Toán 8 chứng minh bất đẳng thức

Phúc Lâm said:

cho a,b,c,d > 0 c/m :[tex]\frac{a}{b+c+d}+\frac{b}{c+d+a}+\frac{c}{d+a+b}+\frac{d}{a+b+c}\geqslant \frac{4}{3}[/tex]

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đặt b+c+d = x
a+c+d=y
a+b+d=z
a+b+c=m
=> x+y+z+m=3.(a+b+c+d)
=> 3x= 3.(b+c+d)
=> 3a= y+z+m-2x
=> a= (y+z+m-2x)/3
=>a/x= (y+z+m-2x)/3x
CMTT => b/y= (x+z+m-2y)/3y
c/z= (x+y+m-2z)/3z
d/m=(x+y+z-2m)/3m
=> a/x + b/y +c/z +d/m = 1/3. [(y+z+m-2x)/x + (x+z+m-2y)/y + (x+y+m-2z)/z + (x+y+z-2m)/m]
cần chứng minh: [tex]\frac{y+z+m-2x}{x} +\frac{x+z+m-2y}{y} +\frac{x+y+m-2z}{z}+\frac{x+y+z-2m}{m}\geq 4\\\\ <=> \frac{y}{x}+\frac{z}{x}+\frac{m}{x}-2+\frac{x}{y}+\frac{z}{y}+\frac{m}{y}-2+\frac{x}{z}+\frac{y}{z}+\frac{m}{z}-2+\frac{x}{m}+\frac{y}{m}+\frac{z}{m}-2\\\\ \geq 12\sqrt[12]{\frac{y}{x}.\frac{z}{x}.\frac{m}{x}.\frac{x}{y}.\frac{z}{y}.\frac{m}{y}.\frac{x}{z}.\frac{y}{z}.\frac{m}{z}.\frac{x}{m}.\frac{y}{m}.\frac{z}{m}}-8=12-8=4 (đpcm)[/tex]
(lười viết tex...bạn chịu khó dịch nha.....)

Phúc Lâm said:

cho a,b,c,d > 0 c/m :[tex]\frac{a}{b+c+d}+\frac{b}{c+d+a}+\frac{c}{d+a+b}+\frac{d}{a+b+c}\geqslant \frac{4}{3}[/tex]

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Bổ đề : Cho [tex]x,y,z,t> 0[/tex]
[tex]\frac{a^2}{x}+\frac{b^{2}}{y}+\frac{c^{2}}{z}+\frac{d^{2}}{t}\geq \frac{(a+b+c+d)^2}{x+y+z+t}[/tex]
Áp dụng
[tex]\sum \frac{a}{b+c+d}=\sum \frac{a^2}{ab+ac+ad}\geq \frac{(a+b+c+d)^2}{\sum (ab+bc+cd+da)}[/tex]
Cần chứng minh
[tex]=\frac{(a+b+c+d)^2}{\sum (ab+ac+ad)}\\=\frac{a^2+b^2+c^2+d^2+2ab+2ac+2ad+2bc+2bd+2cd}{2ab+2ac+2ad+2bc+2bd+2cd}[/tex]
[tex]\geq \frac{4}{3}\\\Leftrightarrow 3(a^2+b^2+c^2+d^2)+3(2ab+2ac+2ad+2bc+2bd+2cd)\\\geq 4(2ab+2ac+2ad+2bc+2bd+2cd)\\\Leftrightarrow 3(a^2+b^2+c^2+d^2)\geq 2ab+2ac+2ad+2bc+2bd+2cd\\\Leftrightarrow (a-b)^2+(a-c)^2+(a-d)^2+(b-c)^2+(b-d)^2+(c-d)^2\geq 0[/tex]
(luôn đúng)
Vậy ta có đpcm
Dấu ''='' xảy ra khi a=b=c=d

theo AM-GM:
(b+c+d)+(c+d+a)+(d+a+b)+(a+b+c)[tex]\geq 4\sqrt[4]{(b+c+d)(c+d+a)(d+a+b)(a+b+c)}[/tex]
[tex]\frac{1}{b+c+d}+\frac{1}{c+d+a}+\frac{1}{d+a+b}+\frac{1}{a+b+c}\geq 4\sqrt[4]{\frac{1}{b+c+d}.\frac{1}{c+d+a}.\frac{1}{d+a+b}.\frac{1}{a+b+c}}[/tex]
nhân vế vs vế
ở vế trái bạn cứ tách cho giống đề bài là dc