2018 AMC 8

Solution 1

The five numbers which cause people to leave the circle are and

The most straightforward way to do this would be to draw out the circle with the people, and cross off people as you count.

Assuming the six people start with , Arn counts so he leaves first. Then, Cyd counts as there are numbers to be counted from this point. Then, Fon, Bob, and Eve, count, and , respectively, so the last one standing is Dan. Hence, the answer would be .

Solution 2

Assign each person a position: Arn is in position , Bob in , 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 ), the person that leaves will stand in position

The five numbers which cause people to leave the circle are and Since there are initially people in the circle, so the person who leaves stands in position , or position .

Arn now leaves, and because Bob is the next person to say a number, we reassign the positions such that Bob is in position , Cyd in , and Fon in . There are more numbers to be said, and someone leaves on the . It follows that the person who leaves stands in position .

Cyd leaves the circle, and people remain, namely Bob, Dan, Eve, and Fon. Assign the position numbers such that Dan has , Eve , etc. There are more numbers to be said, and someone leaves on the . The person standing in position (Fon) now leaves.

There are two people left, Eve and Dan, with Dan in position now. There are more numbers to be said, and Eve is standing in position , so she leaves.

The last person standing is .