Difference between revisions of "2024 AMC 8 Problems/Problem 14"
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We can simply see that path <math>A \rightarrow X \rightarrow M \rightarrow Y \rightarrow C \rightarrow Z</math> will give us the smallest value. Adding, <math>5+2+6+5+10 = \boxed{28}</math>. This is nice as it’s also the smallest value, solidifying our answer. | We can simply see that path <math>A \rightarrow X \rightarrow M \rightarrow Y \rightarrow C \rightarrow Z</math> will give us the smallest value. Adding, <math>5+2+6+5+10 = \boxed{28}</math>. This is nice as it’s also the smallest value, solidifying our answer. | ||
| − | ~MaxyMoosy | + | You can also simply brute-force it or sort of think ahead - for example, getting from A to M can be done <math>2</math> ways; <math>A \rightarrow X \rightarrow M</math> (<math>5+2</math>) or <math>A \rightarrow M (8)</math>, so you should take the shorter route (<math>5+2</math>). Another example is M to C, two ways - one is <math>6+5</math> and the other is <math>14</math>. Take the shorter route. After this, you need to consider a few more times - consider if <math>5+10</math> (<math>Y \rightarrow C \rightarrow Z</math>) is greater than <math>17 (Y \rightarrow Z</math>), which it is not, and consider if <math>25 (M \rightarrow Z</math>) is greater than <math>14+10</math> (<math>M \rightarrow C \rightarrow Z</math>) or <math>6+5+10</math> (<math>M \rightarrow Y \rightarrow C \rightarrow Z</math>) which it is not. TL;DR: <math>5+2=6=5=10 = \boxed{28}</math>. [Note: This is probably just the thinking behind the solution.] {Double-note: As MaxyMoosy said, since this answer is the smallest one, it has to be the right answer.} |
| + | |||
| + | ~MaxyMoosy ~HACKER2022 | ||
==Video Solution 1 (easy to digest) by Power Solve== | ==Video Solution 1 (easy to digest) by Power Solve== | ||
Revision as of 20:12, 29 January 2024
Contents
Problem
The one-way routes connecting towns 
and 
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 form A to Z in kilometers?
Solution 1
We can simply see that path 
will give us the smallest value. Adding, 
. 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 
ways; 
(
) or 
, so you should take the shorter route (). Another example is M to C, two ways - one is 
and the other is 
. Take the shorter route. After this, you need to consider a few more times - consider if 
(
) is greater than 
), which it is not, and consider if 
) is greater than 
(
) or 
(
) which it is not. TL;DR: 
. [Note: This is probably just the thinking behind the solution.] {Double-note: As MaxyMoosy said, since this answer is the smallest one, it has to be the right answer.}
~MaxyMoosy ~HACKER2022
Video Solution 1 (easy to digest) by Power Solve
Video Solution 2 by SpreadTheMathLove
https://www.youtube.com/watch?v=RRTxlduaDs8
Video Solution 3 by NiuniuMaths (Easy to understand!)
https://www.youtube.com/watch?v=V-xN8Njd_Lc
~NiuniuMaths
Video Solution by CosineMethod [🔥Fast and Easy🔥]
https://www.youtube.com/watch?v=bAxLRYT6SCw
See Also
| 2024 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 13 |
Followed by Problem 15 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AJHSME/AMC 8 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
