IMC 2003 · Problem 4

Find all positive integers $n$ for which there exists a family $\mathcal{F}$ of three-element subsets of $S = \{1, 2, \dots, n\}$ satisfying the following two conditions:

(i) for any two different elements $a, b \in S$, there exists exactly one $A \in \mathcal{F}$ containing both $a$, $b$;

(ii) if $a, b, c, x, y, z$ are elements of $S$ such that if $\{a,b,x\}, \{a,c,y\}, \{b,c,z\} \in \mathcal{F}$, then $\{x,y,z\} \in \mathcal{F}$.