Difference between revisions of "2014 AMC 12A Problems/Problem 19"
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==Solution 3== | ==Solution 3== | ||
− | Plug in <math>k=200</math> to find the upper limit. You will find the limit to be a number from <math>0<x<-1</math> and one that is just below <math>-39.</math> All the integer values from <math>-1</math> to <math>-39</math> can be attainable through some value of <math>k</math>. Since the question asks for the absolute value of <math>k</math>, we see that the answer is <math>39\cdot2 = \boxed{\textbf{( | + | Plug in <math>k=200</math> to find the upper limit. You will find the limit to be a number from <math>0<x<-1</math> and one that is just below <math>-39.</math> All the integer values from <math>-1</math> to <math>-39</math> can be attainable through some value of <math>k</math>. Since the question asks for the absolute value of <math>k</math>, we see that the answer is <math>39\cdot2 = \boxed{\textbf{(E) }78.}</math> |
iron | iron | ||
+ | |||
+ | ==Video Solution by SpreadTheMathLove== | ||
+ | https://www.youtube.com/watch?v=BoPnuYKBq30 | ||
==See Also== | ==See Also== | ||
{{AMC12 box|year=2014|ab=A|num-b=18|num-a=20}} | {{AMC12 box|year=2014|ab=A|num-b=18|num-a=20}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 20:33, 4 April 2024
Contents
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
iron
Video Solution by SpreadTheMathLove
https://www.youtube.com/watch?v=BoPnuYKBq30
See Also
2014 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 18 |
Followed by Problem 20 |
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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