.
Let 
be real numbers such that 
.
be a positive integer. Show that there exist a polynomial 
with integer coefficient that satisfy the following
for each 
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 
.
Y by
Let 
be a circle, 
a diameter and 
an arbitrary point on the circle different than 
and 
such that 
. On the radius 
we consider point 
and the circle 
. The extension of the segment 
meets the circle 
at point 
. From we consider the tangents 
and 
to the circle 
. Prove that the lines 
and 
are concurrent.
be a circle, 
a diameter and 
an arbitrary point on the circle different than 
and 
such that 
. On the radius 
we consider point 
and the circle 
. The extension of the segment 
meets the circle 
at point 
. From we consider the tangents 
and 
to the circle 
. Prove that the lines 
and 
are concurrent.This post has been edited 1 time. Last edited by parmenides51, Mar 3, 2020, 5:25 PM
and 
.
is the orthocenter of 
and 
is cyclic. In this vein, it's well-known that 
are tangent to 
(simple angle-chasing), thus
i.e. 
so it lies on its polar 
.
wrt 
at 
at 
,
be the second intersection of 
lie on the circle 
with diameter 
.
maps 
.
,
,
is tangent to 
and 
and 
passes through 
,
.
which means 
.
and 
again at 
and 
,
intersect 
.
and 
are pairwise parallel.
is an isoceles trapezoid, thus 
.
and 
are directly similar. Thus, triangles 
and 
are also directly similar. Furthermore, it's worth noticing that:![\[ EL\cdot EC = EG\cdot EC = ET^2 \]](http://latex.artofproblemsolving.com/5/4/a/54a7dc59b3cc7f9b2cd2fb7d86e76a211e7b12a0.png)
Then, we should be pretty much done now:![\[ \cfrac{JS}{JT} = \cfrac{JS}{JL}\cdot\cfrac{JL}{JT} = \cfrac{LS}{CT}\cdot\cfrac{CS}{LT} = \cfrac{EL}{ET}\cdot\cfrac{EC}{ET} = 1 \]](http://latex.artofproblemsolving.com/1/f/4/1f41b2cfdc740068e14659937ac095dc97d1085a.png)
Implying that 
,
