cho a,b,c\(\ne\)0 thỏa mãn \(\frac{a}{b}=\frac{b}{c}=\frac{c}{a}\)Tính \(\frac{^{a^2}+b^2+c^2}{\left(a+b+c\right)^2}\)
Cho a, b, c\(\ne\)0, thỏa mãn:
\(\frac{a+b}{c}+\frac{b+c}{a}+\frac{c+a}{b}-\frac{a^3+b^3+c^3}{abc}=2\)
Tính \(H=\left(\left(a+b\right)^{2017}-c^{2017}\right)\left(\left(b+c\right)^{2017}-a^{2017}\right)\left(\left(c+a\right)^{2017}-b^{2017}\right)\)
cho a,b,c \(\ne\)0 thỏa mãn a+b+c = 0 thỏa mãm a+b+c = 0 . Tính \(A=\left(1+\frac{a}{b}\right)\left(1+\frac{b}{c}\right)\left(1+\frac{c}{a}\right)\)
\(A=\left(\frac{a+b}{b}\right).\left(\frac{b+c}{c}\right).\left(\frac{c+a}{a}\right)\)
Vì \(a+b+c=0\)
\(\Rightarrow\hept{\begin{cases}a+b=-c\\a+c=-b\\c+b=-a\end{cases}}\)
\(\Rightarrow A=\frac{-c}{b}.\left(\frac{-a}{c}\right).\left(\frac{-b}{a}\right)=-1\)
Ta có: \(a+b+c=0\)
\(\Rightarrow b+a=-c\)
\(\Rightarrow c+b=-a\)
\(\Rightarrow a+c=-b\)
Ta có: \(A=\left(1+\frac{a}{b}\right)\left(1+\frac{b}{c}\right)\left(1+\frac{c}{a}\right)\)
\(\Rightarrow A=\left(\frac{b+a}{b}\right)\left(\frac{c+b}{c}\right)\left(\frac{a+c}{a}\right)\)
\(\Rightarrow A=\left(\frac{-c}{b}\right)\left(\frac{-a}{c}\right)\left(\frac{-b}{a}\right)\)
\(\Rightarrow A=-1\)
~~k cho mik nha~~
1. Cho a,b,c,x,y,z khác 0 thỏa mãn:
\(\frac{7cy-5bz}{x}=\frac{2az-7cx}{y}=\frac{5bx-2ay}{z}\)
CMR: \(\frac{2a}{x}=\frac{5b}{y}=\frac{7c}{z}\)
2.Cho a,b,c,x,y,z khác 0 thỏa mãn: \(\frac{x}{a}=\frac{y}{b}=\frac{z}{c}\)
CMR: \(\frac{x^2+y^2+z^2}{\left(ax+by+cz\right)^2}=\frac{1}{a^2+b^2+c^2}\)
3.Cho a,b,c thỏa mãn \(\frac{a}{2016}=\frac{b}{2017}=\frac{c}{2018}\)
CMR: 4(a-b)(b-c)=(a-c)2
4. Cho a,b,c thỏa mãn:\(\frac{a}{x}=\frac{b}{x+1}=\frac{c}{x+2}\)
CMR: 4(a-b)(b-c)=(a-c)2
5. Cho a,b,c thỏa mãn:
\(\frac{a}{-2017}=\frac{b}{-2016}=\frac{c}{-2015}\)
CMR: 4(a-b)(b-c)=(a-c)2
6. Cho a,b,c khác 0 và \(\frac{b+c+a}{a}=\frac{a+b-c}{b}=\frac{c+a-b}{c}\)
Tính giá trị biểu thức A=\(\frac{\left(a-b\right)\left(c+b\right)\left(c-a\right)}{abc}\)
Cho a,b,c thỏa mãn a+b+c=0
Tính\(G=\frac{1}{b^2+c^2-a^2}+\frac{1}{c^2+a^2-b^2}+\frac{1}{a^2+b^2-c^2}\)
\(D=\left(\frac{a-b}{c}+\frac{b-c}{a}+\frac{c-a}{b}\right)\left(\frac{c}{a-b}+\frac{a}{b-c}+\frac{b}{c-a}\right)\)
a+b+c=0 <=> a+b=-c ; a+c=-b ; b+c=-a
\(\frac{1}{b^2+c^2-a^2}=\frac{1}{\left(b-a\right)\left(a+b\right)+c^2}=\frac{1}{\left(b-a\right)\left(-c\right)+c^2}=\frac{1}{c\left(a-b+c\right)}=\frac{1}{-2bc}\)
Tương tự: \(\frac{1}{c^2+a^2-b^2}=\frac{1}{-2ca};\frac{1}{a^2+b^2-c^2}=\frac{1}{-2ab}\)
=>\(G=\frac{1}{-2bc}+\frac{1}{-2ca}+\frac{1}{-2ab}=\frac{a+b+c}{-2abc}=\frac{0}{-2abc}=0\)
