IMO Shortlist 1996 problem G7
Dodao/la:
arhiva2. travnja 2012. Let
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
be an acute triangle with circumcenter
![O](/media/m/9/6/0/9601b72f603fa5d15addab9937462949.png)
and circumradius
![R](/media/m/4/d/7/4d76ce566584cfe8ff88e5f3e8b8e823.png)
.
![AO](/media/m/d/9/3/d93e4e1fde6437bd5210d0a50abb3ca8.png)
meets the circumcircle of
![BOC](/media/m/6/5/f/65f3c5c1e0f03d9e2c0b09da52314236.png)
at
![A'](/media/m/9/2/6/9267b8bcabe1ad2df0d51dab3364714b.png)
,
![BO](/media/m/0/7/a/07a11b6040ba7091f20d30cbcd7bedf6.png)
meets the circumcircle of
![COA](/media/m/8/e/9/8e93617f33f2ae5859bd7d9cf8e00463.png)
at
![B'](/media/m/a/1/a/a1a88eb7f35fee4f41c66bfb0c902f51.png)
and
![CO](/media/m/c/b/d/cbdd141357ce17ba526f015faa66a19c.png)
meets the circumcircle of
![AOB](/media/m/3/1/c/31c3942c20efa169bde766e004ffeaa2.png)
at
![C'](/media/m/0/0/1/001d1a1af4c90ceda662e79e88845742.png)
. Prove that
![OA'\cdot OB'\cdot OC'\geq 8R^{3}.](/media/m/6/8/c/68c588dfc6381c6c0cb893ddcb4abcf9.png)
Sorry if this has been posted before since this is a very classical problem, but I failed to find it with the search-function.
%V0
Let $ABC$ be an acute triangle with circumcenter $O$ and circumradius $R$. $AO$ meets the circumcircle of $BOC$ at $A'$, $BO$ meets the circumcircle of $COA$ at $B'$ and $CO$ meets the circumcircle of $AOB$ at $C'$. Prove that $$OA'\cdot OB'\cdot OC'\geq 8R^{3}.$$ Sorry if this has been posted before since this is a very classical problem, but I failed to find it with the search-function.
Izvor: Međunarodna matematička olimpijada, shortlist 1996