Difference between revisions of "2016 AIME I Problems/Problem 7"
m (→Solution) |
Mathkiddie (talk | contribs) |
||
| (11 intermediate revisions by 9 users not shown) | |||
| Line 1: | Line 1: | ||
==Problem== | ==Problem== | ||
For integers <math>a</math> and <math>b</math> consider the complex number | For integers <math>a</math> and <math>b</math> consider the complex number | ||
| − | <cmath>\frac{\sqrt{ab+2016}}{ab+100}-({\frac{\sqrt{|a+b|}}{ab+100}})i</cmath> | + | <cmath>\frac{\sqrt{ab+2016}}{ab+100}-\left({\frac{\sqrt{|a+b|}}{ab+100}}\right)i</cmath> |
Find the number of ordered pairs of integers <math>(a,b)</math> such that this complex number is a real number. | Find the number of ordered pairs of integers <math>(a,b)</math> such that this complex number is a real number. | ||
| Line 9: | Line 9: | ||
We consider two cases: | We consider two cases: | ||
| − | Case 1: <math>ab \ge -2016</math> | + | '''Case 1:''' <math>ab \ge -2016</math>. |
In this case, if | In this case, if | ||
| Line 15: | Line 15: | ||
then <math>ab \ne -100</math> and <math>|a + b| = 0 = a + b</math>. Thus <math>ab = -a^2</math> so <math>a^2 < 2016</math>. Thus <math>a = -44,-43, ... , -1, 0, 1, ..., 43, 44</math>, yielding <math>89</math> values. However since <math>ab = -a^2 \ne -100</math>, we have <math>a \ne \pm 10</math>. Thus there are <math>87</math> allowed tuples <math>(a,b)</math> in this case. | then <math>ab \ne -100</math> and <math>|a + b| = 0 = a + b</math>. Thus <math>ab = -a^2</math> so <math>a^2 < 2016</math>. Thus <math>a = -44,-43, ... , -1, 0, 1, ..., 43, 44</math>, yielding <math>89</math> values. However since <math>ab = -a^2 \ne -100</math>, we have <math>a \ne \pm 10</math>. Thus there are <math>87</math> allowed tuples <math>(a,b)</math> in this case. | ||
| − | Case 2: <math>ab | + | '''Case 2:''' <math>ab < -2016</math>. |
In this case, we want | In this case, we want | ||
| − | <cmath>0 = \text{Im}\left({\frac{\sqrt{ab+2016}}{ab+100}-\left({\frac{\sqrt{|a+b|}}{ab+100}}\right)i}\right) = \frac{\sqrt{ab | + | <cmath>0 = \text{Im}\left({\frac{\sqrt{ab+2016}}{ab+100}-\left({\frac{\sqrt{|a+b|}}{ab+100}}\right)i}\right) = \frac{\sqrt{-ab-2016} - \sqrt{|a+b|}}{ab+100}</cmath> |
Squaring, we have the equations <math>ab \ne -100</math> (which always holds in this case) and | Squaring, we have the equations <math>ab \ne -100</math> (which always holds in this case) and | ||
| − | <cmath>-(ab + 2016) = |a + b|.</cmath> | + | <cmath>-(ab + 2016)= |a + b|.</cmath> |
Then if <math>a > 0</math> and <math>b < 0</math>, let <math>c = -b</math>. If <math>c > a</math>, | Then if <math>a > 0</math> and <math>b < 0</math>, let <math>c = -b</math>. If <math>c > a</math>, | ||
<cmath>ac - 2016 = c - a \Rightarrow (a - 1)(c + 1) = 2015 \Rightarrow (a,b) = (2, -2014), (6, -402), (14, -154), (32, -64). </cmath> | <cmath>ac - 2016 = c - a \Rightarrow (a - 1)(c + 1) = 2015 \Rightarrow (a,b) = (2, -2014), (6, -402), (14, -154), (32, -64). </cmath> | ||
| Line 28: | Line 28: | ||
Thus, the answer is <math>87 + 16 = \boxed{103}</math>. | Thus, the answer is <math>87 + 16 = \boxed{103}</math>. | ||
| − | |||
| − | |||
== See also == | == See also == | ||
{{AIME box|year=2016|n=I|num-b=6|num-a=8}} | {{AIME box|year=2016|n=I|num-b=6|num-a=8}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Latest revision as of 17:00, 2 January 2024
Problem
For integers 
and 
consider the complex number
![\[\frac{\sqrt{ab+2016}}{ab+100}-\left({\frac{\sqrt{|a+b|}}{ab+100}}\right)i\]](http://latex.artofproblemsolving.com/e/f/a/efa8870a65344fd19bb1843e0130aa79b3485967.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
![\[0 = \text{Im}\left({\frac{\sqrt{ab+2016}}{ab+100}-\left({\frac{\sqrt{|a+b|}}{ab+100}}\right)i}\right) = -\frac{\sqrt{|a+b|}}{ab+100}\]](http://latex.artofproblemsolving.com/f/4/3/f43eaba271f440e7ef9bbf533889c9069c629d28.png)
then 
and 
. Thus 
so 
. Thus 
, yielding 
values. However since 
, we have 
. Thus there are 
allowed tuples in this case.
Case 2: 
.
In this case, we want
![\[0 = \text{Im}\left({\frac{\sqrt{ab+2016}}{ab+100}-\left({\frac{\sqrt{|a+b|}}{ab+100}}\right)i}\right) = \frac{\sqrt{-ab-2016} - \sqrt{|a+b|}}{ab+100}\]](http://latex.artofproblemsolving.com/9/a/1/9a1b459ba7a248bdd84f6733c2a78c6f697004dd.png)
Squaring, we have the equations (which always holds in this case) and
![\[-(ab + 2016)= |a + b|.\]](http://latex.artofproblemsolving.com/f/d/f/fdfbc21b53b3752e9d53febc52afea59b324d1ac.png)
Then if 
and 
, let 
. If 
,
![\[ac - 2016 = c - a \Rightarrow (a - 1)(c + 1) = 2015 \Rightarrow (a,b) = (2, -2014), (6, -402), (14, -154), (32, -64).\]](http://latex.artofproblemsolving.com/7/1/8/718187e91fc7f1b29feaedb3c779189c45d51d46.png)
Note that for every one of these solutions. If 
, then
![\[ac - 2016 = a - c \Rightarrow (a + 1)(c - 1) = 2015 \Rightarrow (a,b) = (2014, -2), (402, -6), (154, -14), (64, -32).\]](http://latex.artofproblemsolving.com/b/e/b/bebd0f411c6b1612a26a9f12a120300423a4ef55.png)
Again, for every one of the above solutions. This yields 
solutions. Similarly, if 
and 
, there are solutions. Thus, there are a total of 
solutions in this case.
Thus, the answer is 
.
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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
