2025 AIME I Problems/Problem 8
Problem
Let 
be a real number such that the system
\begin{align*}
&|25 + 20i - z| = 5 \\
&|z - 4 - k| = |z - 3i - k|
\end{align*}
has exactly one complex solution 
. The sum of all possible values of can be written as 
, where 
and 
are relatively prime positive integers. Find 
. Here 
.
Solution 1
![[asy] size(300); draw((0, 0) -- (0, 20), EndArrow(10)); label("$y$", (0, 20), NW); dot((25,20)); draw((0, 0) -- (25, 0), EndArrow(10)); label("$x$", (25, 0), SE); draw(circle((25,20),5)); label(scale(0.7)*"$(25,20)$", (25,20), S); draw((7,0) -- (3,3), blue); draw((5,3/2) -- (21,23), dashed); label("$(4+k,0)$", (7,0), S); label("$(k,3)$", (3,3), N); draw(rightanglemark((3,3),(5,3/2),(21,23), 20)); draw(rightanglemark((25,20),(21,23),(5,3/2), 20)); draw((25,20) -- (21,23)); [/asy]](http://latex.artofproblemsolving.com/5/5/b/55b4293d356e1004283f0668760ab19ef31280a5.png)
The complex number must satisfy the following conditions on the complex plane:
The magnitude between and 
is 
This can be represented by drawing a circle with center and radius
It is equidistant from the points 
and 
Hence it must lie on the perpendicular bisector of the line connecting these points.
For to have one solution, the perpendicular bisector of the segment connecting the two points must be tangent to the circle.
This bisector must pass the midpoint, 
and have slope 
The segment connecting the point of tangency to the center of the circle has slope 
meaning the points of tangency can be 
or 
Solving the equation for the slope of the perpendicular bisector gives ![\[\frac{\frac{3}{2}-23}{k+2-21}=\frac{4}{3}\]](http://latex.artofproblemsolving.com/7/9/d/79daae48af26f2a59fbee0a959646a1f97a2a5be.png)
or ![\[\frac{\frac{3}{2}-17}{k+2-29}=\frac{4}{3},\]](http://latex.artofproblemsolving.com/6/b/9/6b991387e76aa0f7ef73984385688104ec4ad1ac.png)
giving 
or 
, having a sum of 
~nevergonnagiveup
There's actually an easier way to do it using this method by utilizing the distance between point and line formula after building off of what is shown above.
First we find the standard form of the perpendicular bisector, which can be found using the point-slope form: 
-
= (
-
), where and are the and coordinates of a point on the line. By plugging in 
k
, we get -\frac{3}{2})
x
k
-
-7-8
(25, 20)
d$= \frac{abs(8*25-6*20-8$ (Error compiling LaTeX. Unknown error_msg)k$-7)}{sqrt(6^2+8^2)}. Plugging in$ (Error compiling LaTeX. Unknown error_msg)d
k
k
k
(Sorry for the bad LaTeX I'm relatively new, Please someone help)
~Mathycoder
See also
| 2025 AIME I (Problems • Answer Key • Resources) | ||
| 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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
