Difference between revisions of "2014 AMC 12A Problems/Problem 19"
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Revision as of 12:00, 7 February 2014
Problem
There are exactly 
distinct rational numbers 
such that 
and ![\[5x^2+kx+12=0\]](http://latex.artofproblemsolving.com/2/a/3/2a3083567ef6e4c63ddf113aad1c6efad4e79b71.png)
has at least one integer solution for 
. What is ?
Solution
Factor the quadratic into
![\[\left(5x + \frac{12}{n}\right)\left(x + n\right) = 0\]](http://latex.artofproblemsolving.com/f/e/5/fe5199e9f960b0c62e2ab49965e0ff96a0bd31f6.png)
where 
is our integer solution. Then,
![\[k = \frac{12}{n} + 5n,\]](http://latex.artofproblemsolving.com/3/e/b/3eb94d8ebb2ae4384f3d40a12c2ae537ea961970.png)
which takes rational values between 
and 
when 
, excluding 
. This leads to an answer of 
.
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 | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
