Marinada '25

[lang=hr] "Prošli zadatak je bio jako zločest!" Marica je uzviknula ljutito. Međutim, to je Ivici bio i plan jer joj je planirao pokazati najnestandardniji zadatak do sada. Budući da je vidio da se Marica mučila s prošlim zadatcima, prije postavljanja zadatka odlučio joj je dati sljedeći naputak: "Zadatak koji ću ti sada reći ima cool geometrijsku interpretaciju". Rekavši to, Ivica napokon postavlja pitanje: Ako su $a,b,c,x,y,z$ realni brojevi za koje vrijedi $a^2+b^2+c^2=27$ i \[ a^2+x^2=b^2+y^2=c^2+z^2=(a+b)^2+(x+y)^2 \] \[ =(b+c)^2+(y+z)^2=(c+a)^2+(z+x)^2 \] Koliko je $x^2+y^2+z^2$? [/lang] [lang=en] “The previous problem was really nasty!” Marica shouted angrily. However, that was part of Ivica’s plan, because he intended to show her the most unconventional problem yet. Since he saw that Marica struggled with the earlier problems, before assigning the new one he decided to give her the following hint: “The problem I’m about to tell you has a cool geometric interpretation.” Saying that, Ivica finally poses the question: If $a, b, c, x, y, z$ are real numbers that satisfy $a^2+b^2+c^2=27$ and \[ a^2+x^2=b^2+y^2=c^2+z^2=(a+b)^2+(x+y)^2 \] \[ =(b+c)^2+(y+z)^2=(c+a)^2+(z+x)^2 \] What is $x^2+y^2+z^2$? [/lang]