2025 SSMO Team Round Problems/Problem 15

Problem

The circles $\omega_1$ and $\omega_2$ have radii $8$ and $7$ , respectively, and intersect at points $A$ and $B$ . A line $\ell$ passing through $A$ intersects $\omega_1$ again at $P$ and $\omega_2$ again at $Q$ , with $PQ = 24$ . There exists a point $T$ on line $AB$ , with $A$ between $T$ and $B$ , such that $PT$ is tangent to $\omega_1$ and $QT$ is tangent to $\omega_2$ . The length of $BT$ can be written as $\frac{m}{\sqrt{n}}$ , where $m$ and $n$ are positive integers such that $n$ is square-free. Find $m+n$ .

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2025 SSMO Team Round Problems/Problem 15

Problem

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