Difference between revisions of "2018 AMC 10B Problems/Problem 14"
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| + | {{duplicate|[[2018 AMC 12B Problems|2018 AMC 12B #10]] and [[2018 AMC 10B Problems|2018 AMC 10B #14]]}} | ||
| + | |||
| + | == Problem == | ||
| + | |||
A list of <math>2018</math> positive integers has a unique mode, which occurs exactly <math>10</math> times. What is the least number of distinct values that can occur in the list? | A list of <math>2018</math> positive integers has a unique mode, which occurs exactly <math>10</math> times. What is the least number of distinct values that can occur in the list? | ||
<math>\textbf{(A)}\ 202\qquad\textbf{(B)}\ 223\qquad\textbf{(C)}\ 224\qquad\textbf{(D)}\ 225\qquad\textbf{(E)}\ 234</math> | <math>\textbf{(A)}\ 202\qquad\textbf{(B)}\ 223\qquad\textbf{(C)}\ 224\qquad\textbf{(D)}\ 225\qquad\textbf{(E)}\ 234</math> | ||
| − | == Solution == | + | == Solution 1 (Statistics) == |
| + | |||
| + | To minimize the number of distinct values, we want to maximize the number of times a number appears. So, we could have <math>223</math> numbers appear <math>9</math> times, <math>1</math> number appear once, and the mode appear <math>10</math> times, giving us a total of <math>223 + 1 + 1 = \boxed{\textbf{(D)}\ 225}.</math> | ||
| + | |||
| + | == 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 <math>10 + 9( x - 1 )\geq2018,</math> where <math>x</math> is the number of values. Notice how we can then rearrange the equation into <math>1 + 9 ( 1 )+9 ( x - 1 )\geq2018,</math> which becomes <math>9 x\geq2017,</math> or <math>x\geq224\frac19.</math> We cannot have a fraction of a value so we must round up to <math>\boxed{\textbf{(D)}\ 225}.</math> | ||
| + | |||
| + | ~Username_taken12 | ||
| + | |||
| + | ==Video Solution (HOW TO THINK CRITICALLY!!!)== | ||
| + | https://youtu.be/M-z7SJwlsLY | ||
| − | + | ~Education, the Study of Everything | |
==See Also== | ==See Also== | ||
{{AMC10 box|year=2018|ab=B|num-b=13|num-a=15}} | {{AMC10 box|year=2018|ab=B|num-b=13|num-a=15}} | ||
| + | {{AMC12 box|year=2018|ab=B|num-b=9|num-a=11}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
| + | |||
| + | [[Category:Introductory Combinatorics Problems]] | ||
Latest revision as of 14:07, 14 June 2023
- The following problem is from both the 2018 AMC 12B #10 and 2018 AMC 10B #14, so both problems redirect to this page.
Contents
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 a number appears. 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 
~Username_taken12
Video Solution (HOW TO THINK CRITICALLY!!!)
~Education, the Study of Everything
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. 
