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)
of positive integers is called central if for every positive integer 
terms of the sequence is equal to 
of positive integers such that for every central sequence 
there are infinitely many positive integers 
.
be the circumcircle of triangle 
,
and 
and 
,
be a transversal of 
be intersections of 
with 
be a tangent line of 
is cyclic
Y by Adventure10, Mango247, Mango247
Let 
be a cyclic quadrilateral with circumcircle 
The lines tangent to 
at 
intersect at 
is the midpoint of the segment 
The line passing through 
and parallel to 
intersects 
at 
Line 
intersects 
at 
and intersects at 
(different from point 
).
Prove that
be a cyclic quadrilateral with circumcircle 
The lines tangent to 
at 
intersect at 
is the midpoint of the segment 
The line passing through 
and parallel to 
intersects 
at 
Line 
intersects 
at 
and intersects at 
(different from point 
).Prove that
This post has been edited 2 times. Last edited by Henry_2001, Aug 12, 2019, 12:07 PM
is 
in 
and 
is 
in 
.
.
,
then we'd be done.
is cyclic we get
Chasing 
using the symmedian properties we have
we get
Finally we go after 
using that 
wich implies that 
is isosceles with 
.
But there is a little problem.GH//BA
