2016 AIME I Problems/Problem 7
Problem
For integers 
and 
consider the complex number
![\[\frac{\sqrt{ab+2016}}{ab+100}-({\frac{\sqrt{|a+b|}}{ab+100}})i\]](http://latex.artofproblemsolving.com/9/4/a/94aed5cb77ab692adad418e0f67918ea8fe39dc6.png)
Find the number of ordered pairs of integers 
such that this complex number is a real number.
== Solution == We consider two cases:
Case 1: 
In this case, if
![\[0 = \text{Im}({\frac{\sqrt{ab+2016}}{ab+100}-({\frac{\sqrt{|a+b|}}{ab+100}})i}) = {\frac{\sqrt{|a+b|}}{ab+100}}\]](http://latex.artofproblemsolving.com/e/5/1/e517c33fee3f76e9a44eaba190c938a383142499.png)
then 
and 
. Thus 
so 
. Thus 
, yielding 
values. However since 
,
See also
| 2016 AIME I (Problems • Answer Key • Resources) | ||
| Preceded by Problem 6 |
Followed by Problem 8 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
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