IMO Shortlist 1979 problem 16
Dodao/la:
arhiva2. travnja 2012. Let
![K](/media/m/e/1/e/e1ed1943d69f4d6a840e99c7bd199930.png)
denote the set
![\{a, b, c, d, e\}](/media/m/d/3/4/d340cf8d0ec61df5be435c9caa061bbd.png)
.
![F](/media/m/3/e/8/3e8bad5df716d332365fca76f53c1743.png)
is a collection of
![16](/media/m/8/7/3/873df6b5c41cd51cd08caeb528116eb4.png)
different subsets of
![K](/media/m/e/1/e/e1ed1943d69f4d6a840e99c7bd199930.png)
, and it is known that any three members of
![F](/media/m/3/e/8/3e8bad5df716d332365fca76f53c1743.png)
have at least one element in common. Show that all
![16](/media/m/8/7/3/873df6b5c41cd51cd08caeb528116eb4.png)
members of
![F](/media/m/3/e/8/3e8bad5df716d332365fca76f53c1743.png)
have exactly one element in common.
%V0
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.
Izvor: Međunarodna matematička olimpijada, shortlist 1979