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2025 SSMO Accuracy Round Problems/Problem 5

Revision as of 11:26, 9 September 2025 by Pinkpig (talk | contribs) (Created page with "==Problem== <math>ABC</math> is an isosceles triangle with base <math>BC = 6</math> and <math>AB = AC</math>. Point <math>M</math> is the midpoint of <math>BC</math> such tha...")
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Problem

$ABC$ is an isosceles triangle with base $BC = 6$ and $AB = AC$. Point $M$ is the midpoint of $BC$ such that $AM = 9$. Circle $\omega_1$ is the circumcircle of $ABC$ with radius $R,$ and $\omega_2$ is a circle passing through $B$ and $C$ with radius $2R$ and center on the opposite side of $BC$ as $A$. Segment $AM$ intersects $\omega_2$ at point $X$ and $\omega_1$ at point $Y,$ where $X$ lies between $A$ and $Y$. The length $XY$ can be expressed as $m - \sqrt{n},$ where $m$ and $n$ are positive integers. Find $m+n$.

Solution

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