[/quote]
,
and 
Prove that
Prove that
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 
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 
.
be the circumcircle of triangle 
,
is the midpoint of 
and 
and 
be a transversal of 
be intersections of 
with 
be a tangent line of 
is cyclic
be positive integers that satisfy the inequality![\[
\frac{a}{b} < \frac{x}{y} < \frac{c}{d}
\]](http://latex.artofproblemsolving.com/8/d/9/8d982d52e655546ac29039fa6e440e68821bfe7f.png)
with 
.
,
.
.
respectively.
and 
as the reflections of 
be the circumcircle of 
.
with 
the intersection of ray 
with 
is the circumcenter of 
can be inscribed within a circle.
Y by Adventure10
Consider an arc 
of a circle 
and a point 
variable in that arc . Let 
be the midpoint of the arc 
that doeas not contain 
and let 
be the midpoint of the arc 
that does not contain 
. Let 
be the circle with center passing through and 
be the circle with center passing through 
Prove that the line that contains the intersection points of and passes through a fixed point.
of a circle 
and a point 
variable in that arc . Let 
be the midpoint of the arc 
that doeas not contain 
and let 
be the midpoint of the arc 
that does not contain 
. Let 
be the circle with center passing through and 
be the circle with center passing through 
Prove that the line that contains the intersection points of and passes through a fixed point.
be the second intersection point of 
.
is the midpoint of the arc 
.
.
.
