AMC8 2005

AMC8 2005 · Q20

AMC8 2005 · Q20. It mainly tests Patterns & sequences (misc), Remainders & modular arithmetic.

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?

爱丽丝和鲍勃玩一个游戏，圆周上均匀分布12个点，从1到12顺时针编号。两人从点12开始。爱丽丝顺时针移动，鲍勃逆时针移动。每回合，爱丽丝顺时针移动5点，鲍勃逆时针移动9点。游戏在他们停在同一点时结束。需要多少回合？

(A) 6 6

(B) 8 8

(D) 14 14

(E) 24 24

Answer

Correct choice: (A)

正确答案：（A）

Solution

(A) Write the points where Alice and Bob will stop after each move and compare points. \[ \begin{array}{c|ccccccc} \text{Move} & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \text{Alice:} & 12 & 5 & 10 & 3 & 8 & 1 & 6 \\ \text{Bob:} & 12 & 3 & 6 & 9 & 12 & 3 & 6 \end{array} \] So Alice and Bob will be together again after six moves.

（A）写出 Alice 和 Bob 每次移动后停下的点，并比较这些点。 \[ \begin{array}{c|ccccccc} \text{移动次数} & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \text{Alice：} & 12 & 5 & 10 & 3 & 8 & 1 & 6 \\ \text{Bob：} & 12 & 3 & 6 & 9 & 12 & 3 & 6 \end{array} \] 因此，Alice 和 Bob 在移动六次后会再次相遇。

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