[/quote]
,
and 
Prove that
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 
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 
.
,
.
Y by Adventure10, Mango247
Let 
be a triangle and let 
be the foot of the 
- altitude. The circle 
with the diameter ![$[AD]$](//latex.artofproblemsolving.com/0/f/3/0f3e4c424371b27673db323ced8ef0777940c0d4.png)
meet again the lines
, 
in the points 
, 
respectively. Denote the meetpoint 
of the tangents to the circle in the points 
, 
. Prove that the ray 
is the -median in 
(Serbia).
be a triangle and let 
be the foot of the 
- altitude. The circle 
with the diameter ![$[AD]$](http://latex.artofproblemsolving.com/0/f/3/0f3e4c424371b27673db323ced8ef0777940c0d4.png)
meet again the lines
, 
in the points 
, 
respectively. Denote the meetpoint 
of the tangents to the circle
in the points 
, 
. Prove that the ray 
is the -median in 
(Serbia).This post has been edited 3 times. Last edited by Virgil Nicula, Jul 6, 2010, 2:38 AM
are orthogonal to 
it follows that 
pass through the midpoints 
of 
Let 
and 
becomes orthocenter of 
Moreover 
since 
is the polar of the intersection 
WRT 
Since 
is the midpoint of 
then 
and the second intersection 
of the circle 
is harmonically and the ray 
is the 
.
, i.e. 
what means the quadrilateral 
is
and 
obtain that 
.
.
. Thus the quadrilateral 
is an harmonical division and 
.
,
is cyclic, hence 
,
is cyclic and, 
,
