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) | |
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