User:Quantum-phantom

By the law of cosines, \[\cos\angle DAC=\frac{221-CD^2}{220}=\cos\angle DBC=\frac{169-CD^2}{120},\] so $CD=\sqrt{\tfrac{533}{5}}$. Similarly, $AB=\tfrac{2 \sqrt{5} \sqrt{533}}{13}$. Let $AD\cap BC=I$, $AB\cap CD=J$, $OE\cap IJ=F$, then $OF\perp IJ$ by Brocard's theorem. Since $ON\perp DC$, $OM\perp AB$, then MPNP=MOsinMOENOsinNOE=sinMOEsinNOE=sinIJAsin(πDJI)=sinIJAsinDJI=JAJIsinIJAIJDJsinDJIDJJA=[IJA][DJI]DJJA=IAIDDJJA. By the law of sines, \[\frac{IA}{ID}=\frac{IA}{IC}\cdot\frac{IC}{ID}=\frac{AB}{CD}\cdot\frac{AC}{BD}=\frac{55}{78},~\frac{DJ}{JA}=\frac{BD}{AC}=\frac{12}{11}.\] So the answer is MPNP=12115578=1013. [img]https://img.picgo.net/2024/04/07/IMG_3135a148c4fcc55ace27.jpeg[/img]