Ta có:
\(\left(a+b+c\right)\left(\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\right)=a+b+c\) (Do \(\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}=1\) )
\(\Leftrightarrow\frac{a^2}{b+c}+a+\frac{b^2}{c+a}+b+\frac{c^2}{a+b}+c=a+b+c\)
\(\Leftrightarrow\frac{a^2}{b+c}+\frac{b^2}{c+a}+\frac{c^2}{a+b}=a+b+c-\left(a+b+c\right)=0\left(đpcm\right)\)
*Lưu ý: Có rút gọn một số bước.
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
cho \(\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}=1\)1 CMR \(\frac{a^2}{b+c}+\frac{b^2}{c+a}+\frac{c^2}{a+b}=0\)
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 > 0 . CMR:
\(\frac{a+b}{bc+a^2}+\frac{b+c}{ac+b^2}+\frac{c+a}{ab+c^2}\le\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\\ \)
Cho \(\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}=1\)
Cmr : \(\frac{a^2}{b+c}+\frac{b^2}{c+a}+\frac{c^2}{a+b}=0\)
cho: \(\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{b+a}=1\)
CMR: \(\frac{a^2}{b+c}+\frac{b^2}{a+c}+\frac{c^2}{b+a}=0\)
Cho \(\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}=1\)
\(CMR:\frac{a^2}{b+c}+\frac{b^2}{c+a}+\frac{c^2}{a+b}=0\)
Cho \(\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}=1\)
cmr : \(\frac{a^2}{b+c}+\frac{b^2}{c+a}+\frac{c^2}{a+b}=0\)
Cho 3 số a,b,c khác 0
\(\frac{a^2}{b^2}+\frac{b^2}{c^2}+\frac{c^2}{a^2}=\frac{a}{c}+\frac{c}{b}+\frac{b}{a}\)
CMR a=b=c
1. CHo \(\frac{1}{a}+\frac{1}{b}=\frac{2}{b}\)(a,b ,c >0 )
CMR: \(\frac{a+b}{2a-b}+\frac{c+b}{2c-b}\ge4\)
2. CHo a,b,c > 0 và a2 + b2 + c2 = 3. CMR: a2b + b2c + c2a < = 3
3. CHo a,b,c thõa mãn a + b + c = 3. CM: \(\frac{a^2}{a+2b^3}+\frac{b^2}{b+2c^3}+\frac{c^2}{c+2a^3}\le1\)
4. CHo a,b,c > 0 thõa mãn a + b + c < = 3/2
CM: \(P=\left(3+\frac{1}{a}+\frac{1}{b}\right)\left(3+\frac{1}{b}+\frac{1}{c}\right)\left(3+\frac{1}{c}+\frac{1}{a}\right)\ge343\)
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