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Cho a,b,c > 0 CMR : \(\frac{a^3}{\left(a+b\right)^2}+\frac{b^3}{\left(b+c\right)^2}+\frac{c^3}{\left(a+c\right)^2}\ge\frac{a+b+c}{4}\)
Bài 1:Cho a,b,c,d là các số dương. Chứng minh rằng :frac{a^4}{left(a+bright)left(a^2+b^2right)}+frac{b^4}{left(b+cright)left(b^2+c^2right)}+frac{c^4}{left(c+dright)left(c^2+d^2right)}+frac{d^4}{left(d+aright)left(d^2+a^2right)}gefrac{a+b+c+d}{4}Bài 2:Cho a0,b0,c0.CM:frac{a}{bc}+frac{b}{ca}+frac{c}{ab}ge2left(frac{1}{a}+frac{1}{b}+frac{1}{c}right)Bài 3: a) Cho x,y,0. CMR:frac{x^3}{x^2+xy+y^2}gefrac{2x-y}{3}b) Chứng minh rằngSigmafrac{a^3}{a^2+ab+b^2}gefrac{a+b+c}{3}
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Bài 1:Cho a,b,c,d là các số dương. Chứng minh rằng :
\(\frac{a^4}{\left(a+b\right)\left(a^2+b^2\right)}+\frac{b^4}{\left(b+c\right)\left(b^2+c^2\right)}+\frac{c^4}{\left(c+d\right)\left(c^2+d^2\right)}+\frac{d^4}{\left(d+a\right)\left(d^2+a^2\right)}\ge\frac{a+b+c+d}{4}\)
Bài 2:Cho \(a>0,b>0,c>0\).\(CM:\frac{a}{bc}+\frac{b}{ca}+\frac{c}{ab}\ge2\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\)
Bài 3: a) Cho x,y,>0. CMR:\(\frac{x^3}{x^2+xy+y^2}\ge\frac{2x-y}{3}\)
b) Chứng minh rằng\(\Sigma\frac{a^3}{a^2+ab+b^2}\ge\frac{a+b+c}{3}\)
Cho a, b, c > 0. CMR :
\(\frac{a^3}{b^2\left(b+c\right)}+\frac{b^3}{c^2\left(c+a\right)}+\frac{c^3}{a^2\left(a+b\right)}\ge\frac{3}{2}\)
Bài 1 :Cho a,b,c dương thỏa mãn a+b+c=2
CMR \(\frac{bc}{\sqrt{3a^2+4}}+\frac{ca}{\sqrt{3b^2+4}}+\frac{ab}{\sqrt{3c^2+4}}\ge\frac{\sqrt{3}}{3}\)
Bài 2:Cho a,b,c>0. CMR
\(\left(a+b\right)\left(b+c\right)\left(c+a\right)\ge\frac{8}{9}\left(a+b+c\right)\left(ab+bc+ca\right)\)
cho a,b,c > 0 cmr: \(\frac{b^2a}{a^3\left(b+c\right)}+\frac{c^2a}{b^3\left(c+a\right)}+\frac{a^2b}{c^3\left(a+b\right)}\ge\frac{1}{2}\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\)
Cho a,b,c>0 và a+b+c=3
CMR: \(\frac{a^3}{\left(a+b\right)\left(a+c\right)}+\frac{b^3}{\left(b+c\right)\left(b+a\right)}+\frac{c^3}{\left(c+a\right)\left(c+b\right)}\ge\frac{3}{4}\)
cho a,b,c>0 va a+b+c=1. CMR
\(\frac{a}{\left(b+c\right)^3}+\frac{b}{\left(a+c\right)^3}+\frac{c}{\left(a+b\right)^3}\ge\frac{27}{8\left(a+b+c\right)^2}\)
áp dụng cô si ta có:+)frac{a^5}{b^3}+frac{a^3}{b}gefrac{2a^4}{b^2};frac{b^5}{c^3}+frac{b^3}{c}gefrac{2b^4}{c^2};frac{c^5}{a^3}+frac{c^3}{a}gefrac{2c^4}{a^2}Leftrightarrowfrac{a^5}{b^3}+frac{b^5}{c^3}+frac{c^5}{a^3}ge2left(frac{a^4}{b^2}+frac{b^4}{c^2}+frac{c^4}{a^2}right)-left(frac{a^3}{b}+frac{b^3}{c}+frac{c^3}{a}right)+)frac{a^4}{b^2}+a^2gefrac{2a^3}{b};frac{b^4}{c^2}+b^2gefrac{2b^3}{c};frac{c^4}{a^2}+c^2gefrac{2C^3}{a}Leftrightarrowfrac{a^4}{b^2}+frac{b^4}{c^2}+frac{c^4}{a^2}ge2left(frac{a^3}...
