[4] - Algebraic Inequalities for Contests

Έστω $a,b,c$ θετικοί πραγματικοί αριθμοί. Να αποδειχθεί ότι 

$${\displaystyle \frac{b+c}{a^2}}+{\displaystyle \frac{c+a}{b^2}}+{\displaystyle \frac{a+b}{c^2}}+{\displaystyle \frac{45}{a+b+c}}\ge {\displaystyle \frac{7}{a}}+{\displaystyle \frac{7}{b}}+{\displaystyle \frac{7}{c}}.$$ Titu Andreescu, University of Texas at Dallas, US