be 8. |
Now, we have the eight numbers are <math>a, b, c, 8, 8, f, g, a+8</math>. | Now, we have the eight numbers are <math>a, b, c, 8, 8, f, g, a+8</math>. |
Revision as of 21:22, 26 September 2021
- 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 the largest one is , then the smallest one is . This means that we already know four of the values: , , , . Since the mean of all the numbers is , their sum must be . Thus the sum of the missing four numbers is . But if is the smallest number, then the sum of the missing numbers must be at least , which is again a contradiction.
If the largest number is , we can easily find the solution . Hence, our answer is .
Note
The solution for is, in fact, unique. As the median must be , this means that both the and the number, when ordered by size, must be s. This gives the partial solution . For the mean to be each missing variable must be replaced by the smallest allowed value. The solution that works is
Solution 2
Let the 8 numbers be , arranged in increasing order. Since the range of the eight numbers is 8, .
I claim that , must both be . Since the median is 8, the mean of and must be 8. Let's assume that and aren't . The mode of the collection is , and if and aren't, then must be between and (i.e. not in the collection). This is a contradiction, so and have to be 8.
Now, we have the eight numbers are .
Since the mean is , we have , giving us .
Since , . Plugging that in, we have . Note that we can't do the same for and , but we can do , giving us , which means . We want to find This is our answer, so
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.