[/quote]
,
and 
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 
be the second intersection of 
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 
.
,
.
.
and 
to 
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.
,
,
at 
,
be the midpoints of 
and 
,
and 
lie on 
so that 
and 
.
and 
intersect on the circumcircle of 
.
,
.
and 
,
is acute, and let 
be its orthocentre. Points 
and 
,
and 
and 
and 
Y by Adventure10, Mango247
Let 
be a point out of circle 
. Let 
and 
be the tangents to the circle drawn from . Choose a point 
on 
. Suppose that the circumcircle of triangle 
intersects again at 
. Let 
be the reflection of with respect to 
. Prove that
![\[ \angle PBT = \angle {P}'KA \]](//latex.artofproblemsolving.com/8/e/8/8e8e84079b26b0674c3e68f5d333e94fbebe2328.png)
be a point out of circle 
. Let 
and 
be the tangents to the circle drawn from . Choose a point 
on 
. Suppose that the circumcircle of triangle 
intersects again at 
. Let 
be the reflection of with respect to 
. Prove that![\[ \angle PBT = \angle {P}'KA \]](http://latex.artofproblemsolving.com/8/e/8/8e8e84079b26b0674c3e68f5d333e94fbebe2328.png)
This post has been edited 5 times. Last edited by dan23, May 25, 2013, 5:07 PM
so 
so 
done.
is parallel to 
and because 
then we need to prove that 
then we have 
so now 
and we are done
and 
and 
for the point S and we will prove that SP=PK.
is distinct from 
.
is a 
.
is bisected at 
which (again after an angle chase) yields us 
intersects ![\[ \angle DBP = \angle QCA \]](http://latex.artofproblemsolving.com/5/9/5/595529a23011b193fbc4bac7c1f81d014562ff89.png)
and 
.
,
.
