2017 AIME I Problems/Problem 10

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Problem 10

Let $z_1=18+83i,~z_2=18+39i,$ and $z_3=78+99i,$ where $i=\sqrt{-1}.$ Let $z$ be the unique complex number with the properties that $\frac{z_3-z_1}{z_2-z_1}~\cdot~\frac{z-z_2}{z-z_3}$ is a real number and the imaginary part of $z$ is the greatest possible. Find the real part of $z$.

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 $\frac{z_3 - z_1}{z_3 - z_2}$ be some imaginary number with form $r_1 (\cos \theta_1 + i \sin \theta_1).$ Similarly, we can write $\frac{z-z_1}{z-z_2}$ as some $r_2 (\cos \theta_2 + i \sin \theta_2).$

The product must be real, so we have that $r_1 r_2 (\cos \theta_1 + i \sin \theta_1) (\cos \theta_2 + i \sin \theta_2)$ is real. Of this, $r_1 r_2$ must be real, so the imaginary parts only arise from the second part of the product. Thus we have

\[(\cos \theta_1 + i \sin \theta_1) (\cos \theta_2 + i \sin \theta_2) = \cos \theta_1 \cos \theta_2 - \sin \theta_1 \sin \theta_2 + i(\cos \theta_1 \sin \theta_2 + \cos \theta_2 \sin \theta_1)\]

is real. The imaginary part of this is $(\cos \theta_1 \sin \theta_2 + \cos \theta_2 \sin \theta_1),$ which we recognize as $\sin(\theta_1 + \theta_2).$ This is only $0$ when $\theta_1 + \theta_2$ is some multiple of $\pi.$ In this problem, this implies $z_1, z_2, z_3$ and $z$ must form a cyclic quadrilateral, so the possibilities of $z$ lie on the circumcircle of $z_1, z_2$ and $z_3.$

To maximize the imaginary part of $z,$ it must lie at the top of the circumcircle, which means the real part of $z$ 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 $56,$ so the real part of $z$ is $56,$ and thus our answer is $056.$

See Also

2017 AIME I (ProblemsAnswer KeyResources)
Preceded by
Problem 7
Followed by
Problem 9
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions