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IMC 2003 Problem 10

Source: IMC 2003 Day 2 Problem 4

November 2, 2020
combinatorics

Problem Statement

Find all the positive integers nn for which there exists a Family F\mathcal{F} of three-element subsets of S={1,2,...,n}S=\{1,2,...,n\} satisfying (i) for any two different elements a,b∈S there exists exactly one A∈F containing both a and b;\text{(i) for any two different elements $a,b \in S$ there exists exactly one $A \in \mathcal{F}$ containing both $a$ and $b$;} (ii) if a,b,c,x,y,z are elements of S such that {a,b,x},{a,c,y},{b,c,z}∈F, then {x,y,z}∈F .\text{(ii) if $a,b,c,x,y,z$ are elements of $S$ such that $\{a,b,x\},\{a,c,y\},\{b,c,z\} \in \mathcal{F}$, then $\{x,y,z\} \in \mathcal{F} $ }.
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