IMO Shortlist 1969 problem 42
Dodao/la:
arhiva2. travnja 2012. ![(MON 3)](/media/m/3/c/6/3c6d8ced2da03588957f96d519d71ca8.png)
Let
![A_k (1 \le k \le h)](/media/m/4/f/c/4fcc15a7cd8b34aad40c13613e583d28.png)
be
![n-](/media/m/f/a/e/fae563323c2368fde7e704b858164853.png)
element sets such that each two of them have a nonempty intersection. Let
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
be the union of all the sets
![A_k,](/media/m/e/f/2/ef2be448db532f3423f779090ba0bb92.png)
and let
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
be a subset of
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
such that for each
![k (1\le k \le h)](/media/m/7/2/4/7245fb2b3b6af583119dacad6f3ebbd7.png)
the intersection of
![A_k](/media/m/b/2/e/b2e11cfbe70ea64468986bde48119d91.png)
and
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
consists of exactly two different elements
![a_k](/media/m/8/f/f/8ffe60c23d3334cc61d0660473bf1b61.png)
and
![b_k](/media/m/1/e/1/1e1016cbade6ef5706a4e3f9c56841bc.png)
. Find all subsets
![X](/media/m/9/2/8/92802f174fc4967315c2d8002c426164.png)
of the set
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
with
![r](/media/m/3/d/f/3df7cc5bbfb7b3948b16db0d40571068.png)
elements satisfying the condition that for at least one index
![k,](/media/m/3/b/3/3b3f59143c4e876b4adc54b666df1462.png)
both elements
![a_k](/media/m/8/f/f/8ffe60c23d3334cc61d0660473bf1b61.png)
and
![b_k](/media/m/1/e/1/1e1016cbade6ef5706a4e3f9c56841bc.png)
belong to
![X](/media/m/9/2/8/92802f174fc4967315c2d8002c426164.png)
.
%V0
$(MON 3)$ Let $A_k (1 \le k \le h)$ be $n-$element sets such that each two of them have a nonempty intersection. Let $A$ be the union of all the sets $A_k,$ and let $B$ be a subset of $A$ such that for each $k (1\le k \le h)$ the intersection of $A_k$ and $B$ consists of exactly two different elements $a_k$ and $b_k$. Find all subsets $X$ of the set $A$ with $r$ elements satisfying the condition that for at least one index $k,$ both elements $a_k$ and $b_k$ belong to $X$.
Izvor: Međunarodna matematička olimpijada, shortlist 1969