Bài 1 :

a) Ta có : \(\left(1-a\right)\left(1-b\right)\left(1-c\right)=\left(a+b\right)\left(b+c\right)\left(c+a\right)\)

Áp dụng bđt Cauchy : \(a+b\ge2\sqrt{ab}\) , \(b+c\ge2\sqrt{bc}\) , \(c+a\ge2\sqrt{ca}\)

\(\Rightarrow\left(a+b\right)\left(b+c\right)\left(c+a\right)\ge8abc\) hay \(\left(1-a\right)\left(1-b\right)\left(1-c\right)\ge8abc\)

Gọi \(\frac{a}{b}=\frac{c}{d}=k\Rightarrow a=kb;c=kd\)(1)

Thay (1) vào ta có :

\(\frac{5a+3b}{5a-3b}=\frac{5kb+3b}{5kb-3b}=\frac{b\left(5k-3\right)}{b\left(5k-3\right)}=\frac{5k+3}{5k-3}\)(2)

\(\frac{5c+3d}{5c-3d}=\frac{5kd+3d}{5kd-3d}=\frac{d\left(5k+3\right)}{d\left(5k-3\right)}=\frac{5k+3}{5k-3}\)(3)

Từ (2) và (3)

\(\Rightarrow\frac{5a+3b}{5a-3b}=\frac{5c+3d}{5c-3d}\)

\(\RightarrowĐPCM\)

ta có:a/b=c/d suy ra a/c=b/d suy ra 5a/5c=3b/3d

áp dụng tính chất dãy tỉ số bằng nhau ta có

a/c=b/d=5a/5c=3b/3d=5a+3b/5c+3d=5a-3b=5c-3d

Suy ra ĐPCM

Đặt\(\frac{a}{b}\)=\(\frac{c}{d}\)= k => a = bk ; c = dk

Xét\(\frac{5a+3b}{5a-3b}=\frac{5bk+3b}{5bk-3b}=\frac{b\left(5k+3\right)}{b\left(5k-3\right)}=\frac{5k+3}{5k-3}\)(1)

Xét\(\frac{5c+3d}{5c-3d}=\frac{5dk+3d}{5dk-3d}=\frac{d\left(5k+3\right)}{d\left(5k-3\right)}=\frac{5k+3}{5k-3}\)(2)

Từ (1) và (2) => \(\frac{5a+3b}{5a-3b}=\frac{5c+3d}{5c-3d}\)(cùng = \(\frac{5k+3}{5k-3}\))

Lời giải

A = -4/5x(1/2+1/3+1/4)= -4/5x1 = -4/5
B = 6/19 x ( 3/4+4/3+-1/2)= 6/19x 19 = 6
C = 2002/2003x(3/4+5/6-19/12)=2003/2002x0=0

Đặt : \(\frac{a}{b}=\frac{c}{d}=k\)

\(\Rightarrow a=bk;c=dk\)

\(\Rightarrow\frac{7b^2k^2+3bkb}{11b^2k^2-8b^2}=\frac{7d^2k^2+3dkd}{11d^2k^2-8d^2}\)

\(\Rightarrow\frac{b^2\left(7k^2+3k\right)}{b^2\left(11k^2-8\right)}=\frac{d^2\left(7k^2+3k\right)}{d^2\left(11k^2-8\right)}\)

\(\Rightarrow\frac{7k^2+3k}{11k^2-8}=\frac{7k^2+3k}{11k^2-8}\left(đpcm\right)\)

\(\Leftrightarrow\left(2a+13b\right)\left(3c-7d\right)=\left(2c+13d\right)\left(3a-7b\right)\)

\(\Leftrightarrow6ac-14ad+39bc-91bd=6ac-14bc+39ad-91bd\)

\(\Leftrightarrow-14ad+14bc=39ad-39bc\)

\(\Leftrightarrow-14\left(ad-bc\right)=39\left(ad-bc\right)\)

=>ad-bc=0

=>ad=bc

hay a/b=c/d

\(^{2^{25}}\) là \(2^{25}\) mé các bạn, mình sợ mọi người nhầm

Đợi tí nha bạn Phạm Mai Linh

Câu 1 :                                               Bài giải

Theo đề bài : \(x\text{ : }y\text{ : }z=5\text{ : }4\text{ : }3\text{ }\Rightarrow\text{ }\frac{x}{5}=\frac{y}{4}=\frac{z}{3}=\frac{x+y-z}{5+4-3}=\frac{x+y-z}{6}=\frac{x-y+z}{5-4+3}=\frac{x-y+z}{4}\)

( Áp dụng t/c dãy tỉ số bằng nhau )

\(\Rightarrow\text{ }x+y-z=x-y+z\)

\(\Rightarrow\text{ }y=x-y+z+z-x=2z+y\)

\(A=\frac{x+2\cdot y-3\cdot z}{x-2\cdot y+3\cdot z}=\frac{\left(x+y-z\right)+\left(y-2z\right)}{\left(x-y+z\right)+\left(2z-y\right)}=\frac{\left(x+y-z\right)+\left(2z+y-2z\right)}{\left(x-y+z\right)+\left(2z-2z-y\right)}=\frac{\left(x+y-z\right)+y}{\left(x-y+z\right)+\left(-y\right)}\)

Đến đây chịu ! Nhưng giải gần xong rồi !