Ta có: \(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=0\)

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

=>\(\frac{1}{a^2}=-\left(\frac{1}{ab}+\frac{1}{ca}\right)\)

cm tương tự: \(\frac{1}{b^2}=-\left(\frac{1}{ab}+\frac{1}{bc}\right)\)

                     \(\frac{1}{c^2}=-\left(\frac{1}{ca}+\frac{1}{bc}\right)\)

=> \(N=-\left[bc\left(\frac{1}{ab}+\frac{1}{ca}\right)+ca\left(\frac{1}{ab}+\frac{1}{bc}\right)+ab\left(\frac{1}{ca}+\frac{1}{bc}\right)\right]\)

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

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

Ta có : \(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=0\)

=>\(\frac{a+b+c}{a}+\frac{a+b+c}{b}+\frac{a+b+c}{c}=0\)

=>\(1+\frac{b+c}{a}+1+\frac{a+c}{b}+1+\frac{a+b}{c}=0\)

=>\(\frac{b+c}{a}+\frac{a+c}{b}+\frac{a+b}{c}=-3\)   (2)

Từ (1) và (2) =>N=3

\(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=0\Leftrightarrow\frac{ab+bc+ca}{abc}=0\Rightarrow ab+bc+ca=0\\ \)

\(\Rightarrow bc=-ab-ac,ca=-ab-bc,ab=-bc-ca\)

\(\Rightarrow\frac{a^2+bc}{a^2+2bc}=\frac{a^2+bc}{a^2+bc+bc}=\frac{a^2+bc}{a^2+bc-ca-ab}=\frac{a^2+bc}{\left(a-b\right).\left(a-c\right)}\)

     Làm tương tự. có: \(\frac{b^2+ca}{b^2+2ca}=\frac{b^2+ca}{b^2+ca-ab-bc}=\frac{b^2+ca}{\left(a-b\right).\left(c-b\right)}\)

 \(\frac{c^2+ab}{c^2+2ab}=\frac{c^2+ab}{c^2+ab-ca-bc}=\frac{c^2+ab}{\left(b-c\right).\left(a-c\right)}\)

\(\Rightarrow A=\frac{a^2+bc}{\left(a-b\right).\left(a-c\right)}+\frac{b^2+ca}{\left(a-b\right).\left(c-b\right)}+\frac{c^2+ab}{\left(b-c\right).\left(a-c\right)}\)

\(=\frac{\left(a^2+bc\right).\left(b-c\right)}{\left(a-b\right).\left(b-c\right).\left(a-c\right)}-\frac{\left(b^2+ca\right).\left(a-c\right)}{\left(a-b\right).\left(b-c\right).\left(a-c\right)}+\frac{\left(c^2+ab\right).\left(a-b\right)}{\left(a-b\right).\left(b-c\right).\left(a-c\right)}\)

Sau đó bạn thực hiện tiếp nhé.

Bài 1: Cho \(a,b,c\ge0:a^2+b^2+c^2=3\). CMR: \(a^4b^4+b^4c^4+c^4a^4\le3\)

Bài 2: Cho \(a,b,c\ge0\). CMR: \(a^2+b^2+c^2+2abc+1\ge2\left(ab+bc+ca\right)\)

Bài 3: Cho \(a,b,c\ge0:a^2+b^2+c^2=a+b+c\). CMR: \(a^2b^2+b^2c^2+c^2a^2\le ab+bc+ca\)

Bài 4: Cho \(a,b,c\ge0\). CMR: \(4\left(a+b+c\right)^3\ge27\left(ab^2+bc^2+ca^2+abc\right)\)

Bài 5: Cho \(a,b,c\ge0:a+b+c=3\).CMR: \(\frac{1}{2bc^2+1}+\frac{1}{2ca^2+1}+\frac{1}{2ab^2+1}\ge1\)

Ta có :\(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=0\)

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

\(\Leftrightarrow\left(\frac{1}{a}+\frac{1}{b}\right)^3=-\frac{1}{c^3}\)

\(\Leftrightarrow\frac{1}{a^3}+\frac{1}{b^3}+\frac{1}{c^3}=-3\cdot\frac{1}{ab}\left(\frac{1}{a}+\frac{1}{b}\right)=3\cdot\frac{1}{abc}\)

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

Khi đó : \(P=abc\left(\frac{1}{a^3}+\frac{1}{b^3}+\frac{1}{c^3}\right)=abc\cdot\frac{3}{abc}=3\)

Với \(a+b+c=0\) thì \(a^3+b^3+c^3=3abc\) 

Chứng minh : với \(a+b+c=0\) thì \(a=-\left(b+c\right)\Leftrightarrow a^3=-\left(b+c\right)^3\)

\(\Leftrightarrow a^3=-\left(b^3+c^3+3b^2c+3bc^2\right)\Leftrightarrow a^3+b^3+c^3=-\left(b^3+c^3+3b^2c+3bc^2\right)+b^3+c^3\)

\(\Leftrightarrow a^3+b^3+c^3=-3bc\left(b+c\right)=-3bc\left(-a\right)=3abc\)vì \(b+c=-a\) =>đpcm

Vì \(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=0\Leftrightarrow\)\(\frac{1}{a^3}+\frac{1}{b^3}+\frac{1}{c^3}=\frac{3}{abc}\)

Vậy \(P=\frac{ab}{c^2}+\frac{bc}{a^2}+\frac{ca}{b^2}=abc\left(\frac{1}{c^3}+\frac{1}{a^3}+\frac{1}{b^3}\right)=abc\frac{3}{abc}=3\)

**** mình nha 

cho 2014=2013+1 thay vào ta có:\(B=x^{2013}-\left(2013+1\right)x^{2012}+\left(2013+1\right)x^{2011}-...-\left(2013+1\right)x^2+\left(2013+1\right)x-1\)

\(=x^{2013}-\left(x+1\right)x^{2012}+\left(x+1\right)x^{2011}-...-\left(x+1\right)x^2+\left(x+1\right)x-1\)

\(=x^{2013}-x^{2013}-x^{2012}+x^{2012}+x^{2011}-...-x^3-x^2+x^2+x-1\)

\(=x-1=2013-1=2012\)

nhiều quá

lớp 7 hả

Từ \(\frac{ab}{a+b}=\frac{bc}{b+c}=\frac{ca}{c+a}\)

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

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

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

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

\(\Rightarrow M=\frac{ab+bc+ca}{a^2+b^2+c^2}=\frac{a\cdot a+a\cdot a+a\cdot a}{a^2+a^2+a^2}=\frac{a^2+a^2+a^2}{a^2+a^2+a^2}=1\)

theo bài ra ta có:

\(\frac{ab}{a+b}=\frac{bc}{b+c}=\frac{ca}{c+a}\)

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

=> \(\frac{abc}{ca+cb}=\frac{abc}{ab+ac}=\frac{abc}{bc+ba}\)

vì a,b,c khác 0 => ca+cb = ab+ac = bc+ba

=> a = b = c

ta có:

\(M=\frac{ab+bc+ca}{a^2+b^2+c^2}=\frac{a^2+a^2+a^2}{a^2+a^2+a^2}=1\)

vậy M = 1

Mik ko bk đúng hay sai đâu nha!