IMO Shortlist 1991 problem 7
Dodao/la:
arhiva2. travnja 2012. ![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
be a set of
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
points in the plane. No three points of
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
are collinear. Prove that there exists a set
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
containing
![2n - 5](/media/m/4/d/4/4d405ab19a7d1a2d509ac8305ec5b248.png)
points satisfying the following condition: In the interior of every triangle whose three vertices are elements of
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
lies a point that is an element of
%V0
$S$ be a set of $n$ points in the plane. No three points of $S$ are collinear. Prove that there exists a set $P$ containing $2n - 5$ points satisfying the following condition: In the interior of every triangle whose three vertices are elements of $S$ lies a point that is an element of $P.$
Izvor: Međunarodna matematička olimpijada, shortlist 1991