Cho các số thực a,b,c thỏa mãn : \(\frac{a}{b-c}+\frac{b}{c-a}+\frac{c}{a-b}=0\). Chứng minh rằng :
\(\frac{a}{\left(b-c\right)^2}+\frac{b}{\left(c-a\right)^2}+\frac{c}{\left(a-b\right)^2}=0\)
Bài 1:Cho a,b,c là các số nguyên đôi 1 khác nhau thỏa mãn a+b+c=2019.tính giá trị biểu thức
\(M=\frac{a^3}{\left(a+b\right)\left(a-c\right)}+\frac{b^3}{\left(b-a\right)\left(b-c\right)}+\frac{c^3}{\left(c-a\right)\left(c-b\right)}\)
Bài 2:Cho \(a+b+c=0;P=\frac{a-b}{c}+\frac{b-c}{a}+\frac{c-a}{b};Q=\frac{c}{a-b}+\frac{a}{b-c}+\frac{b}{c-a}\)
\(CMR\) \(P\cdot Q=9\)
Bài 3:Cho 3 số x;y;z đôi 1 khác nhau thỏa mãn x+y+z=0 và \(A=\frac{4xy-z^2}{xy+2z^2};B=\frac{4yz-x^2}{yz+2x^2};C=\frac{4xz-y^2}{xz+2y^2}\)
CMR A.B.C=1
Đặt \(\left(\frac{a-b}{c},\frac{b-c}{a},\frac{c-a}{b}\right)\rightarrow\left(x,y,z\right)\)
Khi đó:\(\left(\frac{c}{a-b},\frac{a}{b-c},\frac{b}{c-a}\right)\rightarrow\left(\frac{1}{x},\frac{1}{y},\frac{1}{z}\right)\)
Ta có:
\(P\cdot Q=\left(x+y+z\right)\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)=3+\frac{y+z}{x}+\frac{z+x}{y}+\frac{x+y}{z}\)
Mặt khác:\(\frac{y+z}{x}=\left(\frac{b-c}{a}+\frac{c-a}{b}\right)\cdot\frac{c}{a-b}=\frac{b^2-bc+ac-a^2}{ab}\cdot\frac{c}{a-b}\)
\(=\frac{c\left(a-b\right)\left(c-a-b\right)}{ab\left(a-b\right)}=\frac{c\left(c-a-b\right)}{ab}=\frac{2c^2}{ab}\left(1\right)\)
Tương tự:\(\frac{x+z}{y}=\frac{2a^2}{bc}\left(2\right)\)
\(=\frac{x+y}{z}=\frac{2b^2}{ac}\left(3\right)\)
Từ ( 1 );( 2 );( 3 ) ta có:
\(P\cdot Q=3+\frac{2c^2}{ab}+\frac{2a^2}{bc}+\frac{2b^2}{ac}=3+\frac{2}{abc}\left(a^3+b^3+c^3\right)\)
Ta có:\(a+b+c=0\)
\(\Rightarrow\left(a+b\right)^3=-c^3\)
\(\Rightarrow a^3+b^3+3ab\left(a+b\right)=-c^3\)
\(\Rightarrow a^3+b^3+c^3=3abc\)
Khi đó:\(P\cdot Q=3+\frac{2}{abc}\cdot3abc=9\)
Cho a,b,c khác nhau thỏa mãn : \(\frac{a}{b-c}+\frac{b}{c-a}+\frac{c}{a-b}=0\)
cm: \(\frac{1}{\left(b-c\right)^2}+\frac{1}{\left(c-a\right)^2}+\frac{1}{\left(a-b\right)^2}\)
Cho a,b,c đôi một khác nhau và thỏa mãn
\(\frac{a}{b-c}+\frac{b}{c-a}+\frac{c}{a-b}=0\)
CMR: \(\frac{a}{\left(b-c\right)^2}+\frac{b}{\left(c-a\right)^2}+\frac{c}{\left(a-b\right)^2}=0\)
Cho a , b , c là 3 số từng đôi một khác nhau và thỏa mãn :\(\frac{a}{b-c}+\frac{b}{c-a}+\frac{c}{a-b}=0\). CMR :
\(\frac{a}{\left(b-c\right)^2}+\frac{b}{\left(c-a\right)^2}+\frac{c}{\left(a-b\right)^2}=0\).
\(\frac{a}{b-c}=-\frac{b}{c-a}-\frac{c}{a-b}=-\frac{b\left(a-b\right)+c\left(c-a\right)}{\left(c-a\right)\left(a-b\right)}\Rightarrow\frac{a}{\left(b-c\right)^2}=-\frac{b\left(a-b\right)+c\left(c-a\right)}{\left(a-b\right)\left(b-c\right)\left(c-c\right)}\)
sau đó chứng minh tương tự và cộng theo từng vế thôi