0,3mol chứ nhỉ?

a. \(2Al+6HCl\rightarrow2AlCl_3+3H_2\uparrow\left(1\right)\)

\(FeO+2HCl\rightarrow FeCl_2+H_2O\left(2\right)\)

\(Fe_2O_3+6HCl\rightarrow2FeCl_3+3H_2O\left(3\right)\)

b. Theo phương trình \(n_{Al}=\frac{2}{3}n_{H_2}=0,2mol\) và \(n_{HCl\left(1\right)}=0,6mol\)

\(\rightarrow m_{FeO}+m_{Fe_2O_3}=35,8-0,2.27=30,4g\)

Đặt \(\hept{\begin{cases}n_{FeO}=x\\n_{Fe_2O_3}=y\end{cases}}\)

\(\rightarrow72x+160y=30,4\left(1\right)\)

Theo phương trình \(2x+6y=n_{HCl\left(2+3\right)}=1,6.1-0,6=1\left(2\right)\)

Từ (1) và (2) suy ra x = 0,2 và y = 0,1

\(\rightarrow m_{FeO}=0,2.72=14,4g\) và \(m_{Fe_2O_3}=0,1.160=16g\)

\(\rightarrow\%m_{FeO}=\frac{14,4}{35,8}.100\%\approx40,22\%\)

\(\rightarrow\%m_{Fe_2O_3}=\frac{16}{35,8}.100\%\approx44,69\%\)

c. Theo phương trình \(n_{AlCl_3}=0,2mol\) và \(n_{FeCl_2}=0,2mol\) và \(n_{FeCl_3}=0,2mol\)

\(\rightarrow m_{\text{muối}}=0,2.133,5+0,2.127+0,2.162,5=84,6g\)

ĐKXĐ : \(\hept{\begin{cases}x\ge0\\x\ne4\\x\ne9\end{cases}}\)

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

\(=\left(\frac{x-4}{\left(\sqrt{x}-3\right)\left(\sqrt{x}-2\right)}-\frac{x-9}{\left(\sqrt{x}-3\right)\left(\sqrt{x}-2\right)}\right)\times\frac{2\left(\sqrt{x}-2\right)}{1}\)

\(=\left(\frac{x-4-x+9}{\left(\sqrt{x}-3\right)\left(\sqrt{x}-2\right)}\right)\times\frac{2\left(\sqrt{x}-2\right)}{1}\)

\(=\frac{5}{\left(\sqrt{x}-3\right)\left(\sqrt{x}-2\right)}\times\frac{2\left(\sqrt{x}-2\right)}{1}\)

\(=\frac{5\times2\left(\sqrt{x}-2\right)}{\left(\sqrt{x}-3\right)\left(\sqrt{x}-2\right)}=\frac{10}{\sqrt{x}-3}\)

\(\left(\frac{\sqrt{x}+2}{\sqrt{x}-3}-\frac{\sqrt{x}+3}{\sqrt{x}-2}\right):\frac{1}{2\sqrt{x}-4}\)

\(=\left(\frac{x-4}{\left(\sqrt{x}-3\right)\left(\sqrt{x}-2\right)}-\frac{x-9}{\left(\sqrt{x}-3\right)\left(\sqrt{x}-2\right)}\right):\frac{1}{2\sqrt{x}-4}\)

\(=\frac{5}{\left(\sqrt{x}-3\right)\left(\sqrt{x}-2\right)}.\frac{2\left(\sqrt{x}-2\right)}{1}\)

\(=\frac{10\left(\sqrt{x}-2\right)}{\left(\sqrt{x}-3\right)\left(\sqrt{x}-2\right)}=\frac{10}{\sqrt{x}-3}\)

Ta luôn có \(4\left(x^3+y^3\right)\ge\left(x+y\right)^3\)(*)

Thật vậy: (*)\(\Leftrightarrow3\left(x-y\right)^2\left(x+y\right)\ge0\)*Đúng với mọi x, y thực dương*

\(\Rightarrow\sqrt[3]{4\left(x^3+y^3\right)}\ge x+y\)

Tương tự, ta có: \(\sqrt[3]{4\left(y^3+z^3\right)}\ge y+z,\sqrt[3]{4\left(z^3+x^3\right)}\ge z+x\)

Cộng theo vế ba bất đẳng thức trên, ta được: \(\sqrt[3]{4\left(x^3+y^3\right)}+\sqrt[3]{4\left(y^3+z^3\right)}+\sqrt[3]{4\left(z^3+x^3\right)}\ge2\left(x+y+z\right)\)

Ta cần chứng minh \(\left(x+y+z\right)+\left(\frac{x}{y^2}+\frac{y}{z^2}+\frac{z}{x^2}\right)\ge6\)

Thật vậy, ta có: \(\left(x+y+z\right)+\left(\frac{x}{y^2}+\frac{y}{z^2}+\frac{z}{x^2}\right)\ge3\sqrt[3]{xyz}+\frac{3}{\sqrt[3]{xyz}}\ge3.2=6\)

Vậy bất đẳng thức được chứng minh

Đẳng thức xảy ra khi x = y = z 

Áp dụng bđt: 2xy \(\le\)(x + y)²/2

khi đó, ta có: \(\sqrt{\frac{a+b}{2ab}}\ge\sqrt{\frac{a+b}{\frac{\left(a+b\right)^2}{2}}}=\sqrt{\frac{2}{a+b}}=\frac{1}{\sqrt{\frac{a+b}{2}}}\ge\frac{1}{\frac{\frac{a+b}{2}+1}{2}}=\frac{4}{a+b+2}\)

CMTT: \(\sqrt{\frac{b+c}{2bc}}\ge\frac{4}{b+c+2}\)

\(\sqrt{\frac{c+a}{2ca}}\ge\frac{4}{c+a+2}\)

=>Đặt A = \(\sqrt{\frac{a+b}{2ab}}+\sqrt{\frac{b+c}{2bc}}+\sqrt{\frac{a+c}{2ac}}\ge\frac{4}{a+b+2}+\frac{4}{b+c+2}+\frac{4}{a+c+2}\)

Áp dụng bđt svacso : \(\frac{x_1^2}{y_1}+\frac{x_2^2}{y_2}+\frac{x_3^2}{y_3}\ge\frac{\left(x_1+x_2+x_3\right)^2}{y_1+y_2+y_3}\)

 ta có: 

\(A\ge\frac{\left(2+2+2\right)^2}{a+b+2+b+c+2+a+c+2}=\frac{36}{2\left(a+b+c\right)+6}=\frac{36}{12}=3\)

=> Đpcm

Ta có: \(\Sigma_{cyc}\frac{a+1}{1+b^2}=\Sigma_{cyc}\left(\frac{a}{1+b^2}+\frac{1}{1+b^2}\right)=\Sigma_{cyc}\left(a-\frac{ab^2}{1+b^2}\right)+\Sigma_{cyc}\left(1-\frac{b^2}{1+b^2}\right)\)\(\ge\Sigma_{cyc}\left(a-\frac{ab^2}{2b}\right)+\Sigma_{cyc}\left(1-\frac{b^2}{2b}\right)=\left(3-\frac{ab+bc+ca}{2}\right)+\left(3-\frac{a+b+c}{2}\right)\)\(\ge\left(3-\frac{\frac{\left(a+b+c\right)^2}{3}}{2}\right)+\frac{3}{2}=3\)

Đẳng thức xảy ra khi a = b = c = 1

M là điểm nào bạn ơi?