AMC12 2010 B

AMC12 2010 B · Q8

AMC12 2010 B · Q8. It mainly tests Basic counting (rules of product/sum), Casework.

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?

Euclid市的每所高中都派出一个由$3$名学生组成的队伍参加数学竞赛。竞赛中每位参赛者的得分都不同。Andrea的得分在所有学生中是中位数，并且她的得分是她所在队伍中的最高分。Andrea的队友Beth和Carla分别获得第$37$名和第$64$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

There are $x$ schools. This means that there are $3x$ people. Because no one's score was the same as another person's score, that means that there could only have been $1$ median score. This implies that $x$ is an odd number. $x$ cannot be less than $23$, because there wouldn't be a $64$th place if $x$ was. $x$ cannot be greater than $23$ either, because that would tie Andrea and Beth or Andrea's place would be worse than Beth's. Thus, the only possible answer is $\boxed{\mathbf{(B)}\ 23}$.

设有$x$所学校，则共有$3x$名参赛者。由于没有两人的得分相同，因此只能有$1$个中位数得分，这说明$x$为奇数。$x$不能小于$23$，否则就不会有第$64$名。$x$也不能大于$23$，否则Andrea与Beth会并列，或Andrea的名次会在Beth之后。因此唯一可能的答案是$\boxed{\mathbf{(B)}\ 23}$。

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