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 k=200 to find the upper limit. You will find the limit to be a number from 0<x<-1 and one that is just below -39. All the integer values from -1 to -39 can be attainable through some value of k. Since the questions asks for the absolute value of k, we see that the answer is 39*2 = 78
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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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