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, Mango247
In a scalene triangle 
,the bisectors of angle 
intersect their corresponding sides at 
respectively.
denote respectively the incenter,circumcenter and orthocenter of triangle . Prove that 
are concyclic iff 
is the common tangent line of the circumcircles of the three triangles 
and 
.
,the bisectors of angle 
intersect their corresponding sides at 
respectively.
denote respectively the incenter,circumcenter and orthocenter of triangle . Prove that 
are concyclic iff 
is the common tangent line of the circumcircles of the three triangles 
and 
.
