2005 AMC 8 第 20 题
先试着解答 2005 AMC 8 第 20 题，然后核对你的答案与精心整理的解答，解答来自 LIVE by Po-Shen Loh。你也可以参加完整限时模拟考试、查看全部 2005 AMC 8 解答，或核对答案。
所有题目均经美国数学协会（MAA）官方合法授权使用。
20.
爱丽丝和鲍勃在一个圆上玩游戏，圆周上有 个等距点。这些点按顺时针方向编号为 到 。两人都从点 开始。爱丽丝顺时针移动，鲍勃逆时针移动。每轮游戏中，爱丽丝顺时针移动 个点，鲍勃逆时针移动 个点。当他们停在同一点时游戏结束。需要多少轮？
Alice and Bob play a game involving a circle whose circumference is divided by equally-spaced points. The points are numbered clockwise, from to Both start on point Alice moves clockwise and Bob, counterclockwise. In a turn of the game, Alice moves points clockwise and Bob moves points counterclockwise. The game ends when they stop on the same point. How many turns will this take?
小提示：
跟踪爱丽丝和鲍勃在圆上的相对运动
Track the relative motion of Alice and Bob around the circle.
大提示：
每轮两人的相对位移为 个点，按模 计算
Together they close the gap by points each turn, modulo .
解答：
一轮后，爱丽丝顺时针移动 个点，鲍勃逆时针移动 个点。他们每轮的相对位移为 个点。
当 是 的倍数时他们相遇，其中 是轮数。
因为 ，使 成为 的倍数的最小正整数 是 。
所以正确答案是 A。
After one turn, Alice has moved points clockwise and Bob has moved points counterclockwise. Their relative movement is points per turn.
They meet when is a multiple of , where is the number of turns.
Since , the smallest positive making a multiple of is .
Thus, A is the correct answer.
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