Cho \(\hept{\begin{cases}a_1>a_2>...>a_n>0\\1\le k\in Z\end{cases}}\)

CMR : \(a_1+\frac{1}{a_n\left(a_1-a_2\right)^k\left(a_2-a_3\right)^k...\left(a_{n-1}-a_n\right)^k}\ge\frac{\left(n-1\right)k+2}{\sqrt[\left(n-1\right)k+2]{k^{\left(n-1\right)k}}}\)

Cho \(a_1,a_2,..,a_n\) là các số nguyên dương và n>1.

Đặt \(A=a_1a_2...a_n,\)  \(A_i=\dfrac{A}{a_i}\left(i=\overline{1,n}\right)\). CM các đẳng thức sau:

a) \(\left(a_1,a_2,...,a_n\right)\left[A_1,A_2,...,A_n\right]=A\)

b) \(\left[a_1,a_2,..,a_n\right]\left(A_1,A_2,...,A_n\right)=A\)

Chm bđt:

\(\left(a_1+a_2+...+a_n\right)^2\le n\left(a_1^2+a_2^2+...+a_n^2\right)\)

Chm bđt:

\(\left(a_1+a_2+...+a_n\right)^2\le n\left(a_1^2+a_2^2+...+a_n^2\right)\)

Cho a,b,c dương . CMR :
1) \(\frac{x^3}{y+z}+\frac{y^3}{x+z}+\frac{z^3}{x+y}\ge6;x+y+z\ge6\)
2) \(a_1.a_2....a_n\le\frac{1}{\left(n-1\right)^n};\frac{1}{a_1+1}+\frac{1}{a_2+1}+...+\frac{1}{a_n+1}=n-1\)
3) \(\frac{a}{b+c+1}+\frac{b}{a+c+1}+\frac{c}{b+a+1}+\left(1-a\right)\left(1-b\right)\left(1-c\right)\le1\) với a, b, c thuộc \(\left[0;1\right]\)

Cho \(\left(a_n\right)\) thỏa mãn: \(a_{n+1}=a_n+\dfrac{1}{a_1+a_2+...+a_n}\) \(\left(a_1>0\right)\).

Tính \(lim\dfrac{a_{n+1}}{a_n}\).

Cho \(a_1,a_2,...,a_n>0\) . 

CMR : \(\left(a_1+a_2+...+a_n\right)\left(\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_n}\right)\ge n^2\)(*)