a,b,c > 0
Theo bất đẳng thức Schwarz
\(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}-\dfrac{4}{a+b+c}\)
\(\ge\dfrac{\left(1+1+1\right)^2-4}{a+b+c}=\dfrac{5}{a+b+c}>0\)
Cho: \(\dfrac{a}{c}=\dfrac{a-b}{b-c},a\ne0,c\ne0,a-b\ne0,b-c\ne0\). CMR: \(\dfrac{1}{a}+\dfrac{1}{a-b}=\dfrac{1}{b-c}-\dfrac{1}{c}\)
Cho \(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=0\) với \(a,b,c\ne0\). Chứng minh rằng \(\left(a+b+c\right)^2=a^2+b^2+c^2\)
Cho \(\left(a+b+c\right)^2=a^2+b^2+c^2\) và \(a,b,c\ne0.\)
Chứng minh rằng : \(\dfrac{1}{a^3}+\dfrac{1}{b^3}+\dfrac{1}{c^3}=\dfrac{3}{abc}.\)
Cho \(a,b>0;c\ne0\)
CMR: \(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=0\Leftrightarrow\sqrt{a+b}=\sqrt{a+c}+\sqrt{b+c}\)
chứng minh rằng:\(\dfrac{a+b}{ab+c^2}+\dfrac{b+c}{bc+a^2}+\dfrac{c+a}{ac+b^2}\le\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\)
a) Cho a,b,c >0
Chứng minh: \(\dfrac{a^2}{b+c}+\dfrac{b^2}{c+a}+\dfrac{c^2}{a+b}\ge\dfrac{a+b+c}{2}\)
b) Cho a,b \(\ge\)1 , chứng minh:
\(\dfrac{1}{a^2+1}+\dfrac{1}{b^2+1}\ge\dfrac{2}{ab+1}\)
Chứng minh rằng phương trình sau có ba nghiệm phân biệt :
\(\dfrac{x-a}{b}+\dfrac{x-b}{a}=\dfrac{b}{x-a}+\dfrac{a}{x-b}\) ( a, b là hằng số, \(a\ne0\), \(b\ne0,a\pm b\ne0\) )
Cho
a,b,c > 0 . Chứng minh:
\(\dfrac{a}{b^2}+\dfrac{b}{c^2}+\dfrac{c}{a^2}\ge\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\)
Cho a,b,c thỏa mãn
\(\left(a+b\right)\left(b+c\right)\left(c+a\right)\ne0\) và \(\dfrac{a^2}{a+b}+\dfrac{b^2}{b+c}+\dfrac{c^2}{c+a}=\dfrac{a^2}{b+c}+\dfrac{b^2}{c+a}+\dfrac{c^2}{a+b}\)
Chứng minh : a=b=c
