2024 AMC 8 Problem 14

Question

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

Answer: A

Concepts:graph theory

Difficulty rating: 1420

Small hint:

Track the shortest distance from

A

to each town

Big hint:

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

Video solution:

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Written solution:

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.