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20 tháng 3 2018

Áp dụng bất đẳng thức Cauchy-Schwarz:

\(\dfrac{1}{a+2b+c}+\dfrac{1}{c+3a}\ge\dfrac{\left(1+1\right)^2}{a+2b+c+c+3a}=\dfrac{4}{4a+2b+2c}=\dfrac{2}{2a+b+c}\)

Chứng minh tương tự ta được: \(\left\{{}\begin{matrix}\dfrac{1}{b+2c+a}+\dfrac{1}{a+3b}\ge\dfrac{2}{a+2b+c}\\\dfrac{1}{c+2a+b}+\dfrac{1}{b+3c}\ge\dfrac{2}{a+b+2c}\end{matrix}\right.\)

Cộng theo vế:

\(\dfrac{1}{a+2b+c}+\dfrac{1}{b+2c+a}+\dfrac{1}{c+2a+b}+\dfrac{1}{a+3b}+\dfrac{1}{b+3c}+\dfrac{1}{c+3a}\ge\dfrac{2}{a+2b+c}+\dfrac{2}{b+2c+a}+\dfrac{2}{c+2a+b}\)

\(\Rightarrow\dfrac{1}{a+3b}+\dfrac{1}{b+3c}+\dfrac{1}{c+3a}\ge\dfrac{1}{a+2b+c}+\dfrac{1}{b+2c+a}+\dfrac{1}{c+2a+b}\)

p/s: đã sửa đề

20 tháng 3 2018

\(\dfrac{1}{a+2b+c}+\dfrac{1}{b+2c+a}+\dfrac{1}{c+2a+b}< =\dfrac{1}{a+3b}+\dfrac{1}{b+3c}+\dfrac{1}{c+3a}\)

\(\dfrac{1}{b+2c+a}+\dfrac{1}{a+3b}< =\dfrac{4}{2a+4b+2c}=\dfrac{2}{a+2b+c}\)

Cm tương tự, ta có:

\(\dfrac{1}{c+2a+b}+\dfrac{1}{b+3c}< =\dfrac{2}{b+2c+a}\)\(\)

\(\dfrac{1}{a+2b+c}+\dfrac{1}{c+3a}< =\dfrac{2}{c+2a+b}\)

Cộng 2 vế của 3 BĐT với nhau, ta có:

\(\dfrac{1}{b+2c+a}+\dfrac{1}{a+3b}+\dfrac{1}{c+2a+b}+\dfrac{1}{b+3c}+\dfrac{1}{a+2b+c}+\dfrac{1}{c+3a}< =\dfrac{2}{a+2b+c}+\dfrac{2}{b+2c+a}+\dfrac{2}{c+2a+b}\)

\(\Leftrightarrow\left(\dfrac{1}{b+2c+a}+\dfrac{1}{c+2a+b}+\dfrac{1}{a+2b+c}\right)+\left(\dfrac{1}{a+3b}+\dfrac{1}{b+3c}+\dfrac{1}{c+3a}\right)< =\dfrac{2}{a+2b+c}+\dfrac{2}{b+2c+a}+\dfrac{2}{c+2a+b}\)

\(\Leftrightarrow\dfrac{-\left(c+2a+b\right)\cdot\left(a+2b+c\right)-\left(b+2c+a\right)\left(a+2b+c\right)-\left(b+2c+a\right)\left(c+2a+b\right)}{\left(b+2c+a\right)\cdot\left(c+2a+b\right)\cdot\left(a+2b+c\right)}+\dfrac{\left(b+3c\right)\left(c+3a\right)+\left(a+3b\right)\left(c+3a\right)+\left(a+3b\right)\left(b+3c\right)}{\left(a+3b\right)\left(b+3c\right)\left(c+3a\right)}\le0\)

12 tháng 4 2018

\(Ta có:&nbsp;\(\frac{1}{2a+3b+3c}=\frac{1}{\left(a+b\right)+\left(a+c\right)+2\left(b+c\right)}\) Theo Cauchy:&nbsp;\(\frac{1}{x+y}\le\frac{1}{4}\left(\frac{1}{x}+\frac{1}{y}\right)\) =>&nbsp;\(\frac{1}{2a+3b+3c}\le\frac{1}{4}\left(\frac{1}{\left(a+b\right)+\left(a+c\right)}+\frac{1}{2\left(b+c\right)}\right)\le\frac{1}{4}\left(\frac{1} {4}\left(\frac{1}{a+b}+\frac{1}{a+c}\right)+\frac{1}{2\left(b+c\right)}\right)\) =>&nbsp;\(\frac{1}{2a+3b+3c}\le\frac{1}{8}\left(\frac{1}{2\left(a+b\right)}+\frac{1}{2\left(a+c\right)}+\frac{1}{b+c}\right)\) Tương tự:&nbsp;\(\frac{1}{3a+2b+3c}\le\frac{1}{8}\left(\frac{1}{2\left(a+b\right)}+\frac{1}{2\left(b+c\right)}+\frac{1}{a+c}\right)\) Và:&nbsp;\(\frac{1}{3a+3b+2c}\le\frac{1}{8}\left(\frac{1}{2\left(a+c\right)}+\frac{1}{2\left(b+c\right)}+\frac{1}{a+b}\right)\) =>&nbsp;\(P\le\frac{1}{8}\left(\frac{2}{a+b}+\frac{2}{a+c}+\frac{2}{b+c}\right)=\frac{1}{4}.2017\) => Pmax &nbsp;= 2017:4=504,25\)

