Difference between revisions of "2010 AMC 10B Problems/Problem 17"
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== Problem == | == Problem == | ||
| − | 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 | + | 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>th and <math>64</math>th, respectively. How many schools are in the city? |
<math>\textbf{(A)}\ 22 \qquad \textbf{(B)}\ 23 \qquad \textbf{(C)}\ 24 \qquad \textbf{(D)}\ 25 \qquad \textbf{(E)}\ 26</math> | <math>\textbf{(A)}\ 22 \qquad \textbf{(B)}\ 23 \qquad \textbf{(C)}\ 24 \qquad \textbf{(D)}\ 25 \qquad \textbf{(E)}\ 26</math> | ||
== Solution == | == Solution == | ||
| − | 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 | + | 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 place. Since the number of participants divided by three is the number of schools, <math>n\geq\frac{64}3=21\frac13</math>. Andrea received a higher score than her teammates, so <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 |
| − | 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{\ | + | 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{\textbf{(B)}\ 23}</math> is the only other option. |
| + | |||
| + | ==See Also== | ||
| + | {{AMC10 box|year=2010|ab=B|num-b=16|num-a=18}} | ||
Revision as of 14:19, 26 November 2011
Problem
Every high school in the city of Euclid sent a team of 
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 
th and 
th, respectively. How many schools are in the city?
Solution
Let the 
be the number of schools, 
be the number of contestants, and 
be Andrea's place. Since the number of participants divided by three is the number of schools, 
. Andrea received a higher score than her teammates, so 
. Since 
is the maximum possible median, then 
is the maximum possible number of participants. Therefore, 
. This yields the compound inequality: 
. 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, cannot be even. 
is the only other option.
See Also
| 2010 AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by Problem 16 |
Followed by Problem 18 | |
| 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 | ||
