\(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=2\Rightarrow\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)^2=4\)
\(\Rightarrow\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+\frac{2}{ab}+\frac{2}{bc}+\frac{2}{ac}=4\)
\(\Rightarrow\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+\frac{2\left(a+b+c\right)}{abc}=4\)
\(\Rightarrow\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}=4\)
Vậy A=4
Cho a,b,c khác 0 ,a+ b +c=0
Chứng minh :\(\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)^2=\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\)Tính giá trị biểu thức P=\(\frac{a^2}{bc}+\frac{b^2}{ca}+\frac{c^2}{ab}\)
Cho a,b,c > 0 . Cm : \(\frac{a}{b^2}+\frac{b}{c^2}+\frac{c}{a^2}\ge\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\)
cho a,b,c>0.CMR:\(\frac{a}{b^2}+\frac{b}{c^2}+\frac{c}{a^2}\ge\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\)
10,cho a+b+c=\(\frac{1}{9}\)abc và \(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1\) với a,b,c khác 0.CMR:\(\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}=\frac{1}{2}\)
1) Cho \(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=0\) . Tính giá trị biểu thức: D= \(\frac{bc}{a^2}+\frac{ac}{b^2}+\frac{ab}{c^2}\)
2) Cho a+b+c=\(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=0\); abc khác 0. Ch/m \(a^6+b^6+c^6=3a^2b^2c^2\)
Cho \(a+b+c=0\) ; a, b, c \(\ne\) 0. Chứng minh đẳng thức: \(\sqrt{\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}}=|\frac{1}{a}+\frac{1}{b}+\frac{1}{c}|\).
Nhờ các bạn
cho a,b,c>0 . Cmr: \(\frac{a^2}{b^5}+\frac{b^2}{c^5}+\frac{c^2}{d^5}+\frac{d^2}{a^5}\ge\frac{1}{a^3}+\frac{1}{b^3}+\frac{1}{c^3}+\frac{1}{d^3}\)
(sử dụng AM-GM)
Cho a, b, c > 0 và \(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1\)
CMR : \(\frac{a^2}{a+bc}+\frac{b^2}{b+ca}+\frac{c^2}{c+ab}\) ≥ \(\frac{a+b+c}{4}\)
Cho a, b, c > 0 và a + b + c = 3. Tìm GTNN của:
a, \(A=\frac{a^2}{a+1}+\frac{b^2}{b+1}+\frac{c^2}{c+1}\)
b, \(B=\frac{1}{a}+\frac{4}{b+1}+\frac{9}{c+2}\)
c, \(C=\frac{a^2}{a+b}+\frac{b^2}{b+c}+\frac{c^2}{c+a}\)
d, \(D=a^4+b^4+c^4+a^2+b^2+c^2+2020\)
