such that : 
be an infinite sequence of positive integers. Prove that there exists a unique integer 
such that![\[a_n < \frac{a_0+a_1+a_2+\cdots+a_n}{n} \leq a_{n+1}.\]](http://latex.artofproblemsolving.com/9/4/f/94fc5a51b7e68588123c5b527fe75183bc4c4937.png)
be two positive sequences defined by 
and![\[ \begin{cases} {{x}_{n+1}}{{y}_{n+1}}-{{x}_{n}}=0 \\ x_{n+1}^{2}+{{y}_{n}}=2 \end{cases} \]](http://latex.artofproblemsolving.com/2/0/1/2012dc6ff3a41478547cea4aa9b6bccbe17a3623.png)
for all 
.
Y by
Let 
be a non-equilateral triangle, and let 
and 
be its incenter and circumcenter, respectively. Prove that 
if and only if 
, and that both inequalities become equalities simultaneously.
be a non-equilateral triangle, and let 
and 
be its incenter and circumcenter, respectively. Prove that 
if and only if 
, and that both inequalities become equalities simultaneously.
