AMC10 2010 B

AMC10 2010 B · Q17

AMC10 2010 B · Q17. It mainly tests Averages (mean), Logic puzzles.

Every high school in the city of Euclid sent a team of 3 students to a math contest. Each participant received a different score. Andrea’s score was the median among all students, and hers was the highest on her team. Andrea’s teammates Beth and Carla placed 37th and 64th, respectively. How many schools are in the city?

Euclid 市每所高中派一个 3 人学生队参加数学竞赛。每位参赛者得分均不同。Andrea 的得分是所有学生中的中位数，且是她队内最高。Andrea 的队友 Beth 和 Carla 分别排第 37 和第 64 名。该市有多少所学校？

(A) 22 22

(B) 23 23

(D) 25 25

(E) 26 26

Answer

Correct choice: (B)

正确答案：（B）

Solution

Answer (B): If there are $n$ schools in the city, then there are $3n$ contestants, so $3n\ge 64$, and $n\ge 22$. Because Andrea received the median score and each student received a different score, $n$ is odd, so $n\ge 23$. Andrea’s position is $\frac{3n+1}{2}$, and Andrea finished ahead of Beth, so $\frac{3n+1}{2}<37$, and $3n<73$. Because $n$ is an odd integer, $n\le 23$. Therefore $n=23$.

答案（B）：如果城里有 $n$ 所学校，那么共有 $3n$ 名参赛者，所以 $3n\ge 64$，从而 $n\ge 22$。因为 Andrea 得到的是中位数分数且每个学生的分数都不同，$n$ 为奇数，所以 $n\ge 23$。Andrea 的名次是 $\frac{3n+1}{2}$，并且 Andrea 排在 Beth 前面，所以 $\frac{3n+1}{2}<37$，从而 $3n<73$。因为 $n$ 是奇整数，$n\le 23$。因此 $n=23$。

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