IMO Shortlist 1979 problem 12
Dodao/la:
arhiva2. travnja 2012. Let
![R](/media/m/4/d/7/4d76ce566584cfe8ff88e5f3e8b8e823.png)
be a set of exactly
![6](/media/m/e/e/e/eeec330d59a70f8ed1d6882474cb02a3.png)
elements. A set
![F](/media/m/3/e/8/3e8bad5df716d332365fca76f53c1743.png)
of subsets of
![R](/media/m/4/d/7/4d76ce566584cfe8ff88e5f3e8b8e823.png)
is called an
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
-family over
![R](/media/m/4/d/7/4d76ce566584cfe8ff88e5f3e8b8e823.png)
if and only if it satisfies the following three conditions:
(i) For no two sets
![X, Y](/media/m/8/1/2/8127a30abcded51f7b4b18b62137b2a3.png)
in
![F](/media/m/3/e/8/3e8bad5df716d332365fca76f53c1743.png)
is
![X \subseteq Y](/media/m/4/a/6/4a63ba8b29519488af7e84ddda367810.png)
;
(ii) For any three sets
![X, Y,Z](/media/m/e/e/4/ee4449852a793f7deadce6fa244d0a5b.png)
in
![F](/media/m/3/e/8/3e8bad5df716d332365fca76f53c1743.png)
,
![X \cup Y \cup Z \neq R,](/media/m/0/b/6/0b6d25ecd3e1df378bf3471de0811b22.png)
(iii)
%V0
Let $R$ be a set of exactly $6$ elements. A set $F$ of subsets of $R$ is called an $S$-family over $R$ if and only if it satisfies the following three conditions:
(i) For no two sets $X, Y$ in $F$ is $X \subseteq Y$ ;
(ii) For any three sets $X, Y,Z$ in $F$, $X \cup Y \cup Z \neq R,$
(iii) $\bigcup_{X \in F} X = R$
Izvor: Međunarodna matematička olimpijada, shortlist 1979