[13] - Algebraic Inequalities for Contests

Να αποδειχθεί ότι, αν \(\displaystyle a\), \(\displaystyle b\), \(\displaystyle c>0\) και \(\displaystyle a+b+c=1\), τότε

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

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

KöMaL Problems 2024