(Nghi binh 20/09)

Cho \(a_1,a_2,...,a_n>0;3\le n\in N.\)  Đặt:

\(A_1=\frac{a_1}{a_2+a_3}+\frac{a_2}{a_3+a_4}+...+\frac{a_{n-1}}{a_n+a_1}+\frac{a_n}{a_1+a_2}\)

\(A_2=\frac{a_1}{a_n+a_2}+\frac{a_2}{a_1+a_3}+...+\frac{a_{n-1}}{a_{n-2}+a_n}+\frac{a_n}{a_{n-1}+a_1}\)

Chứng minh rằng: \(Max\left\{A_1,A_2\right\}\ge\frac{n}{2}\)

với \(a_1,a_2,a_3,.....,a_n>0;a_1+a_2+a_3+....+a_n=k\)

Chứng minh\(\left(a_1+\frac{1}{a_2}\right)^2+\left(a_2+\frac{1}{a_3}\right)^2+...+\left(a_n+\frac{1}{a_1}\right)^2\ge\frac{1}{n}\left(\frac{k^2+n^2}{k}\right)^2\)

Chứng minh rằng: \(\frac{a_1}{b_1}+\frac{a_2}{b_2}+\frac{a_3}{b_3}+...+\frac{a_n}{b_n}\ge n\left(\frac{a_1+a_2+a_3+...+a_n}{b_1+b_2+b_3+...+b_n}\right)\)

Với \(a_1,a_2...,a_n;b_1,b_2...,b_n>0\)

Cho \(a_1,a_2,a_3,...,a_n\left(n\ge2\right)\) là các số thực thỏa mãn \(a_1a_2+a_2a_3+...+a_{n-1}a=1\)

Chứng minh rằng : \(a_1^2+a_2^2+a_3^2+...+a_n^2\ge\frac{1}{\cos\frac{\pi}{n+1}}\)

Cho \(a_1\le a_2\le....\le a_n\)  thỏa mãn \(\hept{\begin{cases}a_1+a_2+a_3+...+a_n=0\\\left|a_1\right|+\left|a_2\right|+\left|a_3\right|+...+\left|a_n\right|=1\end{cases}}\)

CMR: \(a_n-a_1\ge\frac{2}{n}\)

\(Cho\) \(\dfrac{a_1}{a_2}=\dfrac{a_2}{a_3}=...=\dfrac{a_{n-1}}{a_n}=\dfrac{a_n}{a_1}\). Và \(a_1+a_2+...+a_n\ne0;a_1=-\sqrt{5}\). Tính \(a_2;a_3;...a_n=?\)

Cho n số khác 0 là a₁, a₂, a₃,....,a_n thảo mãn \(a_2^2=a_1.a_3,a_3^2=a_2.a_4,...,a_{n-1}^2=a_{n-2}.a_n\). Chứng minh \(\frac{a_1^3+a_2^3+a_3^3+...+a_{n-1}^3}{a_2^3+a_3^3+a_4^3+...+a_n^3}=\frac{a_1}{a_n}\)

CMR:

Nếu \(\frac{a_1}{a_2}=\frac{a_2}{a_3}=\frac{a_3}{a_4}=...=\frac{a_n}{a_{n+1}}\)thì\(\left(\frac{a_1+a_2+a_3+...+a_n}{a_2+a_3+a_4+..+a_{n+1}}\right)^n=\frac{a_1}{a_{n+1}}\)