Difference between revisions of "2014 AMC 12B Problems/Problem 11"
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| − | == Problem == | + | ==Problem== |
| − | + | A list of <math>11</math> positive integers has a mean of <math>10</math>, a median of <math>9</math>, and a unique mode of <math>8</math>. What is the largest possible value of an integer in the list? | |
| − | < | + | <math> \textbf{(A)}\ 24\qquad\textbf{(B)}\ 30\qquad\textbf{(C)}\ 31\qquad\textbf{(D)}\ 33\qquad\textbf{(E)}\ 35 </math> |
| − | == Solution == | + | ==Solution== |
| + | |||
| + | We start off with the fact that the median is <math>9</math>, so we must have <math>a, b, c, d, e, 9, f, g, h, i, j</math>, listed in ascending order. Note that the integers do not have to be distinct. | ||
| + | |||
| + | Since the mode is <math>8</math>, we have to have at least <math>2</math> occurrences of <math>8</math> in the list. If there are <math>2</math> occurrences of <math>8</math> in the list, we will have <math>a, b, c, 8, 8, 9, f, g, h, i, j</math>. In this case, since <math>8</math> is the unique mode, the rest of the integers have to be distinct. So we minimize <math>a,b,c,f,g,h,i</math> in order to maximize <math>j</math>. If we let the list be <math>1,2,3,8,8,9,10,11,12,13,j</math>, then <math>j = 11 \times 10 - (1+2+3+8+8+9+10+11+12+13) = 33</math>. | ||
| + | |||
| + | Next, consider the case where there are <math>3</math> occurrences of <math>8</math> in the list. Now, we can have two occurrences of another integer in the list. We try <math>1,1,8,8,8,9,9,10,10,11,j</math>. Following the same process as above, we get <math>j = 11 \times 10 - (1+1+8+8+8+9+9+10+10+11) = 35</math>. As this is the highest choice in the list, we know this is our answer. Therefore, the answer is <math>\boxed{\textbf{(E) }35}</math> | ||
| + | |||
| + | == See also == | ||
| + | {{AMC12 box|year=2014|ab=B|num-b=10|num-a=12}} | ||
| + | {{MAA Notice}} | ||
Latest revision as of 09:13, 3 March 2015
Problem
A list of 
positive integers has a mean of 
, a median of 
, and a unique mode of 
. What is the largest possible value of an integer in the list?
Solution
We start off with the fact that the median is , so we must have 
, listed in ascending order. Note that the integers do not have to be distinct.
Since the mode is , we have to have at least 
occurrences of in the list. If there are occurrences of in the list, we will have 
. In this case, since is the unique mode, the rest of the integers have to be distinct. So we minimize 
in order to maximize 
. If we let the list be 
, then 
.
Next, consider the case where there are 
occurrences of in the list. Now, we can have two occurrences of another integer in the list. We try 
. Following the same process as above, we get 
. As this is the highest choice in the list, we know this is our answer. Therefore, the answer is 
See also
| 2014 AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 10 |
Followed by Problem 12 |
| 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. 
