Marinada '25

[lang=hr] 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 $m+n$. [/lang] [lang=en] 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 $m+n$? [/lang]