các bạn làm được ý nào thì làm ý đó nha
1. Cho a,b,c là độ dài 3 cạnh tam giác. Chứng minh:
a) \(\frac{1}{\left(a+b-c\right)^2}+\frac{1}{\left(a-b+c\right)^2}+\frac{1}{\left(b+c-a\right)^2}\ge\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\)
b) \(\frac{1}{\left(a+b-c\right)^3}+\frac{1}{\left(a-b+c\right)^3}+\frac{1}{\left(b+c-a\right)^3}\ge\frac{1}{a^3}+\frac{1}{b^3}+\frac{1}{c^3}\)
c) \(\frac{1}{\left(a+b-c\right)^{200}}+\frac{1}{\left(a-b+c\right)^{200}}+\frac{1}{\left(b+c-a\right)^{200}}\ge\frac{1}{a^{200}}+\frac{1}{b^{200}}+\frac{1}{c^{200}}\)
d) \(\frac{1}{8}\left(a+b\right)\left(b+c\right)\left(c+a\right)\ge\sqrt{abc\left(-a+b+c\right)\left(a-b+c\right)\left(a+b-c\right)}\)
e) \(a+b+c< \sqrt{a\left(b+c\right)}+\sqrt{b\left(a+c\right)}+\sqrt{c\left(a+b\right)}\)
f) \(\sqrt{\frac{a}{b+c}}+\sqrt{\frac{b}{c+a}}+\sqrt{\frac{c}{a+b}}< \sqrt{6}\)
g) \(\sqrt{-a+b+c}+\sqrt{a-b+c}+\sqrt{a+b-c}\le\sqrt{3\left(a+b+c\right)}\)
1, cho a,b,c>0. chứng minh \(a^4+b^4+c^4\ge abc\left(a+b+c\right)\)
2, chứng minh: với mọi a,b \(\ne0\)\(\frac{a^2}{b^2}+\frac{b^2}{a^2}\ge\frac{a}{b}+\frac{b}{a}\)
3,cho các số thực \(\in\)đoạn 0 đến 1. chứng minh:\(a^4+a^3+c^2-ab-bc-ca\le1\)
4,cho a,b,c là các số thực dương tùy ý. chứng minh: \(\frac{a^3+b^3}{ab}+\frac{b^3+c^3}{bc}+\frac{c^3+a^3}{ca}\ge2\left(a+b+c\right)\)
5,cho a,b,c>0. chứng minh\(\frac{a}{bc}+\frac{b}{ac}+\frac{c}{ab}\ge2\left(\frac{1}{a}+\frac{1}{b}-\frac{1}{c}\right)\)
ai làm đk bài nào thì làm hộ e vs ạ
Cho a,b,c>0 tm abc=1.CMR \(\frac{a}{\left(ab+a+1\right)^2}\)+\(\frac{b}{\left(bc+b+1\right)^2}\)+\(\frac{c}{\left(ac+c+1\right)^2}\)\(\ge\)\(\frac{1}{a+b+c}\)
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 a,b,c, là 3 số dương tm đk \(a+b+c=1\)
cmr \(\frac{a^3}{\left(b+c\right)^2}+\frac{b^3}{\left(c+a\right)^2}+\frac{c^3}{\left(a+b\right)^2}\ge\frac{1}{4}\)
Chứng minh rằng \(\frac{1}{2\sqrt[3]{abc}}+\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}\ge\frac{\left(a+b+c+\sqrt[3]{abc}\right)^2}{\left(a+b\right)\left(b+c\right)\left(c+a\right)}\forall a,b,c>0\)
1) Cho a, b, c > 0. Chứng minh: \(\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)^2\ge\left(a+b+c\right)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\)
2) Cho \(a,b,c\in R\).
a) Chứng minh: \(\left(a^2+3\right)\left(b^2+3\right)\left(c^2+3\right)\ge4\left(a+b+c+1\right)^2\)
b) Chứng minh: \(\left(a^2+1\right)\left(b^2+1\right)\left(c^2+1\right)\ge\frac{5}{16}\left(a+b+c+1\right)^2\)
3) Cho \(a,b,c\in R\)Chứng minh: \(\frac{a^3}{b^2}+\frac{b^3}{c^2}+\frac{c^3}{a^2}\ge\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}\)
Cho a, b, c là các số dương biết abc = 1. Chứng minh rằng:
\(\frac{a^3}{\left(b+1\right)\left(c+2\right)}+\frac{b^3}{\left(c+1\right)\left(a+2\right)}+\frac{c^3}{\left(a+1\right)\left(b+2\right)}\ge\frac{1}{2}\)
cho a,b,c>0,abc=1.cmr:\(\frac{1}{a\left(b+1\right)}+\frac{1}{b\left(c+1\right)}+\frac{1}{c\left(a+1\right)}\ge\frac{3}{2}\)