2023 AMC 8 Problems/Problem 13
Problem
Along the route of a bicycle race, 
water stations are evenly spaced between the start and finish lines,
as shown in the figure below. There are also 
repair stations evenly spaced between the start and
finish lines. The 
rd water station is located miles after the 
st repair station. How long is the race
in miles?
![[asy] usepackage("mathptmx"); size(10cm); filldraw((11,4.5)--(171,4.5)--(171,17.5)--(11,17.5)--cycle,mediumgray*0.4 + lightgray*0.6); draw((11,11)--(171,11),linetype("2 2")+white+linewidth(1.2)); draw((0,0)--(11,0)--(11,22)--(0,22)--cycle); draw((171,0)--(182,0)--(182,22)--(171,22)--cycle); draw((31,4.5)--(31,0)); draw((51,4.5)--(51,0)); draw((151,4.5)--(151,0)); label(scale(.85)*rotate(45)*"Water 1", (23,-13.5)); label(scale(.85)*rotate(45)*"Water 2", (43,-13.5)); label(scale(.85)*rotate(45)*"Water 7", (143,-13.5)); filldraw(circle((103,-13.5),.2)); filldraw(circle((98,-13.5),.2)); filldraw(circle((93,-13.5),.2)); filldraw(circle((88,-13.5),.2)); filldraw(circle((83,-13.5),.2)); label(scale(.85)*rotate(90)*"Start", (5.5,11)); label(scale(.85)*rotate(270)*"Finish", (176.5,11)); [/asy]](http://latex.artofproblemsolving.com/2/a/d/2ad6d08a028cea1e8b56cba7529e582c5ed59523.png)
Solution
Suppose that the race is 
miles long. The water stations are located at ![\[\frac{d}{8}, \frac{2d}{8}, \ldots, \frac{7d}{8}\]](http://latex.artofproblemsolving.com/5/d/7/5d75992c71e35e3d504ae9cbb3b9aeeb572d0fbb.png)
miles from the start, and the repair stations are located at ![\[\frac{d}{3}, \frac{2d}{3}\]](http://latex.artofproblemsolving.com/b/d/c/bdc9e9aa2443a5c15309d91a1a860ab82d30094a.png)
miles from the start.
We are given that 
from which
~apex304, SohumUttamchandani, wuwang2002, TaeKim, Cxrupptedpat, MRENTHUSIASM
Video Solution (Animated)
~Star League (https://starleague.us)
Video Solution by Magic Square
https://youtu.be/-N46BeEKaCQ?t=4439
See Also
| 2023 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 12 |
Followed by Problem 14 | |
| 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 | ||
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. Error creating thumbnail: File missing
