[/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 
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 the circumcenter of an acute triangle 
and let 
be the circle with center 
that is tangent to at 
and tangent to side 
at 
. Circle meets 
and 
again at 
and 
respectively. The lines 
and 
meet again at 
and 
. Lines 
and 
intersect at 
. Prove that 
.
be the circumcenter of an acute triangle 
and let 
be the circle with center 
that is tangent to at 
and tangent to side 
at 
. Circle meets 
and 
again at 
and 
respectively. The lines 
and 
meet again at 
and 
. Lines 
and 
intersect at 
. Prove that 
.
,
,
;
is a rectangle and 
a parallelogram, so 
,
( 1 ). It's also easy to see that 
and 
and 
.
and 
are the radii of the circles 
and 
.
( 2 ), 
( 3 )
,
,
( 4 ).
,
and 
ie 
so 
is a parllellogrm
and 
we suffice to prove that 
.
is the angular bisector of 
Let 
intersect 
at 
.But since 
we have 
and we are done
