Marinada '26

[lang=hr] Sada kada su uspješno smjestili sve probleme u kaveze, mentori su si priuštili odmor uz, naravno, još jedan problem. A taj problem glasi: Zadan je skup $\{1,2,3,...,2023\}$ i odabran je podskup $S$ danog skupa. Za $S$ vrijedi: ako su $x$ i $y$ u $S$ onda $x+y$ nije u skupu. Koliko je maksimalno $|S|$? [/lang] [lang=en] Now that the tutors have put all the problems in their cages, they can now rest, with no other than another math problem: You are given a set $\{1,2,3,...,2023\}$ and a subset $S$ of this set was chosen. For $S$ is the following true: if $x$ and $y$ are both in $S$ then $x+y$ are not in $S$. What is the maximal value of $|S|$? [/lang]