1960 AMC 12 第 34 题
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所有题目均经美国数学协会（MAA）官方合法授权使用。
34.
两名游泳者分别位于一个长 英尺的泳池两端，同时开始沿泳池长度方向游泳，一人的速度为每秒 英尺，另一人的速度为每秒 英尺。他们来回游了 分钟。假设转身不耗费时间，求他们相互经过的次数。
Two swimmers, at opposite ends of a -foot pool, start to swim the length of the pool, one at the rate of feet per second, the other at feet per second. They swim back and forth for minutes. Allowing no loss of time at the turns, find the number of times they pass each other.
小提示：
每当游泳者转身时，将泳池作镜像延拓，使两条路径都变成直线
Reflect the pool each time a swimmer turns so both paths become straight
大提示：
两人的位置变化共同以 秒为周期；检查一个完整周期，并计入端点处的位置重合
The joint position pattern repeats every seconds; inspect one full cycle, including endpoint coincidences
解答：
速度较快者的位置每 秒重复一次，速度较慢者的位置每 秒重复一次。因此，两人的位置变化共同以 秒为周期。在一个周期内，他们的位置在 秒时重合。原题答案采用的计数方式把 秒时两人在端点同时转身也计作一次相遇。因此每个周期计 次， 分钟内共有 个周期，得到 。
因此，正确答案是 C。
The faster swimmer’s position repeats every seconds, and the slower swimmer’s position repeats every seconds. Thus their joint position pattern repeats every seconds. During one such cycle, their positions coincide at seconds. The keyed interpretation counts the simultaneous turn at the endpoint when as one of these encounters. Thus there are counted encounters per cycle and cycles in minutes, giving
Therefore, the correct answer is C.