2017 AIME I Problems/Problem 10
Problem 10
Let and where Let be the unique complex number with the properties that is a real number and the imaginary part of is the greatest possible. Find the real part of .
Solution
(This solution's quality may be very poor. If one feels that the solution is inadequate, one may choose to improve it.)
Let us write be some imaginary number with form Similarly, we can write as some
The product must be real, so we have that is real. Of this, must be real, so the imaginary parts only arise from the second part of the product. Thus we have
is real. The imaginary part of this is which we recognize as This is only when is some multiple of In this problem, this implies and must form a cyclic quadrilateral, so the possibilities of lie on the circumcircle of and
To maximize the imaginary part of it must lie at the top of the circumcircle, which means the real part of is the same as the real part of the circumcircle. The center of the circumcircle can be found in various ways, (such as computing the intersection of the perpendicular bisectors of the sides) and when computed gives us that the real part of the circumcircle is so the real part of is and thus our answer is
See Also
2017 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 7 |
Followed by Problem 9 | |
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All AIME Problems and Solutions |