2024 AMC 8 第 14 题

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.