2025 SSMO Accuracy Round Problems/Problem 5

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 $m+n$ .

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2025 SSMO Accuracy Round Problems/Problem 5

Problem

Solution