11 tháng 4 2018

Ta có: \(\frac{1}{2a+3b+3c}=\frac{1}{\left(a+b\right)+\left(a+c\right)+2\left(b+c\right)}\)

Theo Cauchy: \(\frac{1}{x+y}\le\frac{1}{4}\left(\frac{1}{x}+\frac{1}{y}\right)\)

=> \(\frac{1}{2a+3b+3c}\le\frac{1}{4}\left(\frac{1}{\left(a+b\right)+\left(a+c\right)}+\frac{1}{2\left(b+c\right)}\right)\le\frac{1}{4}\left(\frac{1}{4}\left(\frac{1}{a+b}+\frac{1}{a+c}\right)+\frac{1}{2\left(b+c\right)}\right)\)

=> \(\frac{1}{2a+3b+3c}\le\frac{1}{8}\left(\frac{1}{2\left(a+b\right)}+\frac{1}{2\left(a+c\right)}+\frac{1}{b+c}\right)\)

Tương tự: \(\frac{1}{3a+2b+3c}\le\frac{1}{8}\left(\frac{1}{2\left(a+b\right)}+\frac{1}{2\left(b+c\right)}+\frac{1}{a+c}\right)\)

Và: \(\frac{1}{3a+3b+2c}\le\frac{1}{8}\left(\frac{1}{2\left(a+c\right)}+\frac{1}{2\left(b+c\right)}+\frac{1}{a+b}\right)\)

=> \(P\le\frac{1}{8}\left(\frac{2}{a+b}+\frac{2}{a+c}+\frac{2}{b+c}\right)=\frac{1}{4}.2017\)

=> Pmax = 2017:4=504,25

10 tháng 8 2016

Hình như đề sai ,  giả sử a = b = c = 0

=> vế trái bằng 0 , vé phải bằng 24

10 tháng 8 2016

\(\left(3a+b-c\right)^3+\left(3b+c-a\right)^3+\left(3c+a-b\right)^3+24\)
\(=24+27a^3+27b^3+27c^3+3\left(\left(3a+b\right)\left(3a-c\right)\left(b-c\right)+\left(3b+c\right)\left(3b-a\right)\left(c-a\right)+\left(3c+a\right)\left(3c-b\right)\left(a-b\right)\right)\)\(\left(3a+3b+3c\right)^3=27a^3+27b^3+27c^3+81\left(a+b\right)\left(b+c\right)\left(c+a\right)\)
\(\Rightarrow8+A=\left(a+b\right)\left(b+c\right)\left(c+a\right)\)

3 tháng 1 2023

Lời giải:

Đặt ⎧⎪⎨⎪⎩3a+b−c=x3b+c−a=y3c+a−b=z{3a+b−c=x3b+c−a=y3c+a−b=z

Khi đó, điều kiện đb tương đương với:

(x+y+z)3=24+x3+y3+z3⇔3(x+y)(y+z)(x+z)=24(x+y+z)3=24+x3+y3+z3⇔3(x+y)(y+z)(x+z)=24

⇔3(2a+4b)(2b+4c)(2c+4a)=24⇔3(2a+4b)(2b+4c)(2c+4a)=24

⇔(a+2b)(b+2c)(c+2a)=1⇔(a+2b)(b+2c)(c+2a)=1

Do đó ta có đpcm

3 tháng 1 2023

Lời giải:

Đặt ⎧⎪⎨⎪⎩3a+b−c=x3b+c−a=y3c+a−b=z{3a+b−c=x3b+c−a=y3c+a−b=z

Khi đó, điều kiện đb tương đương với:

(x+y+z)3=24+x3+y3+z3⇔3(x+y)(y+z)(x+z)=24(x+y+z)3=24+x3+y3+z3⇔3(x+y)(y+z)(x+z)=24

⇔3(2a+4b)(2b+4c)(2c+4a)=24⇔3(2a+4b)(2b+4c)(2c+4a)=24

⇔(a+2b)(b+2c)(c+2a)=1⇔(a+2b)(b+2c)(c+2a)=1

Do đó ta có đpcm

20 tháng 2 2022

\(\dfrac{1}{a+b}+\dfrac{1}{a+c}+\dfrac{1}{b+c}+\dfrac{1}{b+c}\ge\dfrac{16}{2a+3b+3c}\)

\(\dfrac{1}{b+c}+\dfrac{1}{a+b}+\dfrac{1}{a+c}+\dfrac{1}{a+c}\ge\dfrac{16}{2b+3a+3c}\)

\(\dfrac{1}{a+c}+\dfrac{1}{b+c}+\dfrac{1}{a+b}+\dfrac{1}{a+b}\ge\dfrac{16}{2c+3a+3b}\)

cộng tất cả lại ta được \(4.2017\ge16.\left(\dfrac{1}{2a+3b+3c}+\dfrac{1}{2b+3a+3c}+\dfrac{1}{2c+3a+3b}\right)< =>P\le\dfrac{2017}{4}\)

dấu bằng xảy ra khi \(\left\{{}\begin{matrix}\dfrac{1}{a+b}=\dfrac{1}{b+c}=\dfrac{1}{a+c}\\\dfrac{1}{a+b}+\dfrac{1}{b+c}+\dfrac{1}{a+c}=2017\end{matrix}\right.< =>\left\{{}\begin{matrix}a=b=c\\\dfrac{3}{2a}=\dfrac{3}{2b}=\dfrac{3}{2c}=2017\end{matrix}\right.< =>a=b=c=\dfrac{3}{4034}}\)

20 tháng 2 2022

mấy cái bất đẳng thức ở đầu là như nào v ạ