Question

Euclid 市的每所高中都派出一支 $3$ 人队伍参加数学竞赛。每位参赛者的分数都不同。Andrea 的分数是所有学生中的中位数，并且她是自己队伍中得分最高的。Andrea 的队友 Beth 和 Carla 分别排第 $37^\text{th}$ 和第 $64^\text{th}$ ，这个城市有多少所学校？

Every high school in the city of Euclid sent a team of $3$ 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 $37^\text{th}$ and $64^\text{th},$ respectively. How many schools are in the city?

$22$

$23$

$24$

$25$

$26$

答案：B

知识点：中位数（数据）不等式逻辑推理

难度评级：1600

解答：

设有 $n$ 所学校，则比赛中共有 $3n$ 名参赛者。

因为中位数是某一名学生的分数， $3n$ 是奇数，所以 $n$ 是奇数。Carla 排第 $64^\text{th}$ 要求 $3n\ge64,$ 因此奇数 $n$ 至少为 $23.$

Andrea 的中位名次是 $\dfrac{3n+1}{2}.$ 因为她是队内最高分，所以名次在 Beth 之前，故 $\dfrac{3n+1}{2}\lt37.$ 这给出 $3n\lt73,$ 所以奇数 $n$ 至多为 $23.$

因此共有 $23$ 所学校。

所以正确答案是 B。

Let there be $n$ schools. Then there are $3n$ participants in the contest.

Because the median is one student's score, $3n$ is odd, so $n$ is odd. Carla's $64^\text{th}$ place requires $3n\ge64,$ and hence the odd integer $n$ is at least $23.$

Andrea's median place is $\dfrac{3n+1}{2}.$ Because she had the highest score on her team, she finished ahead of Beth, so $\dfrac{3n+1}{2}\lt37.$ This gives $3n\lt73,$ and therefore the odd integer $n$ is at most $23.$

Thus there are $23$ schools.

Thus, B is the correct answer.

2010 AMC 10B 第 17 题

答案：B

难度评级：1600

解答：

其他年份的第 17 题