Inegalitate conditionata

opincariumihai · Post by **opincariumihai** » Sat May 23, 2009 5:38 pm

Daca \( a,b,c \) sunt pozitive si \( a+b+c=1 \), aratati ca \( \frac{a^3}{a^2+b^2}+\frac{b^3}{b^2+c^2}+\frac{c^3}{c^2+a^2}\geq\frac{1}{2} \)

Mihai Opincariu

Marius Mainea · Post by **Marius Mainea** » Sat May 23, 2009 7:22 pm

\( LHS=\sum a-\sum\frac{ab^2}{a^2+b^2}\ge RHS \) folosind AM-GM

P.S. Se poate arata ca :

\( \frac{a^4}{a^3+b^3}+\frac{b^4}{b^3+c^3}+\frac{c^4}{c^3+a^3}\ge\frac{a+b+c}{2} \)

opincariumihai · Post by **opincariumihai** » Sat May 23, 2009 7:38 pm

Felicitari pt. rapiditatea cu care rezolvi pb.

Mai "smechereste" se redacteaza asa :
Demonstram ca \( \frac{a^3}{a^2+b^2}\geq\frac{2a-b}{2} \) si analoagele care insumate duc la concluzie.