Marinada '24

[lang=hr] Nakon još jednog zanimljivog razgovora o muškoj prirodi, kazao je Platon Sokratu: "Sokrate, pazi ovo! Zamisli $S=\{1,2,...,N\}$, $A_1,A_2,A_3,A_4\subseteq S$, te svaki od danih podskupova ima točno $500$ elemenata. Još ti dakako vrijedi ova formulica: $\forall x,y\in S$, $\exists i\in\{1,2,3,4\}$, $x,y\in A_i$. Odredite maksimalanu vrijednost koju $N$ može poprimiti." Sokrat naravno zna odgovor na ovo pitanje već nakon kratkog mućkanja glavom. Koji mu je broj na umu? [/lang] [lang=en] After yet another intriguing discussion about male nature, Plato said to Socrates: "Socrates, check this out! Imagine \( S = \{1, 2, \ldots, N\} \), with \( A_1, A_2, A_3, A_4 \subseteq S \), and each of the given subsets has exactly \( 500 \) elements. Also, this formula holds: \( \forall x, y \in S \), \( \exists i \in \{1, 2, 3, 4\} \) such that \( x, y \in A_i \). Determine the maximum value that \( N \) can take." Socrates, of course, knew the answer to this question after a brief moment of contemplation. What number was on his mind? [/lang]