IMO Shortlist 1969 problem 5
Dodao/la:
arhiva2. travnja 2012. ![(BEL 5)](/media/m/2/7/8/278186e3e9a13bd99570f9053e717b5a.png)
Let
![G](/media/m/f/e/b/feb7f8fc95cee3c3a479382202e06a86.png)
be the centroid of the triangle
![(a)](/media/m/a/7/f/a7fedf50ce0b917a00dd07d5233906f1.png)
Prove that all conics passing through the points
![O,A,B,G](/media/m/5/e/8/5e8fed3492709929bfdcfb582fb15baf.png)
are hyperbolas.
![(b)](/media/m/9/2/7/92773ef234467079b4efc86655fdc459.png)
Find the locus of the centers of these hyperbolas.
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$(BEL 5)$ Let $G$ be the centroid of the triangle $OAB.$
$(a)$ Prove that all conics passing through the points $O,A,B,G$ are hyperbolas.
$(b)$ Find the locus of the centers of these hyperbolas.
Izvor: Međunarodna matematička olimpijada, shortlist 1969