Question

Alice and Bob play a game involving a circle whose circumference is divided by $12$ equally-spaced points. The points are numbered clockwise, from $1$ to $12.$ Both start on point $12.$ Alice moves clockwise and Bob, counterclockwise. In a turn of the game, Alice moves $5$ points clockwise and Bob moves $9$ points counterclockwise. The game ends when they stop on the same point. How many turns will this take?

$6$

$8$

$12$

$14$

$24$

Solution:

After one turn, Alice has moved $5$ points clockwise and Bob has moved $9$ points counterclockwise. Their relative movement is $5+9=14$ points per turn.

They meet when $14k$ is a multiple of $12$ , where $k$ is the number of turns.

Since $14k\equiv2k\pmod{12}$ , the smallest positive $k$ with $12\mid2k$ is $6$ .

Thus, A is the correct answer.

2005 AMC 8 Problem 20

Solution:

Problem 20 in Other Years