[/quote]
,
such that 
is a perfect square for all integers 
and 
.
and 
Prove that
Prove that
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 
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 
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
In an acute-angled triangle 
, we consider the feet 
and 
of the altitudes from 
and 
, and the intersections 
and 
of the angle bisectors from and with the opposite sides 
and 
respectively. Show that the centre of the incircle 
of triangle lies on the segment 
if and only if the centre of the circumcircle 
of triangle lies on the segment 
.
, we consider the feet 
and 
of the altitudes from 
and 
, and the intersections 
and 
of the angle bisectors from and with the opposite sides 
and 
respectively. Show that the centre of the incircle 
of triangle lies on the segment 
if and only if the centre of the circumcircle 
of triangle lies on the segment 
.
,
and 
be the feet of the altitudes from the vertices 
and 
,
and 
and 
at the points 
and 
,
of triangle 
if and only if the circumcenter 
of triangle 
be the directed distance from P to the line BC. ["Directed" means: positive if P and A lie in the same halfplane with respect to BC, and negative if P and A lie in different halfplanes with respect to BC.] Similarly, let 
and 
be the directed distances from P to the lines CA and AB, respectively. Then, the triple 
will be called the triple of "actual trilinear coordinates" of the point P with respect to triangle ABC. Instead of "trilinear coordinates", we will often just say "trilinears".
,
,
.
;
.
.
;
.
has the line trilinears
.
of triangle 
,
and 
(and consequently, 
,
and 
)
,
and 
.
if and only if the circumcenter 
,
,
,
be three points on the sidelines of a triangle 
,
,
concur at a point 
,
.
if and only if
.
and 
where the angle bisectors of the angles 
where the angle bisector of the angle 
meets the side 
,
,
are the angle bisectors of the three angles of triangle 
.
and 
,
.
.
,
,
meet the sides 
,
,
,
,
,
concur at the point 
.
and 
yields
.
,
,
.
,
,
is isosceles, so that its base angle 
equals
.
.
.
.
.
.
.
of the altitude of triangle 
and 
yield 
and 
,
.
.
,
and 
.
,
and 
.
,
,
,
and 
.
,
,
and 
;
.
.
.
.
,
,
and 
;
.
.
.
.
-
lies on 
must be acute.
or 
.
,
is acute. Otherwise, we will have 
(contradiction).
is between the rays 
and 
.
is the intersection of the ray 
.
the excenter of 
.
touch the sides 
at 
.
are the midpoints of 
respectively. Then 
and 
,
and 
.
is the polar of 
is the polar of 
wrt 
the intersection of 
and 
,
and 
.
.
the circumcenter of the medial triangle of 
.
is the inverse image of 
.
and its isogonals 
respectively. For 
define 
,
is the cevian triangle of 
. Prove that 
,
and 
respectively w.r.t. 
and
