[13] - Algebraic Inequalities for Contests

Να αποδειχθεί ότι, αν ${\displaystyle a}$, ${\displaystyle b}$, ${\displaystyle c>0}$ και ${\displaystyle a+b+c=1}$, τότε

${\displaystyle a)}$ ${\displaystyle \frac{1-a^2}{b+c}+\frac{1-b^2}{c+a}+\frac{1-c^2}{a+b}=4}$,

${\displaystyle b)}$ ${\displaystyle \frac{1-a^3}{b+c}+\frac{1-b^3}{c+a}+\frac{1-c^3}{a+b}\ge \frac{13}{3}}$.

KöMaL Problems 2024