2018 AMC 10B Problems/Problem 14
- The following problem is from both the 2018 AMC 12B #10 and 2018 AMC 10B #14, so both problems redirect to this page.
Problem
A list of positive integers has a unique mode, which occurs exactly times. What is the least number of distinct values that can occur in the list?
Solution 1 (Statistics)
To minimize the number of distinct values, we want to maximize the number of times they appear. So, we could have numbers appear times, number appear once, and the mode appear times, giving us a total of
Solution 2 (Algebra)
As in Solution 1, we want to maximize the number of time each number appears to do so. We can set up an equation where is the number of values. Notice how we can then rearrange the equation into which becomes or We cannot have a fraction of a value so we must round up to
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See Also
2018 AMC 10B (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 10 Problems and Solutions |
2018 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 9 |
Followed by Problem 11 |
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.