Question

Euclid Middle School 女子垒球队的三名队员进行了以下对话。

Ashley：我刚意识到我们的球衣号码都是 $2$ 位质数。

Bethany：而且你们两个球衣号码之和是我本月早些时候生日的日期。

Caitlin：真巧。你们两个球衣号码之和是我本月晚些时候生日的日期。

Ashley：而你们两个球衣号码之和是今天的日期。

Caitlin 穿几号球衣？

Three members of the Euclid Middle School girls' softball team had the following conversation.

Ashley: I just realized that our uniform numbers are all $2$ -digit primes.

Bethany: And the sum of your two uniform numbers is the date of my birthday earlier this month.

Caitlin: That's funny. The sum of your two uniform numbers is the date of my birthday later this month.

Ashley: And the sum of your two uniform numbers is today's date.

What number does Caitlin wear?

$11$

$13$

$17$

$19$

$23$

答案：A

知识点：质数逻辑推理分类讨论

难度评级：1610

解答：

任意两人的号码和必须是月份中的日期，所以至多为 $31$ 。两个两位质数之和不超过 $31$ 的可能有 $11+13=24$ 、 $11+17=28$ 、 $11+19=30$ 和 $13+17=30$ 。

三个日期分别是早些时候、今天和晚些时候，所以必须互不相同且递增。唯一可能的一组三个不同日期是 $24,28,30$ ，对应号码 $11$ 、 $13$ 、 $17$ 。

Bethany 的日期是 Ashley 加 Caitlin，Ashley 的日期是 Bethany 加 Caitlin，Caitlin 的晚些时候日期是 Ashley 加 Bethany。因此 Caitlin 的号码是三者中最小的，即 $11$ 。

所以正确答案是 A。

Any pair-sum must be a date in the month, so it is at most $31$ . The possible sums of two two-digit primes that are at most $31$ are $11+13=24$ , $11+17=28$ , $11+19=30$ , and $13+17=30$ .

The three dates are earlier, today, and later, so they must be distinct and increasing. The only possible set of three distinct dates is $24,28,30$ , coming from the numbers $11$ , $13$ , and $17$ .

Bethany's date is Ashley plus Caitlin, Ashley's date is Bethany plus Caitlin, and Caitlin's later date is Ashley plus Bethany. This means Caitlin's number is the smallest of the three, namely $11$ .

Thus, A is the correct answer.

2014 AMC 8 第 23 题

答案：A

难度评级：1610

解答：

其他年份的第 23 题