\(A_n=\dfrac{\sqrt{2n-1}}{\left(2n+1\right)\left(2n-1\right)}=\dfrac{\sqrt{2n-1}}{2}\left(\dfrac{1}{2n-1}-\dfrac{1}{2n+1}\right)\)

\(=\dfrac{\sqrt{2n-1}}{2}\left(\dfrac{1}{\sqrt{2n-1}}-\dfrac{1}{\sqrt{2n+1}}\right)\left(\dfrac{1}{\sqrt{2n-1}}+\dfrac{1}{\sqrt{2n+1}}\right)\)

\(< \dfrac{\sqrt{2n-1}}{2}\left(\dfrac{1}{\sqrt{2n-1}}-\dfrac{1}{\sqrt{2n+1}}\right)\left(\dfrac{1}{\sqrt{2n-1}}+\dfrac{1}{\sqrt{2n-1}}\right)\)

\(=\dfrac{1}{\sqrt{2n-1}}-\dfrac{1}{\sqrt{2n+1}}\)

\(\Rightarrow A_1+A_2+...+A_n< 1-\dfrac{1}{\sqrt{3}}+\dfrac{1}{\sqrt{3}}-\dfrac{1}{\sqrt{5}}+...+\dfrac{1}{\sqrt{2n-1}}-\dfrac{1}{\sqrt{2n+1}}=1-\dfrac{1}{\sqrt{2n+1}}< 1\)

a) Đặt \(d=\left(a_1,a_2,...,a_n\right)\Rightarrow\left\{{}\begin{matrix}a_1=dx_1\\a_2=dx_2\\...\\a_n=dx_n\end{matrix}\right.\) (với \(\left(x_1,x_2,...,x_n\right)=1\)).

Ta có \(A_i=\dfrac{A}{a_i}=\dfrac{d^nx_1x_2...x_n}{dx_i}=d^{n-1}\dfrac{x_1x_2...x_n}{x_i}=d^{n-1}B_i\forall i\in\overline{1,n}\).

Từ đó \(\left[A_1,A_2,...,A_n\right]=d^{n-1}\left[B_1,B_2,...,B_n\right]\).

Mặt khác do \(\left(x_1,x_2,...,x_n\right)=1\Rightarrow\left[B_1,B_2,...B_n\right]=x_1x_2...x_n\).

Vậy \(\left(a_1,a_2,...,a_n\right)\left[A_1,A_2,...,A_n\right]=d.d^{n-1}x_1x_2...x_n=d^nx_1x_2...x_n=A\).

\(\dfrac{a_1}{2-a_1}+\dfrac{a_2}{2-a_2}+...+\dfrac{a_n}{2-a_n}\ge\dfrac{n}{2n-1}\)

\(\Leftrightarrow\dfrac{a^2_1}{2a_1-a^2_1}+\dfrac{a^2_2}{2a_2-a^2_2}+...+\dfrac{a^2_n}{2a_n-a^2_2}\ge\dfrac{n}{2n-1}\)

Áp dụng bất đẳng thức cộng mẫu số

\(\Rightarrow\dfrac{a^2_1}{2a_1-a^2_1}+\dfrac{a^2_2}{2a_2-a^2_2}+...+\dfrac{a^2_n}{2a_n-a^2_2}\ge\dfrac{\left(a_1+a_2+...+a_n\right)^2}{2\left(a_1+a_2+...+a_n\right)-\left(a^2_1+a^2_2+...+a_n^2\right)}\)

\(\Rightarrow\dfrac{a^2_1}{2a_1-a^2_1}+\dfrac{a^2_2}{2a_2-a^2_2}+...+\dfrac{a^2_n}{2a_n-a^2_2}\ge\dfrac{1}{2-\left(a^2_1+a^2_2+...+a_n^2\right)}\)

Chứng minh rằng \(\dfrac{1}{2-\left(a^2_1+a_2^2+...+a^2_n\right)}\ge\dfrac{n}{2n-1}\)

\(\Leftrightarrow2n-1\ge n\left[2-\left(a^2_1+a^2_2+...+a^2_n\right)\right]\)

\(\Leftrightarrow2n-1\ge2n-n\left(a^2_1+a^2_2+...+a^2_n\right)\)

\(\Leftrightarrow-1\ge-n\left(a^2_1+a^2_2+...+a^2_n\right)\)

\(\Leftrightarrow1\le n\left(a^2_1+a^2_2+...+a^2_n\right)\)

\(\Leftrightarrow\dfrac{1}{n}\le a^2_1+a^2_2+...+a^2_n\)

Áp dụng bất đẳng thức cộng mẫu số

\(\Rightarrow VP=\dfrac{a^2_1}{1}+\dfrac{a^2_2}{1}+...+\dfrac{a^2_n}{1}\ge\dfrac{\left(a_1+a_2+...+a_n\right)^2}{n}=\dfrac{1}{n}\)

\(\Rightarrow\) đpcm

Vậy \(\dfrac{1}{2-\left(a^2_1+a_2^2+...+a^2_n\right)}\ge\dfrac{n}{2n-1}\)

\(\Rightarrow\dfrac{a_1}{2-a_1}+\dfrac{a_2}{2-a_2}+...+\dfrac{a_n}{2-a_n}\ge\dfrac{n}{2n-1}\) ( đpcm )