such that:
with all 
is real number
be the set of integers. Determine all functions 
such that, for all integers 
and 
,
Proposed by Liam Baker, South Africa
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 
.
Y by Adventure10
Given a circle centered at 
and two points 
and 
lying on it. and do not form a diameter. The point 
is chosen on the circle so that the line 
divides the segment 
in half. Let lines 
and 
intersect at 
, and let lines 
and 
intersect at 
. Prove that 
.
and two points 
and 
lying on it. and do not form a diameter. The point 
is chosen on the circle so that the line 
divides the segment 
in half. Let lines 
and 
intersect at 
, and let lines 
and 
intersect at 
. Prove that 
.
be the midpoint of 
,
and 
.
,
,
,
.
,
,
.
according to line 
gives 
,
,
(Because 
)
,
