Difference between revisions of "2025 AIME I Problems/Problem 8"
(→Problem) |
|||
| Line 1: | Line 1: | ||
==Problem== | ==Problem== | ||
Let <math>k</math> be a real number such that the system | Let <math>k</math> 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 <math>z</math>. The sum of all possible values of <math>k</math> can be written as <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m + n</math>. Here <math>i = \sqrt{-1}</math>. | has exactly one complex solution <math>z</math>. The sum of all possible values of <math>k</math> can be written as <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m + n</math>. Here <math>i = \sqrt{-1}</math>. | ||
| + | |||
==Solution 1== | ==Solution 1== | ||
Revision as of 18:28, 13 February 2025
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 
.
