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2 tháng 12 2015

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

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

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

\(\Leftrightarrow\frac{1}{a^3}+3.\frac{1}{a^2}.\frac{1}{b}+3.\frac{1}{a}.\frac{1}{b^2}+\frac{1}{b^3}=-\frac{1}{c}\)

\(\Leftrightarrow\frac{1}{a^3}+\frac{1}{b^3}+\frac{3}{a^2b}+\frac{3}{ab^2}=-\frac{1}{c^3}\)

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

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

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

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

Do đó:

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

 

22 tháng 9 2017

Cậu viết gì vậy?

22 tháng 9 2017

dễ mà

\(A=\frac{abc}{c^3}+\frac{abc}{b^3}+\frac{abc}{a^3}\)

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

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

ta có

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

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

\(A=abc.\frac{3}{abc}=3\)

9 tháng 12 2018

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

\(\Rightarrow ab+bc+ac=0\Rightarrow\hept{\begin{cases}ab=-ac-bc\\bc=-ab-ac\\ac=-ab-bc\end{cases}}\)

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

Tương tự: \(b^2+2ac=\left(b-c\right)\left(b-a\right)\)

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

\(B=\frac{bc+1}{\left(a-b\right)\left(a-c\right)}+\frac{ca+1}{\left(b-a\right)\left(b-c\right)}+\frac{ab+1}{\left(a-c\right)\left(b-c\right)}\)

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

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

\(\left(bc+1\right)\left(b-c\right)-\left(ca+1\right)\left(a-c\right)+\left(ab+1\right)\left(a-b\right)\)

\(=\left(bc+1\right)\left(b-c\right)-\left(ca+1\right)\left(a-b\right)-\left(ca+1\right)\left(b-c\right)+\left(ab+1\right)\left(a-b\right)\)

\(=\left(b-c\right)\left(bc+1-ca-1\right)+\left(a-b\right)\left(ab+1-ca-1\right)\)

\(=\left(b-c\right)\left(bc-ca\right)+\left(a-b\right)\left(ab-ca\right)\)

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

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

Vậy B = 1

30 tháng 6 2018

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

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

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

     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.

2 tháng 8 2021

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: \(\frac{1}{2bc^2+1}+\frac{1}{2ca^2+1}+\frac{1}{2ab^2+1}\ge1\)

25 tháng 11 2016

Chú ý rằng nếu \(x+y+z=0\) thì \(x^3+y^3+z^3=3xyz.\).

Nếu vậy : \(x+y+z=0\) \(\Rightarrow z=-\left(x+y\right).\)

Do đó : \(x^3+y^3+z^3=x^3+y^3-\left(x+y\right)^3=-3x^2y-3xy^2=-3xy\left(x+y\right)=3xyz\)

Từ đây có thể suy ra :

\(x^3+y^3+z^3-3xyz=\left(x+y+z\right)\left(x^2+y^2+z^2-xy-xz-yz\right)\)

Áp dụng nhận xét trên,ta có :

Nếu \(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=0\) thì \(\frac{1}{a^3}+\frac{1}{b^3}+\frac{1}{c^3}=3.\frac{1}{a}.\frac{1}{b}.\frac{1}{c}=\frac{3}{abc}.\)

Do đó : \(N=\frac{bc}{a^2}+\frac{ca}{b^2}+\frac{ab}{c^2}=\frac{abc}{a^3}+\frac{abc}{b^3}+\frac{abc}{c^3}\)

\(=abc\left(\frac{1}{a^3}+\frac{1}{b^3}+\frac{1}{c^3}\right)=abc.\frac{3}{abc}=3\) với \(a,b,c\ne0\).

5 tháng 7 2016

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

        

2 tháng 1 2017

từ giả thiết ta có

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

Nhân hai vế với \(\frac{a}{bc-a^2}\) ta có:

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

làm tương tự với hai số hạng còn lại ta được:

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

cộng ba vế của đẳng thức trên ta được kq là 0 hihi

2 tháng 1 2017

cách kia dài quá

Đặt \(x=bc-a^2;y=ac-b^2;z=ab-c^2\)

Suy ra cần chứng minh \(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=0\) thì \(\frac{a}{x^2}+\frac{b}{y^2}+\frac{c}{z^2}=0\)

Xét \(T=\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)\left(\frac{a}{x}+\frac{b}{y}+\frac{c}{z}\right)\).....

6 tháng 7 2019

Em tham khảo link:Câu hỏi của Conan Kudo - Toán lớp 8 - Học toán với OnlineMath

Ta có bổ đề

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

ÁP DỤNG BỔ ĐỀ VÀO P ta có

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

\(=abc.\frac{3}{abc}=3\)

Vậy P=3

7 tháng 3 2020

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