Cho a,b,c,d > 0.Cmr : $\frac{a-d}{b+d} + \frac{d-b}{c+b}+\frac{b-c}{c+a}+\frac{c-a}{a+d} \geq 0$
$\frac{a-d}{b+d} + \frac{d-b}{c+b}+\frac{b-c}{c+a}+\frac{c-a}{a+d} \geq 0$
Started By caokhanh97, 13-02-2013 - 19:00
#2
Posted 13-02-2013 - 19:08
banhgaongonngon
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- Thành viên
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Thượng úy
Cho a,b,c,d > 0.Cmr : $\frac{a-d}{b+d} + \frac{d-b}{c+b}+\frac{b-c}{c+a}+\frac{c-a}{a+d} \geq 0$
BĐT đã cho $\Leftrightarrow \frac{a-d}{b+d}+1+\frac{d-b}{c+b}+1+\frac{b-c}{c+a}+1+\frac{c-a}{a+d}+1\geq 4$
$\Leftrightarrow \frac{a+b}{b+d}+\frac{d+c}{b+c}+\frac{a+b}{a+c}+\frac{d+c}{a+d}\geq 4$
BĐT trên đúng vì
$\left\{\begin{matrix} (a+b)\left ( \frac{1}{b+d}+\frac{1}{a+c} \right )\geq \frac{4(a+b)}{a+b+c+d}\\ (d+c)\left ( \frac{1}{b+c}+\frac{1}{a+d} \right )\geq \frac{4(d+c)}{a+b+c+d} \end{matrix}\right.$
Edited by banhgaongonngon, 13-02-2013 - 19:09.
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#3
Posted 13-02-2013 - 19:09
vutuanhien
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- ĐHV Toán Cao cấp
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- 691 posts
Thiếu úy
BĐT $\Leftrightarrow (a+b)(\frac{1}{b+d}+\frac{1}{c+a})+(c+d)(\frac{1}{c+b}+\frac{1}{a+d})\geq 4$. Ta có $(a+b)(\frac{1}{b+d}+\frac{1}{c+a})+(c+d)(\frac{1}{c+b}+\frac{1}{a+d})\geq (a+b)(\frac{4}{a+b+c+d})+(c+d)(\frac{4}{a+b+c+d})=4$Cho a,b,c,d > 0.Cmr : $\frac{a-d}{b+d} + \frac{d-b}{c+b}+\frac{b-c}{c+a}+\frac{c-a}{a+d} \geq 0$
Edited by vutuanhien, 13-02-2013 - 19:10.
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"Algebra is the offer made by the devil to the mathematician. The devil says: I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvelous machine." (M. Atiyah)
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