IMO Shortlist 1977 problem 14
Kvaliteta:
Avg: 0,0Težina:
Avg: 0,0 Let
be a finite set of points such that
is not contained in a plane and no three points of
are collinear. Show that at least one of the following alternatives holds:
(i)
contains five points that are vertices of a convex pyramid having no other points in common with ![E;](/media/m/f/0/7/f07d49894a6fd31ad03bef007db6c66a.png)
(ii) some plane contains exactly three points from
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
(i)
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
![E;](/media/m/f/0/7/f07d49894a6fd31ad03bef007db6c66a.png)
(ii) some plane contains exactly three points from
![E.](/media/m/7/1/3/7136cc218be743ebf2a21d6342b1e82f.png)
Izvor: Međunarodna matematička olimpijada, shortlist 1977