:![\[
2 \left( 5^x + 6^x - 3^x \right) = 7^x + 9^x.
\]](http://latex.artofproblemsolving.com/f/b/6/fb6d5ebb4f448d8eeb73cce3288fea3a56d7f528.png)
,
,
,
for all 
,
by 
.
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)
Y by Adventure10, Mango247
Convex quadrilateral 
is cyclic in circle 
, 
is the intersection of the diagonals 
and 
. Circle 
passes through and 
, circle 
passes through and 
, Circles and intersect at and 
. , intersect at another points 
, 
(besides , ), respectively. Prove that 
, 
, 
are concurrent or parallel.
is cyclic in circle 
, 
is the intersection of the diagonals 
and 
. Circle 
passes through and 
, circle 
passes through and 
, Circles and intersect at and 
. , intersect at another points 
, 
(besides , ), respectively. Prove that 
, 
, 
are concurrent or parallel.
