AMC 8 2018 Problem 3
AMC8 2018 年第 3 题
基本计数原理★★☆☆☆
Students Arn, Bob, Cyd, Dan, Eve, and Fon are arranged in that order in a circle. They start counting: Arn first, then Bob, and so forth. When the number contains a 7 as a digit (such as 47) or is a multiple of 7 that person leaves the circle and the counting continues. Who is the last one present in the circle?
Arn，Bob，Cyd，Dan，Eve 和 Fon 以这样的顺序围成一圈。他们开始数数：Arn 先开始，然后 Bob，以此类推。当数包含数字 7(例如 47)或者是 7 的倍数时，那个人就离开圆圈，数数继续进行。最后一个离开圆圈的是谁？
- A.
- B.
- C.
- D.
- E.
正确答案 · Correct Answer
D
解析 · Solution
Solution 1
Assign each person a position: Arn is in position 1, Bob in 2, etc. Note that if there are people standing in a circle, a person will say a number that is congruent to their position modulo . It follows that if there are numbers to be said (with someone leaving on the $kth$ ), the person that leaves will stand in position The five numbers which cause people to leave the circle are 7, 14, 17, 21, and 27. Since there are initially 6 people in the circle, so the person who leaves stands in position , or position 1. Arn now leaves, and because Bob is the next person to say a number, we reassign the positions such that Bob is in position 1, Cyd in 2, and Fon in 5. There are 7 more numbers to be said, and someone leaves on the 7th. It follows that the person who leaves stands in position . Cyd leaves the circle, and 4 people remain, namely Bob, Dan, Eve, and Fon. Assign the position numbers such that Dan has 1, Eve 2, etc. There are 3 more numbers to be said, and someone leaves on the 3rd. The person standing in position 3 (Fon) now leaves. There are two people left, Eve and Dan, with Dan in position 1 now. There are 6 more numbers to be said, and Eve is standing in position , so she leaves. The last person standing is . -Benedict T (countmath1)
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