Ta có: \(a+b+c=0\Leftrightarrow a+b=-c\)
\(\Leftrightarrow\left(a+b\right)^3+c^3=0\)
\(\Leftrightarrow a^3+b^3+c^3+3ab\left(a+b\right)=0\)
\(\Leftrightarrow a^3+b^3+c^3-3abc=0\Leftrightarrow a^3+b^3+c^3=3abc\left(\circledast\right)\)
\(A=\dfrac{a^2}{bc}+\dfrac{b^2}{ac}+\dfrac{c^2}{ab}=\dfrac{1}{abc}\left(a^3+b^3+c^3\right)=\dfrac{3abc}{abc}=3\left(từ\circledast\right)\)
Cho a, b, c khác 0: \(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=0\)
Tính M = \(\dfrac{bc}{a^2}+\dfrac{ac}{b^2}+\dfrac{ab}{c^2}\)
Bài 1: Cho \(\text{a+b+c=ab+bc+ac=abc}\) \(\ne\) \(0\) và \(\dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2}=2\)
Tính \(A=\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\)
Bài 2: Cho \(a,b,c\ne0\). CMR nếu \(x,y\) thỏa mãn :
\(\dfrac{a}{c}x+\dfrac{b}{c}y=\dfrac{b}{a}x+\dfrac{c}{a}y=\dfrac{c}{b}x+\dfrac{a}{b}y=1\)
thì \(\dfrac{a^2}{bc}+\dfrac{b^2}{ac}+\dfrac{c^2}{ab}=3\)
Bài 3: Cho \(ax+by+cz=0\) và \(a+b+c=\dfrac{1}{2019}\)
Tính \(A=\dfrac{a^2x^2+b^2y^2+c^2z^2}{bc\left(y-z\right)^2+ac\left(x-z\right)^2+ab\left(x-y\right)^2}\)
Cho a,b,c đôi một và \(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=0\) . Rút gọn
N=\(\dfrac{bc}{a^2+2bc}+\dfrac{ac}{b^2+2ac}+\dfrac{ab}{c^2+2ab}\)
M=\(\dfrac{a^2}{a^2+2bc}+\dfrac{b^2}{b^2+2ac}+\dfrac{c^2}{c^2+2ab}\)
1)Cho A,B,C>0.CMR:\(\dfrac{AB}{C}+\dfrac{AC}{B}+\dfrac{BC}{A}>A+B+C\)
2)CMR:A2+B2+C2+D2+4\(\ge2\left(A+B+C+D\right)\)
Cho a+b+c=0 (a,b,c≠0). Rút gọn biểu thức:
a) A=\(\dfrac{a^2}{bc}\)+\(\dfrac{b^2}{ca}\)+\(\dfrac{c^2}{ab}\)
b) B=\(\dfrac{a^2}{a^2-b^2-c^2}\)+\(\dfrac{b^2}{b^2-c^2-a^2}\)+\(\dfrac{c^2}{c^2-a^2-b^2}\)
Cho \(\dfrac{a^3}{a^2+ab+b^2}+\dfrac{b^3}{b^2+bc+c^2}+\dfrac{c^3}{c^2+ca+a^2}\).Tính M=\(\dfrac{a^3+b^3}{a^2+ab+b^2}+\dfrac{b^3+c^3}{b^2+bc+c^2}+\dfrac{c^3+a^3}{c^2+ca+a^2}\)
Cho 3 số a , b , c khác 0 thỏa mãn : \(\dfrac{a^2}{b^2}+\dfrac{b^2}{c^2}+\dfrac{c^2}{a^2}=\dfrac{a}{c}+\dfrac{c}{b}+\dfrac{b}{a}\)
Chứng minh rằng : a=b=c
Cho \(\dfrac{a}{b-c}+\dfrac{b}{c-a}+\dfrac{c}{a-b}=0\). Chứng minh rằng
\(\dfrac{a}{\left(b-c\right)^2}+\dfrac{b}{\left(c-a\right)^2}+\dfrac{c}{\left(a-b\right)^2}=0\)
Bài 1: Cho \(a,b,c>0;a+b+c=3\)
Tìm min \(A=\dfrac{a^2}{b+1}+\dfrac{b^2}{c+1}+\dfrac{c^2}{a+1}\)
Bài 2: Cho \(a,b>0;a+b\le0\)
Tìm min \(B=ab+\dfrac{1}{ab}\)
