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Ta có: BĐT phụ sau: \(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\ge\frac{9}{a+b+c}\)( CM bằng BĐT Shwars nha).Áp dụng ta có:
\(\frac{1}{a+3b+5c}+\frac{1}{b+3c+5a}+\frac{1}{3a+2b+4c}\ge\frac{9}{9a+6b+12c}=\frac{3}{3a+2b+4c}\left(1\right)\)
\(\frac{1}{b+3c+5a}+\frac{1}{c+3a+5b}+\frac{1}{3b+2c+4a}\ge\frac{9}{9b+6c+12a}=\frac{3}{3b+2c+4a}\left(2\right)\)
\(\frac{1}{c+3a+5b}+\frac{1}{a+3b+5c}+\frac{1}{3c+2a+4b}\ge\frac{9}{9c+6a+12b}=\frac{3}{3c+2a+4b}\left(3\right)\)
Cộng (1),(2) và (3) có:
\(2\left(\frac{1}{a+3b+5c}+\frac{1}{b+3c+5c}+\frac{1}{c+3a+5b}\right)+\left(\frac{1}{3a+2b+4c}+\frac{1}{3b+2c+4a}+\frac{1}{3c+2a+4b}\right)\ge3\left(\frac{1}{3a+2b+4c}+\frac{1}{3b+2c+4a}+\frac{1}{3c+2a+4b}\right)\)
\(\Rightarrow2VP\ge2VT\)
\(\RightarrowĐPCM\)
![](https://rs.olm.vn/images/avt/0.png?1311)
Ta có : \(P=\frac{2a+3b+3c+1}{2015+a}+\frac{3a+2b+3c}{2016+b}+\frac{3a+3b+2c-1}{2017+c}\)
\(\Rightarrow P+3=\frac{2a+3b+3c+1}{2015+a}+1+\frac{3a+2b+3c}{2016+b}+1+\frac{3a+3b+2c-1}{2017+c}+1\)
\(=\frac{3a+3b+3c+2016}{2015+a}+\frac{3a+3b+3c+2016}{2016+b}+\frac{3a+3b+3c+2016}{2017+c}\)
\(=\left(3a+3b+3c+2016\right)\left(\frac{1}{2015+a}+\frac{1}{2016+b}+\frac{1}{2017+c}\right)\)
\(=4.2016\left(\frac{1}{2015+a}+\frac{1}{2016+b}+\frac{1}{2017+c}\right)\) \(\left(a+b+c=2016\right)\)
\(=8064.\left(\frac{1}{2015+a}+\frac{1}{2016+b}+\frac{1}{2017+c}\right)\)
Vì a ; b ; c dương , áp dụng BĐT phụ \(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\ge\frac{9}{x+y+z}\), ta có :
\(\frac{1}{2015+a}+\frac{1}{2016+b}+\frac{1}{2017+c}\ge\frac{9}{2015+2016+2017+a+b+c}=\frac{9}{8064}\)
\(\Rightarrow P+3\ge8064.\frac{9}{8064}=9\) \(\Rightarrow P\ge6\)
Dấu " = " xảy ra \(\Leftrightarrow\left\{{}\begin{matrix}2015+a=2016+b=2017+c\\a+b+c=2016\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}a=b+1=c+2\\a+b+c=2016\end{matrix}\right.\)
\(\Leftrightarrow a=673;b=672;c=671\)
Vậy ...
![](https://rs.olm.vn/images/avt/0.png?1311)
Áp dụng bất đẳng thức Svác xơ ngược ta có
\(\frac{1}{2a+3b+3c}=\frac{1}{a+b+a+c+2\left(b+c\right)}\le\frac{1}{4}\left(\frac{1}{a+b}+\frac{1}{a+c}+\frac{2}{b+c}\right)\)
tương tự mấy cái kia rồi cộng vào
![](https://rs.olm.vn/images/avt/0.png?1311)
bài này ko khác gì câu 921427 nhé bạn, có điều bạn tìm cách tách a + 3b + 2c = (a + b) + (b + c) + (b + c)
Thêm nữa, áp dụng BĐT \(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\ge\frac{9}{a+b+c}\) với a, b, c > 0
Đẳng thức xảy ra khi và chỉ khi a = b = c.
EZ!!!Sau khi sử dụng 1 số bđt đơn giản, ta sẽ được:
\(\text{Σ}_{cyc}\frac{ab}{a+3b+2c}\le\frac{1}{9}\text{Σ}_{cyc}\left(\frac{ab}{a+c}+\frac{ab}{b+c}+\frac{a}{2}\right)=K\)
\(P\le K=\frac{1}{9}\left[\text{Σ}_{cyc}\left(\frac{ab}{a+c}+\frac{bc}{a+c}\right)+\frac{a+b+c}{2}\right]\)
\(=\frac{1}{9}\left(b+a+c+\frac{a+b+c}{2}\right)=\frac{a+b+c}{6}\le1\)
Dấu "=" xảy ra khi và chỉ khi a = b = c = 2
\(\frac{3a-2b}{5}=\frac{2c-5a}{3}=\frac{5b-3c}{2}=\frac{15a-10b}{25}=\frac{6c-15a}{9}=\frac{10b-6c}{4}\)
\(=\frac{15a-10b+6c-15a+10b-6c}{25+9+4}=0\)
\(\Rightarrow\left\{{}\begin{matrix}3a=2b\\2c=5a\\5b=3c\end{matrix}\right.\Rightarrow\frac{a}{2}=\frac{b}{3}=\frac{c}{5}=\frac{a+b+c}{10}\)
\(\Rightarrow\left\{{}\begin{matrix}a=\frac{a+b+c}{5}\\b=\frac{3\left(a+b+c\right)}{10}\\c=\frac{a+b+c}{2}\end{matrix}\right.\)
\(\Rightarrow P=\frac{\frac{33\left(a+b+c\right)}{10}}{\frac{43\left(a+b+c\right)}{10}}=\frac{33}{43}\)