2002 AMC 12P Problems/Problem 22
- The following problem is from both the 2002 AMC 12P #22 and 2002 AMC 10P #25, so both problems redirect to this page.
Contents
[hide]Problem
Under the new AMC scoring method, points are given for each correct answer, points are given for each unanswered question, and no points are given for an incorrect answer. Some of the possible scores between and can be obtained in only one way, for example, the only way to obtain a score of is to have correct answers and one unanswered question. Some scores can be obtained in exactly two ways, for example, a score of can be obtained with correct answers, unanswered question, and incorrect, and also with correct answers and unanswered questions. There are (three) scores that can be obtained in exactly three ways. What is their sum?
Solution 1
Let denote the amount of correct answers, denote the amount of unanswered questions, and denote the amount of wrong answers. The conversion rate is what differentiates between two ways to get the same score.
There are two ways to realize the conversion rate of wrong answers to unanswered questions and correct answers to unanswered questions.
Sub Solution 1.1 (Cheese)
Notice how the way to get points directly tells us the conversion rate. correct questions, unanswered question, and wrong answers is equivalent to correct answers, unanswered questions, and wrong answers. Since and , the conversion rate is
Sub Solution 1.2
The second way to find out the conversion rate is to actually do the math. If a wrong answer is converted to an unanswered question, the total value will be more than the total value. If a correct answer is converted to an unanswered question, the total value will be less than the total value. Thus, we want the "more than" to be equal to the "less than".
Let denote the amount of correct answers converted to unanswered questions and denote the amount of wrong answers converted to unanswered questions. Since we want "more than" to be equal to the "less than", , where and are integers. Multiply by on both sides, implies and Again, the conversion rate is
Sub Solutions Combine
Either way, we find If there are three ways, this implies the first way the second way is and the third way is Since it follows that our three cases' first ways are as follows:
correct answers, unanswered questions, wrong answers
correct answers, unanswered questions, wrong answers
correct answers, unanswered questions, wrong answers.
Thus, our answer is
Solution 2
This solution is formulated in formal language rather than cheesy methods, though the idea is the same. Additionally, we do not need to take into account the amount of wrong answers, which is easier to deal with.
Let denote the amount of correct answers, denote the amount of unanswered questions. Then We want to find the LCM of and so we can convert between different ways, similar to solution The LCM is equal to , since and
The first way will take the form The second way will take the form The third way will take the form In particular, since we cannot have a negative amount of unanswered questions, Combined with , our three cases are and
Thus, our answer is
Note
The conversion rate can either be or Both solutions use a different conversion rate to illustrate this point. The first solution defines the "first way" as the one with the most correct, while the second solution defines the "first way" as the one with the most unanswered.
See also
2002 AMC 10P (Problems • Answer Key • Resources) | ||
Preceded by Problem 24 |
Followed by Last question | |
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 |
2002 AMC 12P (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 12 Problems and Solutions |
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