Marinada '22

[lang=hr] Nestašni Lovro nenadano je zalutao u podrum svoga djeda. U njemu je našao puno zanimljivih stvari, kao što su dedin poznati sok od šljiva u prozirnoj boci te puno matematickih zadatak. Lovri je odma jasno da je deda bio talentirani matematicar pa se i on uz sok od šljiva odlučio okušati u njima. Lovro je uzeo jedan papir na kojem je pisalo: Neka su $x,y,z,t$ realni brojevi takvi da vrijedi $xy+yz+zt+tx = 1$ odredi minimum izraza $$5x^2+4y^2+5z^2+t^2$$ Pomozi Lovri naći odgovor. Odgovor zaokruži na 4 decimale. [/lang] [lang=en] Mischievous Lovro suddenly wandered into his grandfather's basement. He found a lot of interesting things in it, such as his grandfather's famous plum juice in a transparent bottle and a lot of math problems. It was immediately clear to Lovro that his grandfather was a talented mathematician, so he too decided to try them out with plum juice. Lovro took a piece of paper on which it was written: Let $x,y,z,t$ be real numbers such that $xy+yz+zt+tx = 1$ is valid, determine the minimum of the expression $$5x^2+4y^2+5z^2+t^2$$ Help Lovro find the answer. Round the answer to 4 decimal places. [/lang]