Đặt \(\hept{\begin{cases}a-b=x\\b-c=y\\c-a=z\end{cases}}\)

Thế vào bài toán trở thành 

Cho: \(\frac{x+z}{xz}+\frac{x+y}{xy}+\frac{y+z}{yz}=2013\left(1\right)\)

Tính \(M=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\)

Từ (1) ta có

\(\left(1\right)\Leftrightarrow\frac{xy+yz+zx+yz+xy+zx}{xyz}=2013\)

\(\Leftrightarrow\frac{2\left(xy+yz+zx\right)}{xyz}=2013\)

\(\Leftrightarrow\frac{xy+yz+zx}{xyz}=\frac{2013}{2}\)

Ta lại có

\(M=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{xy+yz+zx}{xyz}=\frac{2013}{2}\)

\(\frac{b-c}{\left(a-b\right)\left(a-c\right)}+\frac{c-a}{\left(b-a\right)\left(b-c\right)}+\frac{a-b}{\left(c-b\right)\left(c-a\right)}\)

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

\(=\frac{1}{a-b}-\frac{1}{a-c}+\frac{1}{b-c}-\frac{1}{b-a}+\frac{1}{c-a}-\frac{1}{c-b}\)

\(=2\left(\frac{1}{a-b}+\frac{1}{b-c}+\frac{1}{c-a}\right)=2013\)

\(\Rightarrow M=\frac{2013}{2}\)

đây là bài trong đề thi tớ mà, lúc đó là 5/12 sao bạn chép ra đây để hỏi?

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

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

\(\Rightarrow P=\left(1+1\right)\left(1+1\right)\left(1+1\right)=8\)

b) Đề bài sai ^^

help me

=>\(\frac{a-b+c}{2b}+1=\frac{c-a+b}{2a}+1=\frac{a-c+b}{2c}+1\)

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

*TH1: nếu a+b+c=0 => a+b=-c; b+c=-a; c+a=-b

=>P=\(\left(\frac{b+c}{b}\right)\left(\frac{a+b}{a}\right)\left(\frac{c+a}{c}\right)\)

=\(\frac{-a}{b}.\frac{-c}{a}.\frac{-b}{c}=\frac{-\left(a.b.c\right)}{a.b.c}=-1\)

*TH2: Nếu a+b+c khác 0: thì a=b=c

Khi đó P=2.2.2=8

Vậy P= -1 hoặc 8

TH1 a+b+c khác 0 

Aps dụng tính......a−b+c2b =c−a+b2a =a−c+b2c =1/2 (tự tính )

* với a-b+c/2b=1/2 suy ra 2a-2b+2c=2b suy ra 2a+2c=4b suy ra 2(a+c)=2.2.b suy ra a+c =2b

tương tự với từng t/h một  thì ta được c+b=2a;a+b=2c

suy raP=(1+cb )(1+ba )(1+ac ) =(b+c/b)(a+b/a)(c+a/c)=2a/b.2c/a.2a/c=8

TH2a+b+c =0 tự cm

a) 9x² - 36

=(3x)²-6²

=(3x-6)(3x+6)

=4(x-3)(x+3)

b) 2x³y-4x²y²+2xy³

=2xy(x²-2xy+y²)

=2xy(x-y)²

c) ab - b²-a+b

=ab-a-b²+b

=(ab-a)-(b²-b)

=a(b-1)-b(b-1)

=(b-1)(a-b)

P/s đùng để ý đến câu trả lời của mình

dễ!Ta có:

\(\frac{b-c}{\left(a-b\right)\left(a-c\right)}=\frac{b-a+a-c}{\left(a-b\right)\left(a-c\right)}=\frac{b-a}{\left(a-b\right)\left(a-c\right)}+\frac{a-c}{\left(a-b\right)\left(a-c\right)}=\frac{1}{a-b}+\frac{1}{c-a}\)

Chứng minh tương tự,Ta được:

\(\hept{\begin{cases}\frac{c-a}{\left(b-c\right)\left(b-a\right)}=\frac{1}{a-b}+\frac{1}{b-c}\\\frac{a-b}{\left(c-a\right)\left(c-b\right)}=\frac{1}{c-a}+\frac{1}{b-c}\end{cases}}\)

\(\Rightarrow\frac{b-c}{\left(a-b\right)\left(a-c\right)}+\frac{c-a}{\left(b-a\right)\left(b-c\right)}+\frac{a-b}{\left(c-a\right)\left(c-b\right)}=\frac{2}{a-b}+\frac{2}{b-c}+\frac{2}{c-a}=2\left(\frac{1}{a-b}+\frac{1}{b-c}+\frac{1}{c-a}\right)=2013\)\(\Rightarrow\frac{1}{a-b}+\frac{1}{b-c}+\frac{1}{c-a}=\frac{2013}{2}\)

Xong!

\(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=0\Rightarrow\frac{1}{a}+\frac{1}{b}=\frac{-1}{c}\Rightarrow\frac{a+b}{ab}=\frac{-1}{c}\)

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

Tương tự : \(b+c=\frac{-bc}{a};a+c=\frac{-ac}{b}\)

thay vào A,ta được :

\(A=\frac{\frac{-ab}{c}.\frac{-bc}{a}.\frac{-ac}{b}}{abc}=\frac{-a^2b^2c^2}{abc}=-abc\)

nhầm đoạn cuối : \(A=\frac{-a^2b^2c^2}{a^2b^2c^2}=-1\)

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

\(\Leftrightarrow\frac{a+b}{c}+1=\frac{b+c}{a}+1=\frac{c+a}{b}+1\)

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

\(\Rightarrow a=b=c\)

\(\Rightarrow\frac{a}{b}=1;\frac{b}{c}=1;\frac{c}{a}=1\)

\(\Rightarrow M=\left(1+\frac{a}{b}\right)\left(1+\frac{b}{c}\right)\left(1+\frac{c}{a}\right)=\left(1+1\right)\left(1+1\right)\left(1+1\right)=2.2.2=8\)

\(\frac{a-b+c}{2b}=\frac{c-a+b}{2a}=\frac{a-c+b}{2c}=\frac{a+b+c}{2\left(a+b+c\right)}=\frac{1}{2}\)

=> 2a-2b+2c=2b <=> a+c=2b. Chia cả 2 vế cho c ta được: \(1+\frac{a}{c}=\frac{2b}{c}\)

Tương tự: \(1+\frac{c}{b}=\frac{2a}{b}\) và \(1+\frac{b}{a}=\frac{2c}{a}\)

=> \(\left(1+\frac{c}{b}\right)\left(1+\frac{b}{a}\right)\left(1+\frac{a}{c}\right)=\frac{2a}{b}.\frac{2c}{a}.\frac{2b}{c}=\frac{8.abc}{abc}=8\)

Đáp số: 8

tại sao 2a-2b+2c=2b lại suy ra a+c=2b vậy bạn

Thì 2a-2b+2c=2b  <=> 2a+2c=2b+2b <=> 2(a+c)=4b => a+c=2b