Difference between revisions of "1991 AJHSME Problems/Problem 14"
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Revision as of 23:07, 4 July 2013
Problem
Several students are competing in a series of three races. A student earns 
points for winning a race, 
points for finishing second and 
point for finishing third. There are no ties. What is the smallest number of points that a student must earn in the three races to be guaranteed of earning more points than any other student?
Solution
There are two ways for a student to get 
: 
and 
. Clearly if someone gets one of these combinations someone else could get the other, so we are not guaranteed the most points with .
There is only one way to get 
points: 
. In this case, the largest score another person could get is 
, so having points guarantees having more points than any other person 
.
See Also
| 1991 AJHSME (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. 
