2024 SSMO Team Round Problems/Problem 5

Problem

Let $ABC$ be a triangle with $AB=AC=5$ and $BC=6$ . Let $\omega_1$ be the circumcircle of $ABC$ and let $\omega_2$ be the circle externally tangent to $\omega_1$ and tangent to rays $AB$ and $AC$ . If the distance between the centers of $\omega_1$ and $\omega_2$ can be expressed as $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers, find $m+n$ .

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2024 SSMO Team Round Problems/Problem 5

Problem

Solution