Question

下图显示了连接城镇 $A$ 、 $M$ 、 $C$ 、 $X$ 、 $Y$ 和 $Z$ 的单向路线（图不按比例）。每条路线的距离以千米标出。沿这些路线行驶，从 $A$ 到 $Z$ 的最短距离是多少千米？

The one-way routes connecting towns $A,$ $M,$ $C,$ $X,$ $Y,$ and $Z$ are shown in the figure below (not drawn to scale). The distances in kilometers along each route are marked. Traveling along these routes, what is the shortest distance from $A$ to $Z$ in kilometers?

$28$

$29$

$30$

$31$

$32$

答案：A

知识点：图论

难度评级：1420

小提示：

记录从 $A$ 到每个城镇的当前最短距离

Track the shortest distance from $A$ to each town

大提示：

一旦确定到某城镇的最短距离，就用它更新从该城镇出发的路线

Once a shortest distance to a town is known, use it to update outgoing routes

视频讲解：

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文字解答：

系统地追踪到 $Z$ 的最短总距离，可以逐一考虑从 $A$ 到每个城镇的最短距离。例如，从 $A$ 到城镇 $X$ 的最短距离显然是 $5$ 千米。

接着看城镇 $M$ ：先经过城镇 $X$ 比直接从 $A$ 过去更短，所以到城镇 $M$ 的最短路长为 $7$ 千米。

到 $Y$ 时，从 $X$ 来需要 $15$ 千米，从 $M$ 来只需 $13$ 千米，所以从 $A$ 到 $Y$ 的最短距离为 $13$ 千米。

对城镇 $C$ 做同样的分析，可得从城镇 $Y$ 来最短，为 $18$ 千米。

最后到 $Z$ 可以分别从 $Y$ 、 $C$ 或 $M$ 来，对应总距离为 $30$ 、 $28$ 和 $32$ 。最短距离是 $28$ 千米，也就是从 $A$ 到 $Z$ 的答案。

所以正确答案是 A。

A systematic way of tracking the shortest overall distance to $Z$ is to consider the shortest distance to get to each town from $A$ . For instance, the shortest distance to get to town $X$ from $A$ is $5$ km, trivially.

Then, for town $M$ , going to town $X$ first will be shorter compared to going directly from $A$ , so the shortest path to town $M$ has a length of $7$ km.

For town $Y$ , it will take us $15$ km if we come from town $X$ and only $13$ km coming from $M,$ so $13$ km is the length of shortest path to $Y$ from $A$ .

Doing the same for town $C$ will give us $18$ km as the shortest distance by coming from town $Y$ .

Finally, for town $Z$ , we can either come from town $Y,$ $C,$ or $M$ . The total distance if we come from each three towns respectively would be $30,$ $28,$ and $32$ . Hence, $28$ km is the shortest distance from $A$ to $Z$ .

Thus, A is the correct answer.

2024 AMC 8 第 14 题