Difference between revisions of "1991 AJHSME Problems/Problem 14"
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There is only one way to get <math>13</math> points: <math>5+5+3</math>. In this case, the largest score another person could get is <math>5+3+3=11</math>, so having <math>13</math> points guarantees having more points than any other person <math>\rightarrow \boxed{\text{D}}</math>. | There is only one way to get <math>13</math> points: <math>5+5+3</math>. In this case, the largest score another person could get is <math>5+3+3=11</math>, so having <math>13</math> points guarantees having more points than any other person <math>\rightarrow \boxed{\text{D}}</math>. | ||
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If someone gets <math>11</math> points, the possible combinations are <math>5,5,1</math> and <math>5,3,3</math> If he gets <math>5,3,3</math> then someone else can be <math>5,3,3</math> which would not guarantee victory. | If someone gets <math>11</math> points, the possible combinations are <math>5,5,1</math> and <math>5,3,3</math> If he gets <math>5,3,3</math> then someone else can be <math>5,3,3</math> which would not guarantee victory. | ||
If we have 13 points, the only way to make this is <math>5,5,3</math>. There is no way to get any number of points higher than this, so the answer is <math>\boxed{\text{D}}</math>.---stjwyl | If we have 13 points, the only way to make this is <math>5,5,3</math>. There is no way to get any number of points higher than this, so the answer is <math>\boxed{\text{D}}</math>.---stjwyl | ||
Latest revision as of 21:33, 15 November 2022
Contents
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 
.
Solution 2
If someone gets points, the possible combinations are 
and 
If he gets then someone else can be which would not guarantee victory.
If we have 13 points, the only way to make this is 
. There is no way to get any number of points higher than this, so the answer is 
.---stjwyl
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. 
