Áp dụng tính chất dãy tỉ số bằng nhau, ta có: 

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

Như vậy, \(\frac{a}{b}=\frac{a+c}{b+d}=\frac{a-c}{b-d}\) (đpcm)

Ta có : a^2+b^2/c^2+d^2 = ab/cd 
=> (a^2+b^2) . cd = (c^2+d^2). ab 
=> a.a.c.d+b.b.c.d = c.c.a.b + d.d.a.b
=> a.a.c.d-c.c.a.b - d.d.a.b + b.b.c.d= 0
=> ac(ad - bc) - bd(ad - bc) = 0 
=> (ac - bd)(ad - bc) = 0 
=> ac - bd = 0 hoặc ad - bc = 0 
=> ac = bd 
=> a/b =c/d  (đpcm)
 

Vì $abc\ne0$ nên từ $a+b+c=\dfrac1a+\dfrac1b+\dfrac1c$

suy ra $abc(a+b+c)=ab+bc+ca$.

Xét hiệu hai vế cần chứng minh:

$b(a^2-bc)(1-ac)-a(1-bc)(b^2-ac)$

$=-(a-b)\left[abc(a+b+c)-(ab+bc+ca)\right]$

$=-(a-b)\cdot0$

$=0$

Suy ra $\boxed{b(a^2-bc)(1-ac)=a(1-bc)(b^2-ac)}$

Ta có $(a+b+c)^2=a^2+b^2+c^2$

$\Rightarrow 2(ab+bc+ca)=0$

$\Rightarrow ab+bc+ca=0$

Ta có $\dfrac1{a^3}+\dfrac1{b^3}+\dfrac1{c^3}
=\dfrac{a^3b^3+b^3c^3+c^3a^3}{a^3b^3c^3}$

Mà $a^3b^3+b^3c^3+c^3a^3-3a^2b^2c^2$

$=(ab+bc+ca)(a^2b^2+b^2c^2+c^2a^2-abc(a+b+c))$

$=0$

Do đó $a^3b^3+b^3c^3+c^3a^3=3a^2b^2c^2$

Suy ra $\dfrac1{a^3}+\dfrac1{b^3}+\dfrac1{c^3}$

$=\dfrac{3a^2b^2c^2}{a^3b^3c^3}$

$=\boxed{\dfrac3{abc}}$

Lời giải:

Ta có:

\((ab+cd)^2=a^2b^2+c^2d^2+2abcd\)

\(=a^2b^2+c^2d^2-2abcd+4abcd\)

\(=(ab-cd)^2+4abcd\geq 4abcd=4\)

Vậy \((ab+cd)^2\geq 4\)

\(\Rightarrow ab+cd\geq \sqrt{4}=2\) (với \(ab+cd>0\))

Vậy......

1) Với x > 0 ta có:

\(x+\dfrac{1}{x}\ge2\\ \Leftrightarrow\dfrac{x^2+1}{x}\ge\dfrac{2x}{x}\\ \Leftrightarrow x^2+1\ge2x\left(\text{vì }x>0\right)\\ \Leftrightarrow x^2-2x+1\ge0\\ \Leftrightarrow\left(x-1\right)^2\ge0\left(\text{luôn đúng }\forall x>0\right)\)

Dấu "=" xảy ra \(\Leftrightarrow x=1\). Vậy BĐT được chứng mình với x > 0.

1: Áp dụng Bđt cosi, ta được:

\(x+\dfrac{1}{x}\ge2\cdot\sqrt{x\cdot\dfrac{1}{x}}=2\)

\(a^2+b^2+c^2+2ab-2ac-2bc=a^2+b^2\)

\(\Rightarrow\left(a+b-c\right)^2=a^2+b^2\)

\(\Rightarrow\hept{\begin{cases}a^2=\left(a+b-c\right)^2-b^2=\left(a+b-c-b\right)\left(a+b-c+b\right)=\left(a-c\right)\left(a+2b-c\right)\\b^2=\left(a+b-c\right)^2-a^2=\left(a+b-c-a\right)\left(a+b-c+a\right)=\left(b-c\right)\left(2a+b-c\right)\end{cases}}\)

\(a^2+\left(a-c\right)^2=\left(a-c\right)\left(a+2b-c\right)+\left(a-c\right)^2\)

\(=\left(a-c\right)\left(a+2b-c+a-c\right)=2\left(a-c\right)\left(a+b-c\right)\)

\(b^2+\left(b-c\right)^2=\left(b-c\right)\left(2a+b-c\right)+\left(b-c\right)^2\)

\(=\left(b-c\right)\left(2a+b-c+b-c\right)=2\left(b-c\right)\left(a+b-c\right)\)

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