\(\dfrac{1+a+b}{2}\ge\dfrac{1+a+b+ab}{2+a+b}\)
\(\Leftrightarrow\left(1+a+b\right)\left(2+a+b\right)\ge2\left(1+a+b+ab\right)\)
\(\Leftrightarrow2+a+b+2a+a^2+ab+2b+ab+b^2\ge2+2a+2b+2ab\)
\(\Leftrightarrow a^2+b^2+2ab+3a+3b+2\ge2ab+2a+2b+2\)
\(\Leftrightarrow a^2+b^2+a+b\ge0\)
cho a , b, c la cac so thuc duong thoa man he thuc a+b+c=6abc
Chung minh rang \(\dfrac{bc}{a^3\left(c+2b\right)}+\dfrac{ac}{b^3\left(a+2c\right)}+\dfrac{ab}{c^3\left(b+2a\right)}\ge2\)
Cho a,b,c la cac so nguyen chung minh\(\sqrt{\dfrac{a}{b+c}}+\sqrt{\dfrac{b}{a+c}}+\sqrt{\dfrac{c}{a+b}}>2\)
Cho a, b>0. Chứng minh rằng:
a) \(\dfrac{3a^2+2ab+3b^2}{a+b}\ge2\sqrt{2\left(a^2+b^2\right)}\)
b) \(\dfrac{2ab}{a+b}+\sqrt{\dfrac{a^2+b^2}{2}}\ge\sqrt{ab}+\dfrac{a+b}{2}\)
c) \(\dfrac{1}{\left(1+a\right)^2}+\dfrac{1}{\left(1+b\right)^2}\ge\dfrac{1}{1+ab}\)
cho a>1,b>1.Chung minh rang :
\(\dfrac{a^2}{b-1}+\dfrac{b^2}{a-1}\ge8\)
Cho 3 số dương a, b, c thỏa mãn: ab+bc+ca=3. Chứng minh: \(\dfrac{1}{a^2+1}+\dfrac{1}{b^2+1}+\dfrac{1}{c^2+1}\ge\dfrac{3}{2}\)
cho 3 so thuc x,y,z khac khong va thoa man hai dieu kien \(ax^3=by^3=cz^3\) va \(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}=1\)
chung minh rang : \(\sqrt[3]{ax^2+by^2+cz^2}=\sqrt[3]{a}+\sqrt[3]{b}+\sqrt[3]{c}\)
Cho a, b, c > 0 thoả mãn: \(a+b+c=1\). Chứng minh: \(\dfrac{ab}{a^2+b^2}+\dfrac{bc}{b^2+c^2}+\dfrac{ca}{c^2+a^2}+\dfrac{1}{4}\left(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\right)\ge\dfrac{15}{4}\)
Cho ba số dương a,b,c thỏa mãn điều kiện \(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=1\)
Chứng minh \(\dfrac{a^2}{a+bc}+\dfrac{b^2}{b+ca}+\dfrac{c^2}{c+ab}\ge\dfrac{a+b+c}{4}\)
Cho a, b, c > 0 và \(a+b+c=1\). Chứng minh: \(\dfrac{a^3}{a^2+ab+b^2}+\dfrac{b^3}{b^2+bc+c^2}+\dfrac{c^3}{c^2+ca+a^2}\ge\dfrac{1}{3}\)
