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| − | == Problem ==
| + | #redirect [[2010 AMC 12B Problems/Problem 8]] |
| − | Every high school in the city of Euclid sent a team of <math>3</math> students to a math contest. Each participant in the contest received a different score. Andrea's score was the median among all students, and hers was the highest score on her team. Andrea's teammates Beth and Carla placed <math>37</math><sup>th</sup> and <math>64</math><sup>th</sup>, respectively. How many schools are in the city?
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| − | <math>\textbf{(A)}\ 22 \qquad \textbf{(B)}\ 23 \qquad \textbf{(C)}\ 24 \qquad \textbf{(D)}\ 25 \qquad \textbf{(E)}\ 26</math>
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| − | == Solution ==
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| − | Let the <math>n</math> be the number of schools, <math>3n</math> be the number of contestants, and <math>x</math> be Andrea's score. Since the number of participants divided by three is the number of schools, <math>n\geq\frac{64}3=21\frac13</math>. Since Andrea received a higher score than her teammates, <math>x\leq36</math>. Since <math>36</math> is the maximum possible median, then <math>2*36-1=71</math>is the maximum possible number of participants. Therefore, <math>3n\leq71\Rightarrow n\leq\frac{71}3=23\frac23</math>. This yields the compound inequality: <math>21\frac13\leq n\leq
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| − | 23\frac23</math>. Since a set with an even number of elements has a median that is the average of the two middle terms, an occurrence that cannot happen in this situation, <math>n</math> cannot be even. <math>\boxed{\mathrm {(B)} 23}</math> is the only other option.
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