BĐT nesbit với n=4. 

chứng minh nó ko hề khó đâu: 
đặt VT =A đi .thì sử dụng BĐT bunhiacopxki ta có: 
A[a(b+c)+b(c+d)+c(d+a)+d(a+b)] 
>=(a+b+c+d)^2 
giờ ta chỉ cần chứng minh: 
(a+b+c+d)^2>=2a(b+c)+b(c+d)+c(d+a)+d(a... 
điều này <=> với:a^2+b^2+c^2+d^2>=2ac+2bd. 
diều này là hiển nhiên theo BĐT cô-si=>đpcm.MinA=2.

Áp dụng BĐT \(\frac{1}{xy}\ge\frac{4}{\left(x+y\right)^2}\left(x;y>0\right)\)

\(\frac{a}{b+c}+\frac{c}{d+a}=\frac{a^2+ad+bc+c^2}{\left(b+c\right)\left(a+d\right)}\ge\frac{4\left(a^2+ad++bc+c^2\right)}{\left(a+b+c+d\right)^2}\left(1\right)\)

Tương tự \(\frac{b}{c+b}+\frac{d}{a+b}\ge\frac{4\left(b^2+ab+cd+d^2\right)}{\left(a+b+c+d\right)^2}\left(2\right)\)

Cộng (1) với (2) \(\frac{a}{b+c}+\frac{b}{c+d}+\frac{c}{d+a}+\frac{d}{a+b}\ge\frac{4\left(a^2+b^2+c^2+d^2+ad+bc+ab+cd\right)}{\left(a+b+c+d\right)^2}=\text{4B}\)

Cần chứng minh \(B\ge\frac{1}{2}\), BDDT này tương đương với

\(2B\ge1\Leftrightarrow2\left(a^2+b^2+c^2+d^2+ad+bc+ab+cd\right)\ge\left(a+b+c+d\right)^2\)

\(\Leftrightarrow a^2+b^2+c^2+d^2-2ac-2bc\ge0\)

\(\Leftrightarrow\left(a-c\right)^2+\left(b-d\right)^2\ge0\)

ta có:

\(\frac{3}{a+b}+\frac{2}{c+d}+\frac{a+b}{\left(a+c\right)\left(b+d\right)}\ge\frac{3}{a+b}+\frac{2}{c+d}+\frac{4\left(a+b\right)}{\left(a+b+c+d\right)^2}\)

xét hiệu:

\(\frac{3}{a+b}+\frac{2}{c+d}+\frac{4\left(a+b\right)}{\left(a+b+c+d\right)^2}-\frac{12}{a+b+c+d}\)

\(=\frac{3}{a+b}+\frac{2}{c+d}-\frac{8\left(a+b\right)+12\left(c+d\right)}{\left(a+b+c+d\right)^2}\)

đặt a+b=x;c+d=y

\(\Rightarrow\frac{3}{a+b}+\frac{2}{c+d}-\frac{8\left(a+b\right)+12\left(c+d\right)}{\left(a+b+c+d\right)^2}=\frac{3}{x}+\frac{2}{y}-\frac{8x+12y}{\left(x+y\right)^2}\ge\frac{3}{x}+\frac{2}{y}-\frac{8x+12y}{4xy}=\frac{3}{x}+\frac{2}{y}-\frac{2}{y}-\frac{3}{x}=0\)

\(\Rightarrow\frac{3}{a+b}+\frac{2}{c+d}+\frac{4\left(a+b\right)}{\left(a+b+c+d\right)^2}\ge\frac{12}{a+b+c+d}\)

\(\Rightarrow\frac{3}{a+b}+\frac{2}{c+d}+\frac{a+b}{\left(a+c\right)\left(b+d\right)}\ge\frac{12}{a+b+c+d}\)

=>đpcm

dấu "=" xảy ra khi a=b=c=d

Dễ mà 

Ta có: \(\frac{a}{b}=\frac{c}{d}\Rightarrow\frac{a}{c}=\frac{b}{d}\)

Áp dụng t/c dãy tỉ số bằng nhau:

Ta có: \(\frac{a}{c}=\frac{b}{d}=\frac{a+b}{c+d}=\frac{a-b}{c-d}\)(1)

