2025 AMC 8 Problems/Problem 17
Contents
Problem
In the land of Markovia, there are three cities: \( A \), \( B \), and \( C \). There are 100 people who live in \( A \), 120 who live in \( B \), and 160 who live in \( C \). Everyone works in one of the three cities, and a person may work in the same city where they live. In the figure below, an arrow pointing from one city to another is labeled with the fraction of people living in the first city who work in the second city. (For example, \( \frac{1}{4} \) of the people who live in \( A \) work in \( B \).) How many people work in \( A \)?
![[asy] /* AMC8 P17 2025, by NUMANA: BUI VAN HIEU, buivanhieu@husc.edu.vn, https://husc.edu.vn */ import graph; unitsize(2cm); real r=0.15; pair A, B, C;B = (0,0);C = (2,0);A = (1,sqrt(3)); // Drawing the nodes draw(circle(A,r)); label("$A$", A); draw(circle(B,r)); label("$B$", B); draw(circle(C,r)); label("$C$", C); guide AB=A+r*dir(-135)..{down}B+r*dir(90), BA=B+r*dir(60)..{up}A+r*dir(-105), BC=B+r*dir(0)..(1,-0.2)..C+r*dir(180), CB=C+r*dir(150)..(1,0.3)..B+r*dir(30), CA=C+r*dir(90){up}..A+r*dir(-45), AC=A+r*dir(-75){down}..C+r*dir(120); draw(AB,L=Label("$1/4$", MidPoint, W),Arrow(HookHead));draw(BA,L=Label("$1/3$", MidPoint, W),Arrow(HookHead));draw(BC,L=Label("$1/6$", MidPoint, S),Arrow(HookHead));draw(CB,L=Label("$1/10$", MidPoint, S),Arrow(HookHead)); draw(CA,L=Label("$1/8$", MidPoint, E),Arrow(HookHead));draw(AC,L=Label("$1/5$", MidPoint, E),Arrow(HookHead)); [/asy]](http://latex.artofproblemsolving.com/f/a/3/fa3520958eb90baa0579f62a228df0b84766ac6b.png)
Solution 1
people do not work in city 
that live in city , meaning 
people that live in city work in city . 
people that live in 
work in and 
people that live in 
work in , so the answer is 
.
~ alwaysgonnagiveyouup
Remark
This model is known as the Markov Chain, a type of stochastic process that models systems where the next state depends only on the current state, not on the sequence of events that preceded it. This is known as the Markov property (memoryless property).
You're welcome for uploading the image. It might be a little big or blurry. If anyone could help fix it, thanks. -- leafy
Video Solution by Pi Academy
https://youtu.be/Iv_a3Rz725w?si=E0SI_h1XT8msWgkK
Video Solution (A Clever Explanation You’ll Get Instantly)
https://youtu.be/VP7g-s8akMY?si=fV-dPbMPVzWTkSV3&t=2020 ~hsnacademy
Video Solution by Thinking Feet
See Also
| 2025 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 16 |
Followed by Problem 18 | |
| 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 | ||
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