1) Cho a,b,c>0 tm a+b+c=3. Cmr \(\frac{1}{2+a^2+b^2}+\frac{1}{2+b^2+c^2}+\frac{1}{2+c^2+a^2}\le\frac{3}{4}\)
2) Cho a,b,c>0 tm \(a^2+b^2+c^2\le abc\).Cmr \(\frac{a}{a^2+bc}+\frac{b}{b^2+ca}+\frac{c}{c^2+ab}\le\frac{1}{2}\)
3) Cho a,b,c>0 tm \(\sqrt{a}+\sqrt{b}+\sqrt{c}=1\).Cmr \(\sqrt{\frac{ab}{a+b+2c}}+\sqrt{\frac{bc}{b+c+2a}}+\sqrt{\frac{ca}{c+a+2b}}\le\frac{1}{2}\)
Giúp mình mới nhé các bạn. Mình đang cần gấp
Giả sử: \(\frac{a+b}{a^2+bc}+\frac{b+c}{b^2+ca}+\frac{c+a}{c^2+ab}\le\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\)
\(\Leftrightarrow\frac{bc+a^2}{a+b}+\frac{ca+b^2}{b+c}+\frac{ab+c^2}{c+a}\ge a+b+c\)
Cho a,b,c dương tm ab+bc+ca=3
Tìm min\(A=\frac{19a+3}{b^2+1}+\frac{19b+3}{c^2+1}+\frac{19c+3}{a^2+1}\)
cho a, b, c>0. CMR a\(\frac{a^3}{b}\ge a^2+ab-b^2\)
CM \(\frac{a^2}{b^2}+\frac{b^2}{c^2}+\frac{c^2}{a^2}\ge\frac{c}{b}+\frac{b}{a}+\frac{a}{c}\)
Cho a, b, c là độ dài 3 cạnh của tam giác CM \(\frac{1}{a+b-c}+\frac{1}{b+c-a}+\frac{1}{c+a-b}\ge\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\)
Cho a, b, c là các số dương và a+b+c=1 chứng minh rằng: \(\frac{ab}{a^2+b^2}+\frac{bc}{b^2+c^2}+\frac{ca}{c^2+a^2}+\frac{1}{4}\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\ge\frac{15}{4}\)
Cho 3 số a;b;c đôi 1 khác nhau.CMR:\(\frac{bc}{\left(b-c\right)^2}+\frac{ca}{\left(c-a\right)^2}+\frac{ab}{\left(a-b\right)^2}\ge\frac{-1}{4}\)
Cho \(a,b,c>0\)
CMR :\(\frac{a^4}{b\left(b+c\right)}+\frac{b^4}{c\left(c+a\right)}+\frac{c^4}{a\left(a+b\right)}\ge\frac{1}{2}\left(ab+bc+ca\right)\)
Áp dụng bđt Svac-xo ta có :
\(VT\ge\frac{\left(a^2+b^2+c^2\right)^2}{a^2+b^2+c^2+ab+bc+ca}\ge\frac{\left(a^2+b^2+c^2\right)^2}{2\left(a^2+b^2+c^2\right)}=\frac{a^2+b^2+c^2}{2}\ge\frac{ab+bc+ca}{2}\)
Dấu "-" xảy ra \(< =>a=b=c\)
Cho a,b,c là số thực dương
CM
\(\frac{a+b}{ab+c^2}\) + \(\frac{b+c}{bc+a^2}\)+ \(\frac{c+a}{ca+b^2}\)\(\ge\)\(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\)
\(cho\)\(a,b,c>0\)\(CM\)
\(\frac{a}{\left(ab+a+1\right)^2}+\frac{b}{\left(bc+b+1\right)^2}+\frac{c}{\left(ca+c+1\right)^2}\ge\frac{1}{a+b+c}\)