áp dụng cauchy-schwarz dạng engel ta có :
\(\dfrac{a^2}{a+2b}+\dfrac{b^2}{b+2c}+\dfrac{c^2}{c+2a}\ge\dfrac{\left(a+b+c\right)^2}{a+2b+b+2c+c+2a}\)
\(=\dfrac{\left(a+b+c\right)^2}{3\left(a+b+c\right)}=\dfrac{3^2}{3.3}=1\) \(\Rightarrow\) (đpcm)
cho a+b+c=3
cmr \(\dfrac{1}{a^2b+2}+\dfrac{1}{b^2c+2}+\dfrac{1}{c^2a+2}\ge1\)
Cho a,b,c là các số thực dương thỏa a+b+c=3
Chứng minh \(\dfrac{1}{2+a^2b}+\dfrac{1}{2+b^2c}+\dfrac{1}{2+c^2a}\ge1\)
Cho a, b, c là các số thực dương. Chứng minh rằng:
\(\dfrac{a}{b+2c}+\dfrac{b}{c+2a}+\dfrac{c}{a+2b}\ge1\)
cho a,b,c>0 Sao cho a+b+c=3
CMR \(\dfrac{a^3}{a+2b^3}+\dfrac{b^3}{b+2c^3}+\dfrac{c^3}{c+2a^3}\ge1\)
Cho a,b,c>0 t/m \(a^2+b^2+c^2=3\). Tìm max
P\(P=\dfrac{a}{a^2+2b+3}+\dfrac{b}{b^2+2c+3}+\dfrac{c}{c^2+2a+3_{ }}\le\dfrac{1}{2}\)
Cho a,b,c là các số dương. CMR
\(\dfrac{a}{\sqrt{2b^2+2c^2-a^2}}+\dfrac{b}{\sqrt{2c^2+2a^2-b^2}}+\dfrac{c}{\sqrt{2a^2+2b^2-c^2}}\ge\sqrt{3}\)
Cho a,b,c >0. Chứng minh rằng
2(\(\dfrac{a}{b+2c}+\dfrac{b}{c+2a}+\dfrac{c}{a+2b}\)) \(\ge\) 1+\(\dfrac{b}{b+2a}+\dfrac{c}{c+2b}+\dfrac{a}{a+2c}\)
Cho a b c>0 tm a+b+c=3
Chứng minh \(\dfrac{a^2}{2a+1}+\dfrac{b^2}{2b+1}+\dfrac{c^2}{2c+1}\le\dfrac{a^2+b^2+c^2}{\sqrt{a^2+b^2+c^2+6}}\)
Cho a, b,c dương. cmr: \(\dfrac{a^3}{2b+3c}+\dfrac{b^3}{2c+3a}+\dfrac{c^3}{2a+3b}\ge\dfrac{1}{5}\left(a^2+b^2+c^2\right)\)
