Question 12
(GRE 2) In a triangle $ABC$, choose any points $K\in BC$, $L\in AC$, $M\in AB$, $N\in LM$, $R\in MK$, and $F\in KL$. If $E_1,E_2,E_3,E_4,E_5,E_6$, and $E$ denote the areas of the triangles $AMR$, $CKR$, $BKF$, $ALF$, $BNM$, $CLN$, and $ABC$ respectively, show that $$E\ge8\sqrt[6]{E_1E_2E_3E_4E_5E_6}.$$ Remark. Points $K,L,M,N,R,F$ lie on segments $BC,AC,AB,LM,MK,KL$ respectively.