3 replies to this topic

[nmtuan2001]

Cho $a,b,c>0$. CMR:$\frac{a^2}{b(b+c)}+\frac{b^2}{c(c+a)}+\frac{c^2}{a(a+b)} \geq \frac{3}{2}$

Edited by nmtuan2001, 14-12-2014 - 17:11.

[hoctrocuanewton]

Cho $a,b,c>0$. CMR:$\frac{a^2}{b(b+c)}+\frac{b^2}{c(c+a)}+\frac{c^2}{a(a+b)} \geq \frac{3}{2}$ 

Áp dụng BĐT cô-si và schwarz ta có :

$\sum \frac{a^{2}}{b(b+c)}=\frac{8a^{2}}{8b(b+c)}\geq \frac{8a^{2}}{(3b+c)^{2}}\geq \frac{8}{3}(\sum \frac{a}{3b+c})^{2}= \frac{8}{3}(\sum \frac{a^{2}}{3ab+ac})^{2}\geq \frac{8}{3}(\frac{(a+b+c)^{2}}{4\sum ab})^{2}\geq \frac{8}{3}.(\sum \frac{(\sum a)^{2}}{\frac{4}{3}(\sum a)^{2}})= \frac{3}{2}$

[Algebra]

Cho $a,b,c>0$. CMR:$\frac{a^2}{b(b+c)}+\frac{b^2}{c(c+a)}+\frac{c^2}{a(a+b)} \geq \frac{3}{2}$

$\frac{a^2}{b(b+c)}+\frac{b^2}{c(c+a)}+\frac{c^2}{a(a+b)}=\frac{a^4}{a^{2}b(b+c)}+\frac{b^4}{b^{2}c(c+a)}+\frac{c^4}{c^{2}a(a+b)}$

$\geq \frac{(a^{2}+b^{2}+c^{2})^{2}}{a^{2}b^{2}+b^{2}c^{2}+c^{2}a^{2}+abc(a+b+c)}\geq \frac{(a^{2}+b^{2}+c^{2})^{2}}{\frac{(a^{2}+b^{2}+c^{2})^{2}}{3}+\frac{(ab+bc+ca)^{2}}{3}}\geq \frac{(a^{2}+b^{2}+c^{2})^{2}}{\frac{(a^{2}+b^{2}+c^{2})^{2}}{3}+\frac{(a^{2}+b^{2}+c^{2})^{2}}{3}}= \frac{3}{2}$

Edited by Algebra, 16-12-2014 - 14:30.

[nmtuan2001]

$\frac{a^2}{b(b+c)}+\frac{b^2}{c(c+a)}+\frac{c^2}{a(a+b)}=\frac{a^4}{a^{2}b(b+c)}+\frac{b^4}{b^{2}c(c+a)}+\frac{c^4}{c^{2}a(a+b)}$

$\geq \frac{(a^{2}+b^{2}+c^{2})^{2}}{a^{2}b^{2}+b^{2}c^{2}+c^{2}a^{2}+3abc(a+b+c)}\geq \frac{(a^{2}+b^{2}+c^{2})^{2}}{\frac{(a^{2}+b^{2}+c^{2})^{2}}{3}+(ab+bc+ca)^{2}}\geq \frac{(a^{2}+b^{2}+c^{2})^{2}}{\frac{(a^{2}+b^{2}+c^{2})^{2}}{3}+(a^{2}+b^{2}+c^{2})^{2}}= \frac{3}{2}$

Chỉ $3abc(a+b+c)$ thôi. Phần cuối sai rồi.