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Nhân 2 vế của 2 ĐT đề bài ta có
\(\left(a+b+c\right)\left(\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{a+c}\right)=\frac{47}{10}\)
<=> \(\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}+\left(\frac{a}{a+b}+\frac{b}{a+b}\right)+\left(\frac{b}{b+c}+\frac{c}{b+c}\right)+\left(\frac{c}{a+c}+\frac{a}{a+c}\right)=\frac{47}{10}\)
=>\(P=\frac{17}{10}\)
Vậy \(P=\frac{17}{10}\)
Ta có: \(\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}+a+b+c=2+2018\)
\(\Leftrightarrow\frac{a+ab+bc}{b+c}+\frac{b+bc+ab}{c+a}+\frac{c+ac+bc}{a+b}=2020\)
\(\Leftrightarrow a\left(\frac{1+b+c}{b+c}\right)+b\left(\frac{1+a+c}{a+c}\right)+c\left(\frac{1+a+b}{a+b}\right)=2020\left(1\right)\)
Vì \(a+b+c=2018\Rightarrow\hept{\begin{cases}a+b=2018-c\\b+c=2018-a\\c+a=2018-b\end{cases}\left(2\right)}\)
Thay (2) vào (1) ta được:
\(a\left(\frac{2019-a}{b+c}\right)+b\left(\frac{2019-b}{a+c}\right)+c\left(\frac{2019-c}{a+b}\right)=2020\)
\(\Leftrightarrow\frac{2019a-a^2}{b+c}+\frac{2019b-b^2}{a+c}+\frac{2019c-c^2}{a+b}=2020\)
\(\Leftrightarrow\frac{2019a}{b+c}-\frac{a^2}{b+c}+\frac{2019b}{a+c}-\frac{b^2}{a+c}+\frac{2019c}{a+b}-\frac{c^2}{a+b}=2020\)
\(\Leftrightarrow2019\left(\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\right)-\left(\frac{a^2}{c+b}+\frac{b^2}{c+a}+\frac{c^2}{a+b}\right)=2020\)
\(\Leftrightarrow4038-\left(\frac{a^2}{c+b}+\frac{b^2}{c+a}+\frac{c^2}{a+b}\right)=2020\)( vì \(\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}=2\))
\(\Leftrightarrow\frac{a^2}{c+b}+\frac{b^2}{c+a}+\frac{c^2}{a+b}=2018\)
\(\Leftrightarrow\frac{a^2}{c+b}+\frac{b^2}{c+a}+\frac{c^2}{a+b}+1=2019\)
Ta có
\(4a^2+b^2=5ab\)
\(\Leftrightarrow4a^2-4ab+b^2-ab=0\)
\(\Leftrightarrow4a\left(a-b\right)-b\left(a-b\right)=0\)
\(\Leftrightarrow\left(a-b\right)\left(4a-b\right)=0\)
\(\Leftrightarrow\orbr{\begin{cases}a-b=0\\4a-b=0\end{cases}}\)
\(\Leftrightarrow\orbr{\begin{cases}a=b\\4a=b\end{cases}}\)
\(TH1:a=b\)
\(\Leftrightarrow\frac{a^2}{4a^2-a^2}=\frac{a^2}{3a^2}=\frac{1}{3}\)
\(TH2:4a=b\)
\(\Leftrightarrow\frac{4a^2}{4a^2-16a^2}=\frac{4a^2}{-12a^2}=\frac{-1}{3}\)
Vậy...............
k mk nha
Ta có:\(10=2xyz\)
=> \(P=\frac{1}{2x+2xz+1}+\frac{2xy}{y+2xy+10}+\frac{10z}{10z+yz+10}\)
\(=\frac{1}{2x+2xz+1}+\frac{2xy}{y+2xy+2xyz}+\frac{2xyz^2}{2xyz^2+yz+2xyz}\)
\(=\frac{1}{2x+2xz+1}+\frac{2x}{1+2x+2xz}+\frac{2xz}{2xz+1+2x}\)
\(=1\)
Vậy P=1