Marinada '24

[lang=hr] Na samom kraju, nakon što je odabrao $13$ djevojaka u prošlom zadatku, Salopekovoj curi je prekipjelo i odlučila je prekinuti s njim. On, vidno pogođen tom informacijom, potrudio se dobro odabrati brucošicu s kojom će biti i u dobru i u zlu dok ih smrt ne rastavi. Kako mu ljepota ipak igra neku ulogu, odlučio je elimirati $2$ brucošice koje su na rubu reda. Ostalih $11$ cura ima zadatak: svaka od njih mora izračunati $\frac{\sin(2^{k + 4} \pi / 89)}{\sin(2^k \pi / 89)}$ za svoj broj $k$ (koji je sad jedan od $1,2,\dots,11$. Na kraju, svoje rezultate zbrajaju i to priopćuju Salopeku. Salopek je zatečen: rezultat koji su dobile je cijeli broj. Salopek to informaciju koristi tako da nađe ostatak koji taj broj daje pri dijeljenju s $11$ (za potrebe zadatka jedan od $1,2,\dots 11$) i odabire brucošicu s tim brojem. Mi vas pitamo: brucošica s kojim brojem mu je suđena? [/lang] [lang=en] In the end, after selecting $13$ girls in the previous task, Salopek’s girlfriend had enough and decided to break up with him. Visibly affected by this, he made an effort to carefully choose a freshman girl with whom he would stay through good and bad times until death do them part. Since beauty still plays a role for him, he decided to eliminate the $2$ girls at the ends of the line. The remaining $11$ girls have a task: each of them must calculate $\frac{\sin(2^{k + 4} \pi / 89)}{\sin(2^k \pi / 89)}$ for her number $k$ (which is now one of $1, 2, \dots, 11$). In the end, they sum up their results and report them to Salopek. Salopek is astonished: the result they got is an integer. He uses that information to find the remainder when that number is divided by $11$ (for the purposes of the task, one of $1, 2, \dots, 11$) and chooses the girl with that number. Our question to you is: which girl’s number is destined for him? [/lang]