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áp dụng cô si ta có:
+)\(\frac{a^5}{b^3}+\frac{a^3}{b}\ge\frac{2a^4}{b^2};\frac{b^5}{c^3}+\frac{b^3}{c}\ge\frac{2b^4}{c^2};\frac{c^5}{a^3}+\frac{c^3}{a}\ge\frac{2c^4}{a^2}\)
\(\Leftrightarrow\frac{a^5}{b^3}+\frac{b^5}{c^3}+\frac{c^5}{a^3}\ge2\left(\frac{a^4}{b^2}+\frac{b^4}{c^2}+\frac{c^4}{a^2}\right)-\left(\frac{a^3}{b}+\frac{b^3}{c}+\frac{c^3}{a}\right)\)
+)\(\frac{a^4}{b^2}+a^2\ge\frac{2a^3}{b};\frac{b^4}{c^2}+b^2\ge\frac{2b^3}{c};\frac{c^4}{a^2}+c^2\ge\frac{2C^3}{a}\)
\(\Leftrightarrow\frac{a^4}{b^2}+\frac{b^4}{c^2}+\frac{c^4}{a^2}\ge2\left(\frac{a^3}{b}+\frac{b^3}{c}+\frac{c^3}{a}\right)-\left(a^2+b^2+c^2\right)\)
+)\(\frac{a^3}{b}+ab\ge2a^2;\frac{b^3}{c}+bc\ge2b^2;\frac{c^3}{a}+ca\ge2c^2\)
\(\Leftrightarrow\frac{a^3}{b}+\frac{b^3}{c}+\frac{c^3}{a}\ge\left(a^2+b^2+c^2\right)+\left(a^2+b^2+c^2-ab-bc-ca\right)\ge\left(a^2+b^2+c^2\right)\)
\(\Leftrightarrow\frac{a^4}{b^2}+\frac{b^4}{c^2}+\frac{c^4}{a^2}\ge\left(\frac{a^3}{b}+\frac{b^3}{c}+\frac{c^3}{a}\right)+\left(\frac{a^3}{b}+\frac{b^3}{c}+\frac{c^3}{a}-a^2-b^2-c^2\right)\ge\frac{a^3}{b}+\frac{b^3}{c}+\frac{c^3}{a}\)
\(\Leftrightarrow\frac{a^5}{b^3}+\frac{b^5}{c^3}+\frac{c^5}{a^3}\ge\left(\frac{a^4}{b^2}+\frac{b^4}{c^2}+\frac{c^4}{a^2}\right)+\left(\frac{a^4}{b^2}+\frac{b^4}{c^2}+\frac{c^4}{a^2}-\frac{a^3}{b}-\frac{b^3}{c}-\frac{c^3}{a}\right)\ge\left(\frac{a^4}{b^2}+\frac{b^4}{c^2}+\frac{c^4}{a^2}\right)\)
Cho a,b,c>0. CMR: \(\frac{a^4}{\left(a+b\right)\left(a^2+b^2\right)}+\frac{b^4}{\left(b+c\right)\left(b^2+c^2\right)}+\frac{c^4}{\left(c+a\right)\left(c^2+a^2\right)}\ge\frac{a+b+c}{4}\)