2019 AIME I Problems/Problem 12
Given , there are complex numbers with the property that , , and are the vertices of a right triangle in the complex plane with a right angle at . There are positive integers and such that one such value of is . Find .
We will use the fact that segments and are perpendicular in the complex plane if and only if . To prove this, when dividing two complex numbers you subtract the angle of one from the other, and if the two are perpendicular, subtracting these angles will yield an imaginary number with no real part.
Now to apply this:
The factorization of the nasty denominator above is made easier with the intuition that must be a divisor for the problem to lead anywhere. Now we know so using the fact that the imaginary part of is and calling the real part r,
solving the above quadratic yields so our answer is
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