,
,
,
for all 
,
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 
.
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)
Y by GeoKing
How to prove: the isogonal image of the Apollonius circle of 
passing through point 
is an oblique strophoid.
passing through point 
is an oblique strophoid.
This post has been edited 2 times. Last edited by luosw, Jun 6, 2021, 7:31 AM
Reason: Spelling mistake.
Reason: Spelling mistake.
on the oblique strophoid, it is easy to know that 
.
on the symmedian of 
is similar to 
,
on the oblique strophoid, 
.