Từ (1),

Ta có: \(\frac{a+b}{c+d}\cdot\frac{a+b}{c+d}=\frac{a+b}{c+d}\cdot\frac{a-b}{c-d}\)(nhân mỗi vế với \(\frac{a+b}{c+d}\))

Vậy \(\frac{\left(a+b\right)^2}{\left(c+d\right)^2}=\frac{\left(a+b\right)\left(a-b\right)}{\left(c+d\right)\left(c-d\right)}=\frac{a^2-b^2}{c^2-d^2}\)(đpcm)

Áp dụng BĐT Cauchy -Schwarz dạng cộng mẫu thôi:

\(\text{VT}=\frac{1^2}{a}+\frac{1^2}{b}+\frac{2^2}{c}+\frac{4^2}{d}\geq \frac{(1+1+2+4)^2}{a+b+c+d}=\frac{64}{a+b+c+d}=\text{VP}\)

Dấu bằng xảy ra khi \(a=b=\frac{c}{2}=\frac{d}{4}>0\)

áp dụng BĐT cauchy-schwazs:

\(\frac{1}{a}+\frac{1}{b}+\frac{4}{c}+\frac{16}{d}\ge\frac{\left(1+1+2+4\right)^2}{a+b+c+d}=\frac{64}{a+b+c+d}\)

dấu = xảy ra khi \(\frac{1}{a}=\frac{1}{b}=\frac{2}{c}=\frac{4}{d}\Leftrightarrow a=b=\frac{c}{2}=\frac{d}{4}\)

áp dụng bđt \(\frac{a^2}{x}+\frac{b^2}{y}\ge\frac{\left(a+b\right)^2}{x+y}\)(bđt svacxo) ta có :

VT= \(\frac{1}{a}+\frac{1}{b}+\frac{4}{c}+\frac{16}{d}\ge\frac{\left(1+1+2+4\right)^2}{a+b+c+d}\)= \(\frac{64}{a+b+c+d}\)=VP (đpcm)

dấu = xảy ra <=>a=b=1; c=2 ; d=4

Dễ dàng CM BĐT phụ sau: \(\frac{1}{a}+\frac{1}{b}\ge\frac{4}{a+b},\forall a,b>0\)

Áp dụng liên tục ta có:

\(\frac{1}{a}+\frac{1}{b}+\frac{4}{c}+\frac{16}{d}\ge\frac{4}{a+b}+\frac{4}{c}+\frac{16}{d}\ge4.\frac{4}{a+b+c}+\frac{16}{d}\ge16.\frac{4}{a+b+c+d}=\frac{64}{a+b+c+d}\)

dấu = xảy ra <=> a+b=c, a+b+c=d, a=b

ĐPCM

+) Chứng minh: \(\frac{a}{b}>\frac{a+c}{b+d}\)   (1)

Xét hiệu: \(\frac{a}{b}-\frac{a+c}{b+d}=\frac{a\left(b+d\right)-b\left(a+c\right)}{b\left(b+d\right)}=\frac{ab+ad-ab-bc}{b\left(b+d\right)}=\frac{ad-bc}{b\left(b+d\right)}\)

Vì a/b > c/d ; b; d > 0 =>  ad > bc => ad - bc > 0 .T a có b(b +d) > 0 nên Hiệu trên > 0 => \(\frac{a}{b}>\frac{a+c}{b+d}\)

+) Chứng minh: \(\frac{a+c}{b+d}>\frac{c}{d}\)

Xét hiệu: \(\frac{a+c}{b+d}-\frac{c}{d}=\frac{\left(a+c\right)d-c\left(b+d\right)}{b\left(b+d\right)}=\frac{ad-bc}{b.\left(b+d\right)}>0\)

=> \(\frac{a+c}{b+d}>\frac{c}{d}\)  (2)

Từ (1)(2 ta có đpcm

\(\frac{a+b}{c+d}=\frac{b+c}{a+d}\)

\(\Rightarrow\left(a+b\right)\left(a+d\right)=\left(b+c\right)\left(c+d\right)\)

\(\Rightarrow\left(a+b\right)a+\left(a+b\right)d=\left(b+c\right)c+\left(b+c\right)d\)

