Question

某个有 $31$ 天的月份中，星期一和星期三的天数相同。这个月的第一天可能是七天中的多少种？

A month with $31$ days has the same number of Mondays and Wednesdays. How many of the seven days of the week could be the first day of this month?

$2$

$3$

$4$

$5$

$6$

答案：B

知识点：日期与时间分类讨论

难度评级：1240

小提示：

$31=4\cdot7+3$ ，所以恰好有三个星期几会出现五次

$31=4\cdot7+3,$ so exactly three weekdays occur five times

大提示：

这三个星期几是这个月的前三天；检查星期一和星期三何时次数相同

Those three days are the first three days of the month; check when Monday and Wednesday match

解答：

因为 $31=4\cdot7+3$ ，这个月的前三天各出现五次，其余四天各出现四次。

星期一和星期三次数相同，恰好发生在它们都属于出现五次的一组，或都属于出现四次的一组时。

如果第一天是星期一，出现五次的是星期一、星期二、星期三。若第一天是星期四或星期五，出现五次的三天都不包含星期一和星期三。其他起始日都会只包含星期一和星期三中的一个。

所以第一天可以是星期一、星期四或星期五，共 $3$ 种可能。

因此，正确答案是 B。

Since $31=4\cdot7+3,$ the first three days of the month each occur five times, and the other four days occur four times.

Mondays and Wednesdays are equal in number exactly when both fall in the five-time group or both fall in the four-time group.

If the first day is Monday, the five-time days are Mon, Tue, Wed (both appear five times). If the first day is Thursday or Friday, the five-time days miss both Monday and Wednesday (both appear four times). Every other starting day includes exactly one of Monday or Wednesday.

So the first day can be Monday, Thursday, or Friday, giving $3$ possibilities.

Thus, the correct answer is B.

2010 AMC 12B 第 4 题

答案：B

难度评级：1240

小提示：

大提示：

解答：

其他年份的第 4 题