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2022 SSMO Team Round Problems/Problem 15

Revision as of 12:07, 14 December 2023 by Pinkpig (talk | contribs) (Created page with "==Problem== Consider two externally tangent circles <math>\omega_1</math> and <math>\omega_2</math> with centers <math>O_1</math> and <math>O_2</math>. Suppose that <math>\ome...")
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Problem

Consider two externally tangent circles $\omega_1$ and $\omega_2$ with centers $O_1$ and $O_2$. Suppose that $\omega_1$ and $\omega_2$ have radii of $1$ and $3$ respectively. There exist points $A, B$ on $\omega_1$ and points $C, D$ on $\omega_2$ such that $AC$ and $BD$ are the external tangents of $\omega_1$ and $\omega_2$. The circumcircle of $\triangle BO_2D$ intersects $AC$ at two points $X$ and $Y$ such that $AX < AY$. If $CX$ can be expressed as $\frac{\sqrt{m}}{n}$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.

Solution

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