2014 AMC 8 第 23 题
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所有题目均经美国数学协会（MAA）官方合法授权使用。
23.
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 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 -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?
小提示：
两个两位质数之和必须是一个月份中的日期
Pair sums of two two-digit primes must be dates in a month.
大提示：
利用“早些时候、今天、晚些时候”的顺序判断哪个球衣号码最小
Use the earlier-today-later order to determine which uniform number is smallest.
解答：
任意两人的号码和必须是月份中的日期，所以至多为 。两个两位质数之和不超过 的可能有 、、 和 。
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 ，对应号码 、、。
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 。
所以正确答案是 A。
Any pair-sum must be a date in the month, so it is at most . The possible sums of two two-digit primes that are at most are , , , and .
The three dates are earlier, today, and later, so they must be distinct and increasing. The only possible set of three distinct dates is , coming from the numbers , , and .
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 .
Thus, A is the correct answer.
其他年份的第 23 题
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