\(\Rightarrow a^2+ab+ad+bd=bc+c^2+bd+cd\)

\(\Rightarrow\left(a^2+ab+ad\right)+bd=\left(c^2+bc+cd\right)+bd\)

\(\Rightarrow a.\left(a+b+d\right)=c.\left(c+b+d\right)\)

xét a< c =>a.(a+b+d)<c(c+b+d)

xét a>c =>a.(a+b+d)>c(c+b+d)

=>a=c

=>đpcm

Bài 1:

a) Ta có: \(\frac{a}{b}=\frac{c}{d}.\)

\(\Rightarrow\frac{b}{a}=\frac{d}{c}\)

\(\Rightarrow\frac{b^2}{a^2}=\frac{d^2}{c^2}.\)

\(\Rightarrow\frac{b^2}{a^2}+1=\frac{d^2}{c^2}+1\)

\(\Rightarrow\frac{b^2}{a^2}+\frac{a^2}{a^2}=\frac{d^2}{c^2}+\frac{c^2}{c^2}.\)

\(\Rightarrow\frac{b^2+a^2}{a^2}=\frac{d^2+c^2}{c^2}\)

\(\Rightarrow\frac{a^2}{a^2+b^2}=\frac{c^2}{c^2+d^2}\left(đpcm\right).\)

Bài 4:

Bài 1:

Đặt \(\frac{a}{b}=\frac{c}{d}=t\Rightarrow a=bt; c=dt\). Khi đó:

a)

\(\frac{a^2}{a^2+b^2}=\frac{(bt)^2}{(bt)^2+b^2}=\frac{b^2t^2}{b^2(t^2+1)}=\frac{t^2}{t^2+1}(1)\)

\(\frac{c^2}{c^2+d^2}=\frac{(dt)^2}{(dt)^2+d^2}=\frac{d^2t^2}{d^2(t^2+1)}=\frac{t^2}{t^2+1}(2)\)

Từ $(1);(2)$ suy ra đpcm.

b)

\(\left(\frac{a+c}{b+d}\right)^2=\left(\frac{bt+dt}{b+d}\right)^2=\left(\frac{t(b+d)}{b+d}\right)^2=t^2(3)\)

\(\frac{a^2+c^2}{b^2+d^2}=\frac{(bt)^2+(dt)^2}{b^2+d^2}=\frac{t^2(b^2+d^2)}{b^2+d^2}=t^2(4)\)

Từ $(3);(4)\Rightarrow \left(\frac{a+c}{b+d}\right)^2=\frac{a^2+c^2}{b^2+d^2}$ (đpcm)

Bài 2:

Từ $a^2=bc\Rightarrow \frac{a}{c}=\frac{b}{a}$

Đặt $\frac{a}{c}=\frac{b}{a}=t\Rightarrow a=ct; b=at$. Khi đó:

a)

$\frac{a^2+c^2}{b^2+a^2}=\frac{(ct)^2+c^2}{(at)^2+a^2}=\frac{c^2(t^2+1)}{a^2(t^2+1)}=\frac{c^2}{a^2}=(\frac{c}{a})^2=\frac{1}{t^2}(1)$

Và:

$\frac{c}{b}=\frac{a}{tb}=\frac{a}{t.at}=\frac{1}{t^2}(2)$

Từ $(1);(2)$ suy ra đpcm.

b)

$\left(\frac{c+2019a}{a+2019b}\right)^2=\left(\frac{c+2019a}{ct+2019at}\right)^2=\left(\frac{c+2019a}{t(c+2019a)}\right)^2=\frac{1}{t^2}(3)$

Từ $(2);(3)$ suy ra đpcm.

ý a ko cần giải đâu nha mk ra òi

Dễ thôi

Dùng mẹo nhé bạn

\(a+c=2b\)

\(\Rightarrow2bd=\left(a+c\right).d=cb+cd\)

\(\Rightarrow ad+cd=cb+cd\)

\(\Rightarrow ad+cd-cd=cb\)

\(ad=cb\Rightarrow\frac{a}{b}=\frac{c}{d}\left(đpcm\right)\)