2014 AMC 12A Problems/Problem 19
Problem
There are exactly distinct rational numbers such that and has at least one integer solution for . What is ?
Solution 1
Factor the quadratic into where is our integer solution. Then, which takes rational values between and when , excluding . This leads to an answer of .
Solution 2
Solve for so Note that can be any integer in the range so is rational with . Hence, there are
Solution 3
Plug in to find the upper limit. You will find the limit to be a number from and one that is just below All the integer values from to can be attainable through some value of . Since the question asks for the absolute value of , we see that the answer is
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See Also
2014 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 18 |
Followed by Problem 20 |
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All AMC 12 Problems and Solutions |
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