[/quote]
,
that satisfy 
are positive real numbers, then
and 
Prove that
such that 
is a perfect square for all integers 
and 
.
Prove that
Prove that
and 
Prove that
satisfied 
Find minimum and maximum of 
such that:
with all 
is real number
be the set of integers. Determine all functions 
such that, for all integers 
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
The angle 
of a triangle 
is 
. Let 
be a point on the side 
and 
a point on the extension of the side 
, such that 
is the internal bisector of the angle 
and 
is the external bisector corresponding to the angle 
. The segment 
intersects 
at point 
. Prove that 
.
of a triangle 
is 
. Let 
be a point on the side 
and 
a point on the extension of the side 
, such that 
is the internal bisector of the angle 
and 
is the external bisector corresponding to the angle 
. The segment 
intersects 
at point 
. Prove that 
.
