1991 AMC 8 第 14 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。
14.
几名学生参加三场赛跑。每场比赛第一名得 分，第二名得 分，第三名得 分。没有并列名次。一个学生在三场比赛中至少要得到多少分，才能保证他的总分高于任何其他学生？
Several students are competing in a series of three races. A student earns points for winning a race, points for finishing second, and point for finishing third. There are no ties. What is the smallest number of points that a student must earn in the three races to be guaranteed of earning more points than any other student?
答案：D
小提示：
要保证领先，要考虑最坏情况：另一个学生也尽可能得高分
To be guaranteed the lead, plan for the worst case where another student does as well as possible
大提示：
依次检查可能的三场总分；对每个总分，判断剩余名次是否允许另一名学生追平或超过它
Check the attainable three-race totals in increasing order; for each one, see whether the remaining places allow a rival to tie or exceed it
解答：
总分 并不能保证第一，例如 可能被另一个学生也得到 分。
但如果一个学生得到 分，那么剩下的名次使任何其他学生最多得到 分。因此 分能保证领先。
所以正确答案是 D。
A total of (for example ) does not guarantee first place, since another student could also reach
But if one student scores the remaining places give every other student at most So points guarantees the lead.
Thus, the correct answer is D .
其他年份的第 14 题
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