\(a,\dfrac{16+x}{x^2-2x}+\dfrac{18}{2x-x^2}\)

\(=\dfrac{16+x}{x^2-2x}-\dfrac{18}{x^2-2x}\)

\(=\dfrac{16+x-18}{x^2-2x}\)

\(=\dfrac{x-2}{x\left(x-2\right)}\)

\(=\dfrac{1}{x}\)

\(b,\dfrac{2y}{2x^2-xy}+\dfrac{4x}{xy-2x^2}\)

\(=\dfrac{2y}{2x^2-xy}-\dfrac{4x}{2x^2-xy}\)

\(=\dfrac{2y-4x}{2x^2-xy}\)

\(=\dfrac{2\left(y-2x\right)}{x\left(2x-y\right)}\)

\(=\dfrac{-2\left(2x-y\right)}{x\left(2x-y\right)}\)

\(=-\dfrac{2}{x}\)

\(c,\dfrac{4-x^2}{x-3}+\dfrac{2x-2x^2}{3-x}+\dfrac{5-4x}{x-3}\)

\(=\dfrac{4-x^2}{x-3}-\dfrac{2x^2-2x}{x-3}+\dfrac{5-4x}{x-3}\)

\(=\dfrac{4-x^2-2x^2+2x+5-4x}{x-3}\)

\(=\dfrac{-3x^2-2x+9}{x-3}\)

Đặt \(A=6^{2n+1}+5^{n+2}\)

Với n=0

=>\(A\left(0\right)=6^{2.0+1}+5^{0+2}=6+5^2=31\) chia hết cho 31

Giả sử n=k thì A sẽ chia hết cho 31

=>\(A\left(k\right)=6^{2k+1}+5^{k+2}\) chia hết cho 31

Chứng minh n=k+1 cũng chia hết cho 31 hay \(A\left(k+1\right)=6^{2\left(k+1\right)+1}+5^{\left(k+1\right)+2}\) chia hết cho 31

 thật vậy

\(A\left(k+1\right)=6^{2k+3}+5^{k+3}=6^{2k+1}.36+5^{k+2}.5\)

\(=5\left(6^{2k+1}+5^{k+2}\right)+3.6^{2k+1}\)

Theo giả thiết ta có

\(6^{2k+1}+5^{k+2}\) chia hết cho 31

=>\(5\left(6^{2k+1}+5^{k+2}\right)\) chia hết cho 31

mà\(31.6^{2k+1}\) chia hết cho 31

=>\(5\left(6^{2k+1}+5^{k+2}\right)+31.6^{2k+1}\) chia hết cho 31

Hay \(A\left(k+1\right)\) chia hết cho 31

Vậy \(^{6^{2n+1}+5^{n+2}}\) chia hết cho 31

\(2Fe+6H_2SO_4\rightarrow Fe_2\left(SO_4\right)_3+3SO_2+6H_2O\)

\(M=\dfrac{B}{A}=\dfrac{\dfrac{\sqrt{x}+1}{\sqrt{x}+3}}{\dfrac{x-\sqrt{x}+2}{\sqrt{x}+3}}=\dfrac{\sqrt{x}+1}{x- \sqrt{x}+2}\)\(=\dfrac{\sqrt{x}+1}{\left(\sqrt{x}-\dfrac{1}{2}\right)^2+\dfrac{7}{4}}\)

Dễ thấy: \(\left(\sqrt{x}-\dfrac{1}{2}\right)^2+\dfrac{7}{4}>0\forall x\)

Và \(\sqrt{x}+1\ge1\forall x\)\(\Rightarrow M=\dfrac{\sqrt{x}+1}{\left(\sqrt{x}-\dfrac{1}{2}\right)^2+\dfrac{7} {4}}\ge1\)

Xảy ra khi \(x=1\)

Bài 1:

\(x^2+2x\sqrt{x+\dfrac{1}{x}}=8x-1\)

ĐK: \(x\ge 0\)

\(\Leftrightarrow4x^2\left(x+\dfrac{1}{x}\right)=x^4-16x^3+66x^2-16x+1\)

\(\Leftrightarrow -x^4+20x^3-66x^2+20x-1=0\)

\(\Leftrightarrow -(x^2-16x+1)(x^2-4x+1)=0\)

