IMO 1979 SL 16

Origin: ISR

Problem

Let K denote the set {a, b, c, d, e}. F is a collection of 16 different subsets of K, and it is known that any three members of F have at least one element in common. Show that all 16 members of F have exactly one element in common.

Solution

Obviously, no two elements of F can be complements of each other. If one of the sets has one element, then the conclusion is trivial. If there exist two different 2-element sets, then they must contain a common element, which in turn must then be contained in all other sets. Thus we can assume that there exists at most one 2-element subset of K in F. Since there can be at most 6 subsets of more than 3 elements of a 5-element set, it follows that at least 9 out of 10 possible 3-element subsets of K belong to F. Let us assume, without loss of generality, that all sets but {c, d, e} belong to F. Then sets {a, b, c}, {a, d, e}, and {b, c, d} have no common element, which is a contradiction. Hence it follows that all sets have a common element.