Question

$N$ 年的第 $300$ 天是星期二， $N + 1$ 年的第 $200$ 天也是星期二。那么 $N - 1$ 年的第 $100$ 天是星期几？

In year $N,$ the $300$ th day of the year is a Tuesday. In year $N + 1,$ the $200$ th day is also a Tuesday. On what day of the week did the $100$ th day of year $N - 1$ occur?

星期四

Thursday

星期五

Friday

星期六

Saturday

星期日

Sunday

星期一

Monday

答案：A

知识点：日期与时间模运算

难度评级：1870

解答：

从 $N$ 年第 $300$ 天到 $N + 1$ 年第 $200$ 天，若 $N$ 不是闰年，相隔 $365 - 300 + 200 = 265$ 天。但 $265 = 7 \cdot 37 + 6$ ，会落在星期一，而不是星期二。

所以 $N$ 是闰年，相隔 $266 = 7 \cdot 38$ 天，正好仍为星期二。因此 $N - 1$ 年不是闰年。

$N - 1$ 年第 $100$ 天比 $N$ 年第 $300$ 天早 $365 - 100 + 300 = 565$ 天。因为 $565 = 7 \cdot 80 + 5$ ，所以那一天比星期二早 $5$ 天，即星期四。

因此，正确答案是 A。

From day $300$ of year $N$ to day $200$ of year $N + 1,$ the number of days is $365 - 300 + 200 = 265$ if $N$ is not a leap year. But $265 = 7 \cdot 37 + 6,$ which would land on a Monday, not a Tuesday.

So year $N$ is a leap year, and the gap is $266 = 7 \cdot 38$ days, giving a Tuesday as stated. It follows that year $N - 1$ is not a leap year.

The $100$ th day of year $N - 1$ precedes the Tuesday on day $300$ of year $N$ by $365 - 100 + 300 = 565$ days. Since $565 = 7 \cdot 80 + 5,$ that day is $5$ weekdays before Tuesday, namely Thursday.

Thus, the correct answer is A.

2000 AMC 12 第 18 题

答案：A

难度评级：1870

解答：

其他年份的第 18 题