Question 16
G7 (GBR) Let $ABC$ be an acute-angled triangle with circumcenter $O$ and circumradius $R$. Let $AO$ meet the circle $BOC$ again in $A'$, let $BO$ meet the circle $COA$ again in $B'$, and let $CO$ meet the circle $AOB$ again in $C'$. Prove that
$$OA'\cdot OB'\cdot OC'\ge8R^3.$$
When does equality hold?
$$OA'\cdot OB'\cdot OC'\ge8R^3.$$
When does equality hold?