Difference between revisions of "2016 AMC 10B Problems/Problem 22"
m (→Problem) |
|||
Line 1: | Line 1: | ||
==Problem== | ==Problem== | ||
− | A set of teams held a round- | + | A set of teams held a round-robn tournament in which every team played every other team exactly once. Every team won <math>10</math> games and lost <math>10</math> games; there were no ties. How many sets of three teams <math>\{A, B, C\}</math> were there in which <math>A</math> beat <math>B</math>, <math>B</math> beat <math>C</math>, and <math>C</math> beat <math>A?</math> |
<math>\textbf{(A)}\ 385 \qquad | <math>\textbf{(A)}\ 385 \qquad |
Revision as of 11:46, 8 January 2017
Problem
A set of teams held a round-robn tournament in which every team played every other team exactly once. Every team won games and lost
games; there were no ties. How many sets of three teams
were there in which
beat
,
beat
, and
beat
Solution
There are teams. Any of the
sets of three teams must either be a fork (in which one team beat both the others) or a cycle:
But we know that every team beat exactly
other teams, so for each possible
at the head of a fork, there are always exactly
choices for
and
. Therefore there are
forks, and all the rest must be cycles.
Thus the answer is which is
.
See Also
2016 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 21 |
Followed by Problem 23 | |
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 AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.