2021 AMC 12A Fall 第 16 题
先试着解答 2021 AMC 12A Fall 第 16 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 2021 AMC 12A Fall 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。
16.
一个组织有 名员工，其中 人使用 A 品牌电脑，另外 人使用 B 品牌电脑。出于安全原因，电脑只能彼此相连，且只能用电缆连接。电缆只能把一台 A 品牌电脑和一台 B 品牌电脑相连。如果两名员工的电脑直接由电缆相连，或可以通过一系列相连的电脑转发消息，则他们可以相互通信。最初没有任何电脑与其他电脑相连。一名技术员任意选择一台每种品牌的电脑，并在它们之间安装电缆，前提是这对电脑之间还没有电缆。技术员在每名员工都能相互通信时停止。最多可能使用多少根电缆？
An organization has employees, of whom have a brand A computer while the other have a brand B computer. For security, the computers can only be connected to each other and only by cables. The cables can only connect a brand A computer to a brand B computer. Employees can communicate with each other if their computers are directly connected by a cable or by relaying messages through a series of connected computers. Initially, no computer is connected to any other. A technician arbitrarily selects one computer of each brand and installs a cable between them, provided there is not already a cable between that pair. The technician stops once every employee can communicate with each other. What is the maximum possible number of cables used?
小提示：
为了尽量延迟连通，尽可能久地保持图不连通，然后最后一根电缆把所有部分连接起来
To delay connectivity, keep the graph disconnected as long as possible, then one final cable joins everything
大提示：
在仍不连通的情况下，最多边数来自 台 A 品牌和 台 B 品牌形成的完全二分图，另留一台 A 品牌电脑孤立
The most edges while still disconnected is a complete bipartite graph on brand A and brand B, leaving one brand A computer isolated
解答：
技术员不断加电缆，直到图变为连通。为了最大化电缆数，要让网络尽可能久地保持不连通：留下一台 A 品牌电脑孤立，并把其余 台 A 品牌电脑与全部 台 B 品牌电脑完全相连。
这样在仍不连通时使用了 根电缆。下一根电缆连接最后一台 A 品牌电脑，使所有人连通，总数为 。
所以正确答案是 B。
The technician keeps adding cables until the graph becomes connected. To maximize the count, keep the network disconnected for as long as possible: leave a single brand A computer isolated and fully connect the remaining brand A computers to all brand B computers.
That uses cables while still disconnected. The next cable connects the last brand A computer, joining everyone, for a total of
Thus, the correct answer is B.
其他年份的第 16 题
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