\(\Rightarrow x=\dfrac{4\pm\sqrt{12}}{2}\) ( thỏa mãn )

Bài 2:

Từ \(x+y+z=0\Leftrightarrow x^2+y^2+z^2+2\left(xy+yz+xz\right)=0\)

\(\Leftrightarrow y^2+z^2=-x^2-2\left(xy+yz+xz\right)\)

\(\Rightarrow\dfrac{x^2}{y^2+z^2-x^2}=\dfrac{x^2}{-2x^2-2\left(xy+yz+xz\right)}=-\dfrac{x^2} {2\left(x+y\right)\left(x+z\right)}\)

\(\Rightarrow P=-\dfrac{1}{2}\left(\dfrac{x^2}{\left(x+y\right)\left(x+z\right)}+\dfrac{y^2} {\left(x+y\right)\left(y+z\right)}+\dfrac{z^2}{\left(x+z\right)\left(y+z\right)}\right)\)

\(=-\dfrac{1}{2}\cdot\dfrac{3xy\left(x+y\right)}{xy\left(x+y\right)}=-\dfrac{3}{2}\)

\(Ư(80) = { 1;80;2;40;4;20;10;8;5;16 }\)

Ta có:

\(\text{VT}=1-\frac{2ab^2}{2ab^2+1}+1-\frac{2bc^2}{2bc^2+1}+1-\frac{2ca^2}{2ca^2+1}\)

\(\text{VT}=3-\underbrace{\left( \frac{2ab^2}{2ab^2+1}+\frac{2bc^2}{2bc^2+1}+\frac{2ca^2} {2ca^2+1}\right)}_{N}\) (1)

Áp dụng BĐT Am-Gm:

\(2ab^2+1=ab^2+ab^2+1\geq 3\sqrt[3]{a^2b^4}\)

\(\Rightarrow \frac{2ab^2}{2ab^2+1}\leq \frac{2ab^2}{3\sqrt[3]{a^2b^4}}=\frac{2}{3}\sqrt[3]{ab^2}\)

Tương tự với các phân thức còn lại và cộng theo vế, suy ra :

\(N\leq \frac{2}{3}(\sqrt[3]{ab^2}+\sqrt[3]{bc^2}+\sqrt[3]{ca^2})\)

Áp dụng BĐT AM-GM:

\(\sqrt[3]{ab^2}\leq \frac{a+b+b}{3}\); \(\sqrt[3]{bc^2}\leq \frac{b+c+c}{3}; \sqrt[3]{ca^2}\leq \frac{c+a+a}{3}\)

\(\Rightarrow N\leq \frac{2}{3}\left(\frac{a+b+b}{3}+\frac{b+c+c}{3}+\frac{c+a+a}{3}\right)\)

\(\Rightarrow N\leq \frac{2}{3}\left(\frac{a+b+b}{3}+\frac{b+c+c}{3}+\frac{c+a+a}{3}\right)\) (2)

Từ \((1),(2)\Rightarrow \text{VT}\geq 1\)

Dấu bằng xảy ra khi \(a=b=c=1\)

Áp dụng BĐT AM-GM ta có:

\(\frac{a^3}{a^3+b^3}+\frac{1}{2}+\frac{1}{2}\geq 3\sqrt[3]{\frac{a^3}{4(a^3+b^3)}}\)

\(\frac{b^3}{a^3+b^3}+\frac{1}{2}+\frac{1}{2}\geq 3\sqrt[3]{\frac{b^3}{4(a^3+b^3)}}\)

Cộng theo vế:

\(\Rightarrow 3\geq 3\sqrt[3]{\frac{a^3}{4(a^3+b^3)}}+3\sqrt[3]{\frac{b^3}{4(a^3+b^3)}}\)

\(\Leftrightarrow 1\geq \frac{a+b}{\sqrt[3]{4(a^3+b^3)}}\Leftrightarrow \sqrt[3]{4(a^3+b^3)}\geq a+b\)

Thực hiện tương tự với các biểu thức còn lại và cộng theo vế suy ra:

\(\sqrt[3]{4(a^3+b^3)}+\sqrt[3]{4(b^3+c^3)}+\sqrt[3]{4(c^3+a^3)}\geq 2(a+b+c)\)

(đpcm)

Dấu bằng xảy ra khi \(a=b=c\)