[/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 
,
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 
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.
Y by
Let 
be an isosceles trapezoid, with a large base 
and a small base 
. Let 
be any point on side and 
be the line through and perpendicular to . Two rays 
and 
are said to satisfy the condition 
if they are symmetric about each other through and intersect the two rays 
and 
at 
and 
respectively. Find the locus of the midpoint of the segment 
when the two rays and change and satisfy condition .
be an isosceles trapezoid, with a large base 
and a small base 
. Let 
be any point on side and 
be the line through and perpendicular to . Two rays 
and 
are said to satisfy the condition 
if they are symmetric about each other through and intersect the two rays 
and 
at 
and 
respectively. Find the locus of the midpoint of the segment 
when the two rays and change and satisfy condition .This post has been edited 1 time. Last edited by parmenides51, Sep 17, 2021, 1:48 PM
;
.
and 
.
.
in the point 
.
of 
.
,
,
