1 reply to this topic

[pnam2511]

cho a,b,c>0 thỏa a+b+c=2

chứng minh

*** Cannot compile formula:
\frac{a}{\sqrt{4a+3bc}}+ \frac{b}{\sqrt{4b+3ac}}+ \frac{c}{\sqrt{4c+3ab}} \leqslant 1

*** Error message:
Error: Nothing to show, formula is empty

 

 

 

 

Edited by HappyLife, 17-01-2016 - 19:42.

[quoccuonglqd]

Đổi biến $(a,b,c)\rightarrow (\frac{2x}{3},\frac{2y}{3},\frac{2z}{3})$($x+y+z=3$)

Ta cần chứng minh $\sum \frac{x}{\sqrt{6x+3yz}}\leq 1$

Áp dụng C-S $(\sum \frac{x}{\sqrt{6x+3yz}})^{2}\leq (x+y+z)(\sum \frac{x}{6x+3yz})$

Ta chứng minh $\sum \frac{x}{2x+yz}\leq 1$

$\Leftrightarrow \sum \frac{yz}{2x+yz}\geq 1$

Mà $\sum \frac{yz}{2x+yz}\geq \frac{(zy+xy+zx)^{2}}{6xyz+x^{2}y^{2}+y^{2}z^{2}+z^{2}x^{2}}= \frac{(zy+xy+zx)^{2}}{2xyz(x+y+z)+x^{2}y^{2}+y^{2}z^{2}+z^{2}x^{2}}=1$

Edited by quoccuonglqd, 18-01-2016 - 21:26.