2016 AIME I Problems/Problem 7
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 |
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