2 , 

Áp dụng BĐT AM - GM ta có :

\(2a+b+c=\left(a+b\right)+\left(a+c\right)\ge2\sqrt{\left(a+b\right)\left(a+c\right)}\)

\(\Rightarrow\left(2a+b+c\right)^2\ge4\left(a+b\right)\left(a+c\right)\)

\(\Rightarrow\frac{1}{\left(2a+b+c\right)^2}\le\frac{1}{4\left(a+b\right)\left(a+c\right)}\)

còn lại 

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

\(\Leftrightarrow M< \frac{1}{4}.\frac{\left(b+c\right)+\left(c+a\right)+\left(a+b\right)}{\left(a+b\right)\left(b+c\right)\left(c+a\right)}\)

\(\Leftrightarrow M\le\frac{a+b+c}{2\left(a+b\right)\left(b+c\right)\left(c+a\right)}\)

Lại có \(\left(a+b\right)\left(b+c\right)\left(c+a\right)\ge2\sqrt{ab}.2\sqrt{bc}.2\sqrt{ac}=8abc\)( theo AM - GM )

\(\Rightarrow M\le\frac{a+b+c}{2.8abc}=\frac{a+b+c}{16abc}\left(1\right)\)

Tiếp tục áp dụng BĐT AM - GM :

\(\frac{1}{a^2}+\frac{1}{b^2}\ge\frac{2}{ab};\frac{1}{b^2}+\frac{1}{c^2}\ge\frac{2}{bc};\frac{1}{c^2}+\frac{1}{a^2}\ge\frac{2}{ac}\)

\(\Rightarrow2\left(\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\right)\ge2\left(\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ac}\right)\)

\(\Leftrightarrow3\ge\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ca}=\frac{a+b+c}{abc}\)

\(\Rightarrow a+b+c\le3abc\left(2\right)\)

Từ ( 1 ) , ( 2 ) \(\Rightarrow M\le\frac{3abc}{16abc}=\frac{3}{16}\)\(M\le\frac{3}{16}< \frac{9}{16}\)

\(\Rightarrow M\le\frac{9}{16}\)

Bổ sung đề : Tìm x,y nguyên :\(\left(x-2\right)\left(y+1\right)=-2\)

Vì \(-2=\left(-2\right).1=2.\left(-1\right)\)nên ta có :

+, \(\orbr{\begin{cases}x-2=-2\\y+1=1\end{cases}}\Rightarrow\orbr{\begin{cases}x=0\\y=0\end{cases}}\)

+, \(\orbr{\begin{cases}x-2=1\\y+1=-2\end{cases}}\Rightarrow\orbr{\begin{cases}x=3\\y=-3\end{cases}}\)

+, \(\orbr{\begin{cases}x-2=2\\y+1=-1\end{cases}}\Rightarrow\orbr{\begin{cases}x=4\\y=-2\end{cases}}\)

+, \(\orbr{\begin{cases}x-2=-1\\y+1=2\end{cases}}\Rightarrow\orbr{\begin{cases}x=1\\y=1\end{cases}}\)

Tuần thứ 2 bán được số bút là:

1028 x 3= 3084(cái)

Đ/s: 3084 cái bút

\(P=\left(\frac{\sqrt{x}}{\sqrt{x}-1}+\frac{\sqrt{x}}{x-\sqrt{x}}\right):\frac{\sqrt{x+1}}{3}\)

\(P=\left(\frac{\left(\sqrt{x}\right)^2}{\sqrt{x}.\left(\sqrt{x}-1\right)}+\frac{\sqrt{x}}{\sqrt{x}.\left(\sqrt{x}-1\right)}\right).\frac{3}{\sqrt{x}+1}\)

\(P=\frac{x+\sqrt{x}}{\sqrt{x}\left(\sqrt{x}-1\right)}.\frac{3}{\sqrt{x}+1}\)

\(P=\frac{\sqrt{x}.\left(\sqrt{x}+1\right)}{\sqrt{x}\left(\sqrt{x}-1\right)}.\frac{3}{\sqrt{x}+1}\)

\(P=\frac{3}{\sqrt{x}-1}\)

Chiều rộng là :

85 x 4/5 = 68 ( m )

Diện tích là :

85 x 68 = 5780 ( m² )