2012 AMC 8 Problems/Problem 14
Problem
In the BIG N, a middle school football conference, each team plays every other team exactly once. If a total of 21 conference games were played during the 2012 season, how many teams were members of the BIG N conference?
Solution 1
This problem is very similar to a handshake problem. We use the formula 
to usually find the number of games played (or handshakes). Now we have to use the formula in reverse.
So we have the equation 
. Solving, we find that the number of teams in the BIG N conference is 
.
Solution 2
(If someone understands what I'm trying to do here and can explain it better, please edit it)We know that every team has to play a game with every other team, so we just need to find out how many consecutive numbers, 
to 
, can fit into 21. We know that 
, and since this doesn't count to 
team that shook hands with the other 
, we know that there are teams in the BIG N conference.
Video Solution
https://youtu.be/zzU98Bk1TrE ~savannahsolver
See Also
| 2012 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. 
