Đặt a/b=b/3c=c/9a=k

Ta có: a/b=b/3c=c/9a

=>(a/b)³=(b/3c)³=(c/9a)³=(a.b.c)/(b.3c.9c)=1/27=k³

=>k= (1/3)

Ta có: b/3c=1/3

=>b=c (đpcm)

\(\frac{a}{b}=\frac{b}{3c}=\frac{c}{9a}=k\Leftrightarrow\left(\frac{a}{b}\right)^3=\frac{a.b.c}{b.3c.9a}=\frac{1}{27}=k^3\Leftrightarrow k=\frac{1}{3}\)

\(\frac{b}{3c}=\frac{1}{3}\Leftrightarrow b=c\)

Sửa đề: Cho ba số thực a,b,c dương

Áp dụng BĐT Cauchy Schwarz, ta được:

\(VT=\left(a+b+c\right)\left(\frac{9}{bc}+\frac{25}{c+a}+\frac{64}{a+b}\right)-98\ge\left(a+b+c\right)\left(\frac{256}{2\left(a+b+c\right)}\right)-98=30\)

\(\Leftrightarrow VT\ge30\)

Dấu '=' xảy ra khi \(\frac{8}{a+b}=\frac{5}{c+a}=\frac{3}{b+c}\)

\(\Leftrightarrow\frac{8}{a+b}=\frac{8}{a+b+2c}\)

hay c=0(vô lý)

=> Dấu bằng không xảy ra

=>ĐPCM

Lời giải:
Đặt \(\frac{a}{b}=\frac{c}{d}=t\Rightarrow a=bt; c=dt\)

Khi đó:

a) Đề bài sai. Bạn xem lại đề.

b) Cần thêm điều kiện $a\neq \pm b; c\neq \pm d$

Khi đó \(t=\frac{a}{b}=\frac{c}{d}\neq \pm 1\)

\(\frac{a+b}{c+d}=\frac{bt+b}{dt+d}=\frac{b(t+1)}{d(t+1)}=\frac{b}{d}\)

\(\frac{a-b}{c-d}=\frac{bt-b}{dt-d}=\frac{b(t-1)}{d(t-1)}=\frac{b}{d}\)

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

Đặt \(\hept{\begin{cases}b+c=x>0\\c+a=y>0\\a+b=z>0\end{cases}}\Rightarrow\hept{\begin{cases}a=\frac{y+z-x}{2}\\b=\frac{z+x-y}{2}\\x=\frac{x+y-z}{2}\end{cases}}\)

Bất đẳng thức cần chứng minh tương đương:

\(\frac{9\left(y+z-x\right)}{2x}+\frac{25\left(z+x-y\right)}{2y}+\frac{64\left(x+y-z\right)}{2z}>30\)

Ta có: \(VP=\frac{9y}{2x}+\frac{9z}{2x}-\frac{9}{2}+\frac{25z}{2y}+\frac{25x}{2y}-\frac{9}{2}+\frac{32x}{z}+\frac{32y}{z}-32\)

\(=\left(\frac{9y}{2x}+\frac{25x}{2y}\right)+\left(\frac{9z}{2x}+\frac{32x}{z}\right)+\left(\frac{25z}{2y}+\frac{32y}{z}\right)-41\)

\(\ge2\cdot\frac{15}{2}+2\cdot12+2\cdot20-41=38>30\)

\(\Rightarrow\frac{9a}{b+c}+\frac{25b}{c+a}+\frac{64c}{a+b}>30\)

Đặt \(\frac{a}{b}=\frac{b}{3c}=\frac{c}{9a}=k\)

Ta có: \(\frac{a}{b}=\frac{b}{3c}=\frac{c}{9a}\)

\(\Rightarrow\left(\frac{a}{b}\right)^3=\left(\frac{b}{3c}\right)^3=\left(\frac{c}{9a}\right)^3=\frac{a.b.c}{b.3c.9a}=\frac{1}{27}=k^3\)

\(\Rightarrow k=\frac{1}{3}\)

Ta có: \(\frac{b}{3c}=\frac{1}{3}\)

\(\Rightarrow b=\frac{1}{3}.3c=c\)

Vậy \(b=c\left(đpcm\right)\)

đáp số 

a=b

hok tốt

Áp dụng BĐT Cauchy cho 2 số dương ta được :

\(\dfrac{a^2}{b+3c}+\dfrac{b+3c}{16}\ge2\sqrt{\dfrac{a^2}{b+3c}\times\dfrac{b+3c}{16}}=\dfrac{2a}{4}\)

Suy ra \(\dfrac{a^2}{b+3c}\ge\dfrac{2a}{4}-\dfrac{b+3c}{16}\)

Cmtt ta cũng được :

\(\dfrac{b^2}{c+3a}\ge\dfrac{2b}{4}-\dfrac{c+3a}{16}\) \(\dfrac{c^2}{a+3b}\ge\dfrac{2c}{4}-\dfrac{a+3b}{16}\)

Khi đó :

\(\dfrac{a^2}{b+3c}+\dfrac{b^2}{c+3a}+\dfrac{c^2}{a+3b}\ge\dfrac{2a}{4}-\dfrac{b+3c}{16}+\dfrac{2b}{4}-\dfrac{c+3a}{16}+\dfrac{2c}{4}-\dfrac{a+3b}{16}\)

mà \(\dfrac{2a}{4}-\dfrac{b+3c}{16}+\dfrac{2b}{4}-\dfrac{c+3a}{16}+\dfrac{2c}{4}-\dfrac{a+3b}{16}=\dfrac{a+b+c}{4}\)

Vậy \(\dfrac{a^2}{b+3c}+\dfrac{b^2}{c+3a}+\dfrac{c^2}{a+3b}\ge\dfrac{a+b+c}{4}\)

Áp dụng bất đẳng thức Cauchy - Schwarz dạng phân thức

\(\Rightarrow\dfrac{a^2}{b+3c}+\dfrac{b^2}{c+3a}+\dfrac{c^2}{a+3b}\ge\dfrac{\left(a+b+c\right)^2}{4\left(a+b+c\right)}=\dfrac{a+b+c}{4}\) (đpcm)

Dấu " = " xảy ra khi \(a=b=c\)