[/quote]
,
and two fixed points 
on that circle, let 
be a moving point on 
is acute and scalene. Let 
be the midpoint of 
and let 
be the three heights of 
,
such that 
.
be the intersection of 
and 
,
be the second intersection of 
and 
.
always goes through a fixed point.
intersects 
.
of 
,
such that 
.
be the center of 
.
always goes through a fixed point.
where 
where 
.
cannot be a perfect cube of a positive integer.
is given. Let 
segments of real line of type ![$[i, j]$](http://latex.artofproblemsolving.com/a/6/4/a644edd67caa59eaded765c417b6d9822f782ec2.png)
,
and 
are integers, 
.
is said to be valuable if the intersection of any two segments from 
is either empty, or is a segment of nonzero length belonging to 
with real coefficients obeying![\[P(x) P(x+1) = P(x^2 + x + 1)\]](http://latex.artofproblemsolving.com/b/e/f/beff3420fa8394f4f98c031d81eea6a5a3b8d28c.png)
for all real numbers 
.
and![\[(x-y)[f(x)+f(y)]\leqslant f(x^2-y^2), \forall x,y \in \mathbb{R}\]](http://latex.artofproblemsolving.com/8/0/b/80bb1d7a992c9683a0c2a07b2febe6271f4e60f5.png)
such that 
,
and 
.
is prime, 
with 
and 
?
be the circumcircle of isosceles triangle 
)
lie on 
.
to the incircle of 
(different from
such that:
is the Gergon point and the incircle 
touch 
,
at 
,
respectively.
,
,
be 
, 
, 
respectively.
and then we get the circle 
and then the circle 
and then the circle 
be non-negative real numbers such that 
Prove that
Y by
Given integers 
, the points 
are chosen independently and uniformly at random on a circle of circumference 
. That is, for each 
, for any 
, and for any arc 
of length 
on the circle, we have 
. What is the probability that there exists an arc of length 
on the circle that contains all the points ?
, the points 
are chosen independently and uniformly at random on a circle of circumference 
. That is, for each 
, for any 
, and for any arc 
of length 
on the circle, we have 
. What is the probability that there exists an arc of length 
on the circle that contains all the points ?
