TW-TST 1997 Problem 8

Let $O$ be the circumcenter and $R$ be the circumradius of an acute triangle $ABC$ . Let $AO$ meet the circumcircle of $OBC$ again at $D$ , $BO$ meet the circumcircle of $OCA$ again at $E$ , and $CO$ meet the circumcircle of $OAB$ again at $F$ . Show that $OD.OE.OF\geq 8R^{3}$ .