AMC12 2000 A

AMC12 2000 A · Q18

AMC12 2000 A · Q18. It mainly tests Patterns & sequences (misc), Remainders & modular arithmetic.

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

在年份$N$中，第$300^{\text{th}}$天是星期二。在年份$N+1$中，第$200^{\text{th}}$天也是星期二。年份$N-1$的第$100$天是星期几？

(A) Thursday 星期四

(B) Friday 星期五

(D) Sunday 星期日

(E) Monday 星期一

Answer

Correct choice: (A)

正确答案：（A）

Solution

There are either \[65 + 200 = 265\] or \[66 + 200 = 266\] days between the first two dates depending upon whether or not year $N+1$ is a leap year (since the February 29th of the leap year would come between the 300th day of year $N$ and 200th day of year $N + 1$). Since $7$ divides into $266$ but not $265$, for both days to be a Tuesday, year $N$ must be a leap year. Hence, year $N-1$ is not a leap year, and so since there are \[265 + 300 = 565\] days between the date in years $N,\text{ }N-1$, this leaves a remainder of $5$ upon division by $7$. Since we are subtracting days, we count 5 days before Tuesday, which gives us $\boxed{\mathbf{(A)} \ \text{Thursday}.}$

两日期之间相隔的天数可能是 \[65 + 200 = 265\] 或 \[66 + 200 = 266\] 取决于年份$N+1$是否为闰年（因为闰年的2月29日会落在年份$N$的第300天与年份$N + 1$的第200天之间）。由于$7$能整除$266$但不能整除$265$，要使两天都为星期二，则年份$N$必须是闰年。因此年份$N-1$不是闰年。又因为年份$N$与$N-1$中对应日期之间相隔 \[265 + 300 = 565\] 天，$565$除以$7$余$5$。由于是在往前推日期，我们从星期二往前数$5$天，得到$\boxed{\mathbf{(A)} \ \text{Thursday}.}$

Topics