2000 AMC 12 第 18 题

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

小提示：

两个日期在同一星期几，当且仅当它们之间相隔的天数是

7

的倍数。

Two dates fall on the same weekday exactly when the number of days between them is a multiple of

7

大提示：

用两个给定的星期二判断

N - 1, N, N + 1

中哪一年是闰年。

Use the two given Tuesdays to decide which of

N - 1, N, N + 1

is a leap year

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

小提示：

大提示：

解答：

其他年份的第 18 题