2016 AMC 8 Problems/Problem 18
18. In an All-Area track meet, 
sprinters enter a 
meter dash competition. The track has 
lanes, so only sprinters can compete at a time. At the end of each race, the five non-winners are eliminated, and the winner will compete again in a later race. How many races are needed to determine the champion sprinter?
Solution
From any nth race, only 
will continue on. Since we wish to find the total number of races, a column representing the races over time is ideal.
Starting with the first race:
![\[\frac{216}{6}=36\]](http://latex.artofproblemsolving.com/7/3/7/7374e5d18f64255ab216a8e82a352a114b66f4fd.png)
![\[\frac{36}{6}=6\]](http://latex.artofproblemsolving.com/a/9/1/a916513079199978782d4485856c6baa38221974.png)
![\[\frac{6}{6}=1\]](http://latex.artofproblemsolving.com/8/8/4/884f0b91255d6b75d42375947ec42a5e307d0bf9.png)
Adding all of the numbers in the second column yields 
| 2016 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 17 |
Followed by Problem 19 | |
| 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. 
\end{align}
