[80] - Geometric Inequalities from and for Math Contests

Έστω $a,b,c$ τα μήκη των πλευρών ενός τριγώνου. Να αποδείξετε ότι: \[ \left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)^{2} + 15 \ge 16\!\left(\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\right). \] Proposed by Titu Andreescu, USA and Marius Stănean, Romania