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 bé hơn hoặc bằng 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 a+b+c<=3. Cmr \(\frac{ab}{\sqrt{3+c}}+\frac{bc}{\sqrt{3+a}}+\frac{ca}{\sqrt{3+b}}\le\frac{3}{2}\)
4) Cho a,b,c>0 tm a+b+c=2. Cmr \(\frac{a}{\sqrt{4a+3bc}}+\frac{b}{\sqrt{4b+3ca}}+\frac{c}{\sqrt{4c+3ab}}\le1\)
5) Cho a,b,c>0. Cmr \(\sqrt{\frac{a^3}{5a^2+\left(b+c\right)^2}}+\sqrt{\frac{b^3}{5b^2+\left(c+a\right)^2}}+\sqrt{\frac{c^3}{5c^2+\left(a+b\right)^2}}\le\sqrt{\frac{a+b+c}{3}}\)
6) Cho a,b,c>0. Cmr \(\frac{a^2}{\left(2a+b\right)\left(2a+c\right)}+\frac{b^2}{\left(2b+a\right)\left(2b+c\right)}+\frac{c^2}{\left(2c+a\right)\left(2c+b\right)}\le\frac{1}{3}\)
Giúp mình với nhé các bạn
Cho ba số dương a,b,c thỏa mãn ab+ac+bc=1
CMR: P=\(\frac{2a}{\sqrt{1+a^2}}+\frac{b}{\sqrt{1+b^2}}+\frac{c}{\sqrt{1+c^2}}\le\frac{9}{4}\)
Giúp mình mấy câu này với nhé các ban.
1) Cho a,b,c>0 cmr:\(\frac{a}{\sqrt{a^2+b^2}}+\frac{b}{\sqrt{b^2+c^2}}+\frac{c}{\sqrt{c^2+a^2}}\le\frac{3}{\sqrt{2}}\)
2)Cho a,b,c>0 và abc=1. Cmr:\(\sqrt{\frac{a}{4a+4b+1}}+\sqrt{\frac{b}{4b+4c+1}}+\sqrt{\frac{c}{4c+4a+1}}\le1\)
3)Cho a,b,c>0 tm a+b+c=3 Cmr \(\frac{1}{2+a^2+b^2}+\frac{1}{2+b^2+c^2}+\frac{1}{2+c^2+a^2}\le\frac{3}{4}\)
Mình cảm ơn các bạn nhiều
Cho a,b,c>0; có a+b+c\(\le\)3.
Chứng minh rằng:
\(\frac{a}{\sqrt{2a^2+b^2}+\sqrt{3}}+\frac{b}{\sqrt{2b^2+c^2}+\sqrt{3}}+\frac{c}{\sqrt{2c^2+a^2}+\sqrt{3}}\le\frac{\sqrt{3}}{2}\)
Cho a,b,c thực dương thỏa mãn: \(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\le2\)
CMR: \(\frac{1}{\sqrt{5a^2+2ab+2b^2}}+\frac{1}{\sqrt{5b^2+2bc+2c^2}}+\frac{1}{\sqrt{5c^2+2ca+2a^2}}\le\frac{2}{3}\)
Cho a,b,c>0
CMR:
\(\sqrt{\frac{2}{a}}+\sqrt{\frac{2}{b}}+\sqrt{\frac{2}{c}}\le\sqrt{\frac{a+b}{ab}}+\sqrt{\frac{b+c}{bc}}\sqrt{\frac{c+a}{ca}}\)
cho a,b,c>0;ab+bc+ac\(\le\)3abc
cmr\(\sqrt{\frac{a^2+b^2}{a+b}}+\sqrt{\frac{b^2+c^2}{b+c}}+\sqrt{\frac{a^2+c^2}{a+c}}+3\le\sqrt{2}\left(\sqrt{a+b}+\sqrt{b+c}+\sqrt{a+c}\right)\)
Cho a,b,c>0 Cmr
\(\sqrt{\frac{2}{a}}+\sqrt{\frac{2}{b}}+\sqrt{\frac{2}{c}}\le\sqrt{\frac{a+b}{ab}}+\sqrt{\frac{b+c}{bc}}+\sqrt{\frac{c+a}{ca}}\)
Chờ a,b,c>0 và a+b+c=6
CMR \(\frac{ab}{\sqrt{c^2+3}}+\frac{bc}{\sqrt{a^2+3}}+\frac{ca}{\sqrt{b^2+3}}\le\frac{3}{2}\)
