Cho $a,b,c,d$ là các số thực dương.Chứng minh $\frac{b(a+c)}{c(a+b)}+\frac{c(b+d)}{d(b+c)}+\frac{d(c+a)}{a(c+d)}+\frac{a(d+b)}{b(d+a)}\geq 4$
Cho $a,b,c,d$ là các số thực dương.Chứng minh $\frac{b(a+c)}{c(a+b)}+\frac{c(b+d)}{d(b+c)}+\frac{d(c+a)}{a(c+d)}+\frac{a(d+b)}{b(d+a)}\geq 4$
Đặt P là biểu thức đề bài
Ta có: $P=(a+c)[\frac{b}{c(a+b)}+\frac{d}{a(c+d)}]+(b+d)[\frac{c}{d(b+c)}+\frac{a}{b(d+a)}]$
$=(abc+bcd+cda+dab)[\frac{a+c}{ac(a+b)(c+d)}+\frac{b+d}{bd(b+c)(a+d)}]$
$=(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d})[\frac{\frac{1}{a}+\frac{1}{c}}{(\frac{1}{a}+\frac{1}{b})(\frac{1}{c}+\frac{1}{d})}+\frac{\frac{1}{b}+\frac{1}{d}}{(\frac{1}{b}+\frac{1}{c})(\frac{1}{a}+\frac{1}{d})}]$
$\geq (\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d})[\frac{4(\frac{1}{a}+\frac{1}{c})}{(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d})^{2}}+\frac{4(\frac{1}{b}+\frac{1}{d})}{(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d})^{2}}]=4$
Vậy đpcm
"Life would be tragic if it weren't funny"
-Stephen Hawking-
Đặt P là biểu thức đề bài
Ta có: $P=(a+c)[\frac{b}{c(a+b)}+\frac{d}{a(c+d)}]+(b+d)[\frac{c}{d(b+c)}+\frac{a}{b(d+a)}]$
$=(abc+bcd+cda+dab)[\frac{a+c}{ac(a+b)(c+d)}+\frac{b+d}{bd(b+c)(a+d)}]$
$=(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d})[\frac{\frac{1}{a}+\frac{1}{c}}{(\frac{1}{a}+\frac{1}{b})(\frac{1}{c}+\frac{1}{d})}+\frac{\frac{1}{b}+\frac{1}{d}}{(\frac{1}{b}+\frac{1}{c})(\frac{1}{a}+\frac{1}{d})}]$
$\geq (\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d})[\frac{4(\frac{1}{a}+\frac{1}{c})}{(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d})^{2}}+\frac{4(\frac{1}{b}+\frac{1}{d})}{(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d})^{2}}]=4$
Vậy đpcm
cho mình hỏi làm sao bạn biết cách đưa $P$ về như trên vậy
"Life would be tragic if it weren't funny"
-Stephen Hawking-
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