be a positive integer. Find all real solutions 
to the system:![\[a_1^2 + a_1 - 1 = a_2\]](http://latex.artofproblemsolving.com/a/4/e/a4e26b83c929b82800576bac11e12de0c3aa1cf6.png)
![\[ a_2^2 + a_2 - 1 = a_3\]](http://latex.artofproblemsolving.com/9/9/5/995e8eec93828b82901b0639463f907f5ff95d08.png)
![\[\hspace*{3.3em} \vdots \]](http://latex.artofproblemsolving.com/e/e/1/ee15e1fefe2ad550d3c9b1c51207eaf66fbfa41f.png)
![\[a_{n}^2 + a_n - 1 = a_1\]](http://latex.artofproblemsolving.com/7/c/7/7c727d1f2044c1bf7c80981e1074bb3cde807b8f.png)
,
,
,
for all 
,
in a non-isosceles triangle 
with incenter 
intersect its circumcircle 
at 
and 
,
at 
is tangent to 
Y by Mango247
In triangle 
. Circle 
passes through 
and meets sides 
and 
at 
and 
, respectively, and the side 
at 
and 
such that lies in between 
and . Suppose that 
, and 
. Find 
.
. Circle 
passes through 
and meets sides 
and 
at 
and 
, respectively, and the side 
at 
and 
such that lies in between 
and . Suppose that 
, and 
. Find 
.This post has been edited 3 times. Last edited by parmenides51, Apr 11, 2020, 5:07 PM
