AMC8 2024

AMC8 2024 · Q14

AMC8 2024 · Q14. It mainly tests Graphs & networks (basic), Optimization (basic).

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?

连接城镇 $A,M,C,X,Y,$ 和 $Z$ 的一条道路线如图所示（未按比例绘制）。每条路线的公里数已标明。沿这些路线旅行，从 A 到 Z 的最短距离是多少公里？

(A) \ 28 \ 28

(B) \ 29 \ 29

(D) \ 31 \ 31

(E) \ 32 \ 32

Answer

Correct choice: (A)

正确答案：（A）

Solution

We can simply see that path $A \rightarrow X \rightarrow M \rightarrow Y \rightarrow C \rightarrow Z$ will give us the smallest value. Adding, $5+2+6+5+10 = \boxed{28}$. This is nice as it’s also the smallest value, solidifying our answer. You can also simply brute-force it or sort of think ahead - for example, getting from A to M can be done $2$ ways; $A \rightarrow X \rightarrow M$ ($5+2$) or $A \rightarrow M (8)$, so you should take the shorter route ($5+2$). Another example is M to C, two ways - one is $6+5$ and the other is $14$. Take the shorter route. After this, you need to consider a few more times - consider if $5+10$ ($Y \rightarrow C \rightarrow Z$) is greater than $17 (Y \rightarrow Z$), which it is not, and consider if $25 (M \rightarrow Z$) is greater than $14+10$ ($M \rightarrow C \rightarrow Z$) or $6+5+10$ ($M \rightarrow Y \rightarrow C \rightarrow Z$) which it is not. TLDR: $5+2+6+5+10 = \boxed{28}$.

可以看出路径 $A \rightarrow X \rightarrow M \rightarrow Y \rightarrow C \rightarrow Z$ 给出最小值。相加：$5+2+6+5+10 = \boxed{28}$。这是最小值，确认答案。也可以暴力枚举或提前思考——例如，从 A 到 M 有 2 种方式：$A \rightarrow X \rightarrow M$ ($5+2$) 或 $A \rightarrow M (8)$，取较短路径 ($5+2$)。M 到 C 有两种：$6+5$ 和 $14$，取较短。之后再考虑：$5+10$ ($Y \rightarrow C \rightarrow Z$) 与 $17 (Y \rightarrow Z$) 比较，前者更短；$25 (M \rightarrow Z$) 与 $14+10$ 或 $6+5+10$ 比较，前者更大。总之：$5+2+6+5+10 = \boxed{28}$。

Topics