1970 AMC 12 第 32 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。
32.
和 从圆形跑道上一对直径相对的点同时出发，以匀速沿相反方向绕跑道行进。他们在 行进 码后第一次相遇，并在 距离完成一圈还有 码时第二次相遇。则跑道的周长（单位为码）是：
and travel around a circular track at uniform speeds in opposite directions, starting from diametrically opposite points. If they start at the same time, meet first after has travelled yards, and meet a second time yards before completes one lap, then the circumference of the track in yards is:
小提示：
设跑道周长为 ，比较两次相遇时两人的路程之比
Let the circumference be and compare the two runners’ traveled-distance ratios at each meeting
大提示：
第一次相遇时两人的路程为 和 ；第二次相遇时为 和
At the first meeting their distances are and ; at the second they are and
解答：
设跑道周长为 。第一次相遇时， 已行进 码， 已行进 ，所以他们的速度之比为 第二次相遇时， 距离完成一圈还差 码，所以它已行进 。此时两人的路程之和为 ，所以 已行进 。因此 交叉相乘可得 ，所以跑道周长为 码。
因此，正确答案是 C。
Let the circumference be At the first meeting, has traveled yards and has traveled so their speed ratio is At the second meeting, is yards short of one lap, so it has traveled By then their combined distance is so has traveled Hence Cross-multiplying gives so the circumference is yards.
Therefore, the correct answer is C.