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AIME 2025 I / Let k be real numbers such that the system |25+20i−z|=5 and…

Problem

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question

Let kk be real numbers such that the system |25+20i−z|=5|25+20i-z|=5 and |z−4−k|=|z−3i−k||z-4-k|=|z-3i-k| has exactly one complex solution zz. The sum of all possible values of kk can be written as mn\frac{m}{n}, where mm and nn are relatively prime positive integers. Find m+nm+n. Here i=−1i=\sqrt{-1}.
Plain-text mathematical notation (without MathML)
Let k be real numbers such that the system |25+20i−z|=5 and |z−4−k|=|z−3i−k| has exactly one complex solution z. The sum of all possible values of k can be written as (m)/(n), where m and n are relatively prime positive integers. Find m+n. Here i=√(−1).
Original LaTeX notation
Let $k$ be real numbers such that the system $|25+20i-z|=5$ and $|z-4-k|=|z-3i-k|$ has exactly one complex solution $z$. The sum of all possible values of $k$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. Here $i=\sqrt{-1}$.

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Source and history

task_id
cc633a4e-f53d-53be-87a0-280ea4b78388
task_key
AIME2025~2dI--test--cc633a4e-f53d-53be-87a0-280ea4b78388
task_revision_id
3
config / split
AIME2025-I / test

Official source

initial import