\(xy+yz+zx=8xyz\Rightarrow\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}=8\)

\(\Rightarrow\dfrac{8}{x}+\dfrac{8}{y}+\dfrac{8}{z}=64\)

Ta có: \(\dfrac{8}{x}+\dfrac{8}{y}+\dfrac{8}{z}\)

\(=\left(\dfrac{1}{x}+...+\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}\right)+\left(\dfrac{1}{y}+...+\dfrac{1}{y}+\dfrac{1}{z}+\dfrac{1}{x}\right)+\left(\dfrac{1}{z}+...+\dfrac{1}{z}+\dfrac{1}{x}+\dfrac{1}{y}\right)\)

(sau dấu chấm là bốn số tương tự).

\(\ge^{Cauchy-Schwarz}\dfrac{8^2}{6x+y+z}+\dfrac{8^2}{6y+z+x}+\dfrac{8^2}{6z+x+y}\)

\(\Rightarrow64\ge\dfrac{8^2}{6x+y+z}+\dfrac{8^2}{6y+z+x}+\dfrac{8^2}{6z+x+y}\)

\(\Rightarrow\dfrac{1}{6x+y+z}+\dfrac{1}{6y+z+x}+\dfrac{1}{6z+x+y}\le1\)

Dấu "=" xảy ra khi \(x=y=z=\dfrac{3}{8}\)

Vậy \(Max\) của biểu thức đã cho là 1.

- Đặt \(\left\{{}\begin{matrix}n_{Al}=a\left(mol\right)\\n_{Mg}=b\left(mol\right)\end{matrix}\right.\Rightarrow27a+24b=10,2\left(1\right)\)

Khí thu được sau p/ứ là khí H₂: \(n_{H_2}=\dfrac{11,2}{22,4}=0,5\left(mol\right)\)

\(2Al+6HCl\rightarrow2AlCl_3+3H_2\uparrow\)

2                                         3       (mol)

a                                        3/2 a   (mol)

\(Mg+2HCl\rightarrow MgCl_2+H_2\uparrow\)

1                                        1   (mol)

b                                         b   (mol)

Từ hai PTHH trên ta có: \(\dfrac{3}{2}a+b=0,5\left(2\right)\)

\(\left(1\right),\left(2\right)\) ta có hệ: \(\left\{{}\begin{matrix}27a+24b=10,2\\\dfrac{3}{2}a+b=0,5\end{matrix}\right.\)

Giải ra ta có \(\left\{{}\begin{matrix}a=0,2\left(mol\right)\\b=0,2\left(mol\right)\end{matrix}\right.\)

a) \(\%Al=\dfrac{m_{Al}}{m_{hh}}.100\%=\dfrac{0,2.27}{10,2}.100\%\approx52,94\%\)

\(\%Mg=100\%-\%Al=100\%-52,94=47,06\%\)

b)

 \(3H_2+Fe_2O_3\rightarrow^{t^0}2Fe+3H_2O\)

3               1              2  (mol)

0,5            1/6         1/3  (mol)

\(m_{Fe}=\dfrac{1}{3}.56=\dfrac{56}{3}\left(g\right)\)

\(m_{Fe_2O_3\left(pứ\right)}=\dfrac{1}{6}.160=\dfrac{80}{3}\left(g\right)\)

\(m_{Fe_2O_3\left(dư\right)}=60-m_{Fe}=60-\dfrac{56}{3}=\dfrac{124}{3}\left(g\right)\)

\(a=\dfrac{124}{3}+\dfrac{80}{3}=68\left(g\right)\)

Ta có bất đẳng thức sau 

a² + b² + c² \(\ge\) ab + bc + ca (1)

Dấu "=" xảy ra <=> a = b = c

Thật vậy (1) <=> 2a² + 2b² + 2c² - 2ab - 2bc - 2ca \(\ge0\) 

  <=> (a - b)² + (b - c)² + (c - a)² \(\ge0\) (bđt này luôn đúng)

Khi đó ta được (1) <=> 2(a² + b² + c²) \(\ge\) 2(ab + bc + ca) 

<=> 3(a² + b² + c²) \(\ge\) 2ab + 2bc + 2ca + a² + b² + c² 

<=> 3(a² + b² + c²) \(\ge\) (a + b + c)² 

=> -(a² + b² + c²) \(\le\dfrac{(a+b+c)^2}{3}\)

Ta có \(P=\dfrac{b+c}{b+c-a}+\dfrac{c+a}{c+a-b}+\dfrac{a+b}{a+b-c}\)

\(=\dfrac{a}{b+c-a}+\dfrac{b}{a+c-b}+\dfrac{c}{a+b-c}+3\)

\(=\dfrac{a^2}{ab+ac-a^2}+\dfrac{b^2}{ab+bc-b^2}+\dfrac{c^2}{ac+bc-c^2}+3\)

\(\ge\dfrac{\left(a+b+c\right)^2}{ab+ac-a^2+ab+bc-b^2+ac+bc-c^2}+3\) (BĐT Schwarz)

\(=\dfrac{\left(a+b+c\right)^2}{2ab+2ac+2bc-a^2-b^2-c^2}+3\)

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

\(\ge\dfrac{\left(a+b+c\right)^2}{\left(a+b+c\right)^2-\dfrac{2}{3}\left(a+b+c\right)^2}+3=\dfrac{1}{1-\dfrac{2}{3}}+3=6\) (đpcm) 

B = x² + 9y² + 2011

= x² + (3y)² + 2011

= x² + (5 - 2x)² + 2011 (do 2x + 3y = 5)

= x² + 4x² - 20x + 25 + 2011

= 5x² - 20x + 2036

= 5x² - 20x + 20 + 2016

= 5(x² - 4x + 4)  + 2016

= 5(x - 2)² + 2016 \(\ge2016\)

=> Min B = 2016 khi x - 2 = 0 <=> x = 2 

khi đó y = \(\dfrac{1}{3}\)

Vậy B_min = 2016 khi x = 2 ; \(y=\dfrac{1}{3}\)