AoPS Academy
. Prove that
Let 
be real numbers such that 
. Prove that
be a positive integer. Show that there exist a polynomial 
with integer coefficient that satisfy the following is at most 
for each 
of positive integers is called central if for every positive integer , the arithmetic mean of the first 
terms of the sequence is equal to .
of positive integers such that for every central sequence 
there are infinitely many positive integers with 
.
. Recall that 
, 
, and the line through 
parallel to 
are tangent to 
. Therefore, is actually the incircle of 
.![[asy]
size(8cm);
defaultpen(fontsize(9pt));
pair B = dir(110), A = dir(180+30), C = dir(-30), B0 = (A+C)/2, H = A+B+C, A1 = foot(A, B, C), C1 = foot(C, A, B), Ap = extension(A1, B0, B, rotate(90, B)*H), Cp = extension(C1, B0, B, rotate(90, B)*H), X = extension(A, Ap, C, Cp);
draw(A--B--C--cycle, red+linewidth(0.9)); draw(Ap--Cp--B0--cycle, blue+linewidth(0.9)); draw(C--Cp^^A--Ap, gray); draw(A1--C1, heavygreen); draw(A--A1^^C--C1, gray); draw(circumcircle(A1, C1, B), fuchsia);
string[] names = {"$A$", "$B$", "$C$", "$B_0$", "$H$", "$A_1$", "$C_1$", "$A'$", "$C'$", "$X$"};
pair[] points = {A, B, C, B0, H, A1, C1, Ap, Cp, X};
pair[] ll = {A, B, C, B0, N, dir(10), C1, Ap, Cp, N};
for (int i=0; i<names.length; ++i)
dot(names[i], points[i], dir(ll[i]));
[/asy]](http://latex.artofproblemsolving.com/b/8/3/b83d62ab7e49d890ebf5425505daa0897e13da01.png)
. Note that 
are collinear by Pappus Theorem on 
and 
. Moreover, since a homothety at 
sends 
to 
, it also sends 
to the midpoint 
of ; in particular, 
are collinear. Now it is well-known that must lie on the perpendicular line to 
through .
is the incircle of 
. Let 
be the center of .
where 
and 
. Indeed, check![\[CC_1^2 - 2\ell AC_1 = CC_1^2 - AC_1\cdot BC_1 = CC_1^2 - HC_1\cdot CC_1 = CH \cdot CC_1\]](http://latex.artofproblemsolving.com/f/f/f/fffd1dca6a824a2839b9bb261d548cf306c64b5c.png)
as desired. 
. So 
, and since 
, it follows that 
is the polar of 
. Similarly, 
is the polar of 
, hence lies on both so is the polar of . Thus, 
. 
then we know that 
then by computing them(calculating 
is easy and 
and the rest is easy) we get that 
which is symmetric with respect to and which means 
Show that 
of acute-angles 
meet at point 
; 
at points 
respectively. Prove that 
and 
concur.
are tangent to 
is the polar of 
we have 
are collinear , so are 
be the feet of altitude from 
.
are collinear. Since 
,
.
and 
.
,![\[\frac{BA'}{BC'} = \frac{BA_1}{CA_1} \times \frac{AC_1}{BC_1} = \frac{B_1A}{B_1C}\]](http://latex.artofproblemsolving.com/9/d/8/9d8b1536e8f89cfba202a73f4c781464c2817147.png)
where the last equality follows from Ceva.