[72] - Algebraic Inequalities from and for Math Contests

Έστω $a,b,c,d\ge 0$ τέτοιοι ώστε \[ 3(a+b+c+d)=2(ab+ac+ad+bc+bd+cd)>0. \] Να αποδείξετε ότι \[ 8\,(a^{3}+b^{3}+c^{3}+d^{3})+49\,abcd \ge 81. \]Proposed by Marius Stănean, Zalău, Romania