2010 AMC 10B 第 17 题

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

和第

64

。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

th and

64

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

小提示：

如果有

n

所学校，则共有

3n

名参赛者。

If there are

n

schools, there are

3n

contestants

大提示：

Andrea 的中位名次为

\dfrac{3n+1}{2}

。

Andrea’s median place is

\dfrac{3n+1}{2}

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

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

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

小提示：

大提示：

解答：

其他年份的第 17 题