Question

Daphne 的三位好友 Alice、Beatrix 和 Claire 会定期来拜访她。Alice 每三天来一次，Beatrix 每四天来一次，Claire 每五天来一次。

昨天三人都拜访了 Daphne。接下来 $365$ 天中，恰好有两位好友来访的天数是多少？

Daphne is visited periodically by her three best friends: Alice, Beatrix, and Claire. Alice visits every third day, Beatrix visits every fourth day, and Claire visits every fifth day.

All three friends visited Daphne yesterday. How many days of the next $365$ -day period will exactly two friends visit her?

$48$

$54$

$60$

$66$

$72$

答案：B

知识点：最小公倍数容斥原理区间内整数计数

难度评级：1420

解答：

两两同时来访的周期分别为 $\operatorname{lcm}(3,4)=12$ 、 $\operatorname{lcm}(3,5)=15$ 、 $\operatorname{lcm}(4,5)=20$ 天。

在接下来 $365$ 天中，这三种两两同时来访的天数分别为 $\left\lfloor\frac{365}{12}\right\rfloor=30$ 、 $\left\lfloor\frac{365}{15}\right\rfloor=24$ 、 $\left\lfloor\frac{365}{20}\right\rfloor=18$ 。

三人都来访的周期为 $60$ 天，共 $\left\lfloor\frac{365}{60}\right\rfloor=6$ 天。恰好两人来访的天数为 $(30-6)+(24-6)$ $+(18-6)=54$ 。

所以正确答案是 B。

Pairwise visits occur every $\operatorname{lcm}(3,4)=12$ , $\operatorname{lcm}(3,5)=15$ , and $\operatorname{lcm}(4,5)=20$ days.

In the next $365$ days, these pair counts are $\left\lfloor\frac{365}{12}\right\rfloor=30$ , $\left\lfloor\frac{365}{15}\right\rfloor=24$ , and $\left\lfloor\frac{365}{20}\right\rfloor=18$ .

All three visit every $60$ days, which happens $\left\lfloor\frac{365}{60}\right\rfloor=6$ times. Subtracting those days from each pair count gives $(30-6)+(24-6)$ $+(18-6)=54$ .

Thus, B is the correct answer.

2013 AMC 10A 第 17 题

答案：B

难度评级：1420

解答：

其他年份的第 17 题