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Kineska matura | Chinese Final Exam #5

Trokut ABC upisan je u kružnicu \omega pri čemu AB = 5, BC = 7 i AC = 3. Simetrala unutarnjeg kuta A siječe BC u D i kružnicu drugi put \omega u točki E. Neka je \gamma kružnica promjera DE. Kružnice \omega i \gamma sijeku se u E i F. Imamo AF^2 = \frac mn, gdje m i n su relativno prosti prirodni brojevi. Nađi m + n.
In the triangle ABC, which is inscribed in the circle \omega, it is known that AB = 5, BC = 7 and AC = 3. The two points D and E are the points where the bisector of angle A intersects side BC and circle \omega, respectively. Let \gamma be the circle with a diameter DE. The circles \omega and \gamma intersect at points E and F. Additionally, AF^2 = \frac mn, where m and n are relatively prime natural numbers. What is the value of m+n?