2013 AMC 12A Problems/Problem 25
Problem
Let be defined by
. How many complex numbers
are there such that
and both the real and the imaginary parts of
are integers with absolute value at most
?
Solution
Suppose . We look for
with
such that
are integers where
.
First, use the quadratic formula:
Generally, consider the imaginary part of a radical of a complex number: , where
.
.
Now let , then
,
,
.
Note that if and only if
. The latter is true only when we take the positive sign, and that
,
or ,
, or
.
In other words, for all ,
satisfies
, and there is one and only one
that makes it true. Therefore we are just going to count the number of ordered pairs
such that
,
are integers of magnitude no greater than
, and that
.
When , there is no restriction on
so there are
pairs;
when , there are
pairs.
So there are in total.
See also
2013 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 24 |
Followed by Last Question |
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