Marinada '23

[lang=hr] Neka je $APQBR$ konveksan peterokut upisan u kružnicu čiji je promjer $\overline{AB}$. Tangenta na kružnicu u $Q$ siječe pravce $BP$ i $BR$ u $U$ i $V$, redom. Pretpostavimo da $\overline{AQ}$ raspolavlja $\angle UAR$ i da vrijedi $AQ=QR$. Nađi minimalnu vrijednost izraza: \[ \frac{AV}{AP} + \left( \frac{AU}{AB} \right)^2 \] Odgovor zaokruži na $3$ decimale. [/lang] [lang=en] Let $APQBR$ be a convex pentagon inscribed in a circle whose diameter is $\overline{AB}$. The tangent to the circle in $Q$ intersects the lines $BP$ and $BR$ in $U$ and $V$, respectively. Assume that $\overline{AQ}$ bisects $\angle UAR$ and $AQ=QR$. Find the minimum value of the expression: \[ \frac{AV}{AP} + \left( \frac{AU}{AB} \right)^2 \] Round the answer to $3$ decimal places. [/lang]