Difference between revisions of "2016 AIME I Problems/Problem 7"
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Revision as of 17:02, 5 March 2016
Problem
For integers and
consider the complex number
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
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
Squaring, we have the equations
(which always holds in this case) and
Then if
and
, let
. If
,
Note that
for every one of these solutions. If
, then
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.