Difference between revisions of "2010 AMC 12B Problems/Problem 8"
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− | Let <math>a</math> be Andrea's place. We know that she was the highest on | + | Let <math>a</math> be Andrea's place. We know that she was the highest on her team, so <math>a < 37</math>. |
Since <math>a</math> is the median, there are <math>a-1</math> to the left and right of the median, so the total number of people is <math>2a-1</math> and the number of schools is <math>(2a-1)/3</math>. This implies that <math>2a-1 \equiv 0 \pmod{3} \implies a \equiv 2 \pmod{3}</math>. | Since <math>a</math> is the median, there are <math>a-1</math> to the left and right of the median, so the total number of people is <math>2a-1</math> and the number of schools is <math>(2a-1)/3</math>. This implies that <math>2a-1 \equiv 0 \pmod{3} \implies a \equiv 2 \pmod{3}</math>. |
Latest revision as of 13:40, 11 July 2021
- The following problem is from both the 2010 AMC 12B #8 and 2010 AMC 10B #17, so both problems redirect to this page.
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 1
There are schools. This means that there are
people. Because no one's score was the same as another person's score, that means that there could only have been
median score. This implies that
is an odd number.
cannot be less than
, because there wouldn't be a
th place if x was.
cannot be greater than
either, because that would tie Andrea and Beth or Andrea's place would be worse than Beth's. Thus, the only possible answer is
.
Solution 2
Let be Andrea's place. We know that she was the highest on her team, so
.
Since is the median, there are
to the left and right of the median, so the total number of people is
and the number of schools is
. This implies that
.
Also, since is the rank of the last-place person, and one of Andrea's teammates already got 64th place,
.
Putting it all together: and
, so clearly
, and the number of schools as we got before is
.
~adihaya
Video Solution
https://youtu.be/FQO-0E2zUVI?t=297
~IceMatrix
See also
2010 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 7 |
Followed by Problem 9 |
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 |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.