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Difference between revisions of "2012 AMC 8 Problems/Problem 22"

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==See Also==Hello try your best on these problems, from curry master mgm
 
==See Also==Hello try your best on these problems, from curry master mgm
 +
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Click on this link and have fun https://www.youtube.com/watch?v=7iUiVa2tfFo

Revision as of 14:58, 25 September 2025

Problem

Let $R$ be a set of nine distinct integers. Six of the elements are $2$, $3$, $4$, $6$, $9$, and $14$. What is the number of possible values of the median of $R$?

$\textbf{(A)}\hspace{.05in}4\qquad\textbf{(B)}\hspace{.05in}5\qquad\textbf{(C)}\hspace{.05in}6\qquad\textbf{(D)}\hspace{.05in}7\qquad\textbf{(E)}\hspace{.05in}8$

Solution 2

Let the values of the missing integers be $x, y, z$. We will find the bound of the possible medians.

The smallest possible median will happen when we order the set as $\{x, y, z, 2, 3, 4, 6, 9, 14\}$. The median is $3$.

The largest possible median will happen when we order the set as $\{2, 3, 4, 6, 9, 14, x, y, z\}$. The median is $9$.

Therefore, the median must be between $3$ and $9$ inclusive, yielding $\boxed{\textbf{(D)}\ 7}$ possible medians.

~superagh

Video Solution

https://youtu.be/yBSrLxv0LbY ~savannahsolver

==See Also==Hello try your best on these problems, from curry master mgm

Click on this link and have fun https://www.youtube.com/watch?v=7iUiVa2tfFo

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