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Kineska matura | Chinese Final Exam #4

Upisana kružnica \omega od \triangle ABC dira \overline{BC} u X. Neka je Y \neq X drugo sjecište \overline{AX} sa \omega. Točke P i Q leže na \overline{AB} i \overline{AC}, redom, tako da \overline{PQ} je tangenta na \omega u Y. Pretpostavi da AP=3, PB = 4, AC=8 i AQ = \tfrac{m}{n}, gdje m i n su relativno prosti prirodni brojevi. Nađi m+n.
The inscribed circle \omega of the triangle \triangle ABC is tangent to \overline{BC} at X. Let Y \neq X be the other intersection of \overline{AX} and \omega. The points P and Q lie on \overline{AB} and \overline{AC}, respectively, so that \overline{PQ} is tangent to \omega at Y. It is known that AP=3, PB = 4, AC=8 and AQ = \tfrac{m}{n}, where m and n are relatively prime natural numbers. What is the value of m+n?