AoPS Academy
![$P(x) \in \mathbb{R}[x]$](http://latex.artofproblemsolving.com/4/5/3/453a624c3b002c1b0e78e0023b24dd22ddd03557.png)
such that:
with 
is an 
th prime and 
is an arbitrary positive integer.
of 
touch 
at 
, 
, let 
denote the line tangent to through 
. Define 
such that 
. Prove that the circumcenter 
of lies on 
.
be a cubic polynomial given by 
, where 
are integers and 
. Suppose that 
for infinitely many pairs 
of integers with 
. Prove that the equation 
has an integer root.
be real numbers such that 
. Determine the minimum value of 
and determine all values of 
such that the minimum value is achived.
be the open unit disk in the complex plane and let 
be a holomorphic function such that 
. Let the Taylor series of 
be 
. Prove that the number of zeroes of (counted with multiplicities) equals 
.
be an abelian additive group such that all nonzero elements have infinite order and for each prime number 
we have the inequality 
, where 
, 
(where the sum has summands) and 
is the order of the quotient group 
(the index of the subgroup 
). of finite index is isomorphic to .
![$\{x_n\} \subseteq [0,1]$](http://latex.artofproblemsolving.com/8/e/4/8e4206325387485d73765c9d7070ecb1ce18ff60.png)
, there exists some constant 
, for any 
, among the sequence 
and 
could be chosen to satisfy 
and 
.
Such That 
is even?
satisfy 
for all 
. Prove that
be a fixed circle, and let 
be a point outside such that 
. A variable line through intersects the circle at two points 
and 
, such that the quadrilateral 
is cyclic, where 
are fixed points on the circle.
,
,
is the tangent from to the circle , and 
is the point of tangency.
, 
, and are collinear.
is perpendicular to chord 
. varies, the point traces a fixed straight line, which is parallel to the isogonal conjugate axis (the so-called *isotropic line*) of the centers 
and .
lie on a circle, their cross-ratio is harmonic:![\[
(ANMB) = -1.
\]](http://latex.artofproblemsolving.com/b/d/5/bd5301ba76f493aec0821c454dffc46dea93c0f2.png)
- The intersection points , and form a well-known harmonic setup along the diagonals of the quadrilateral., , :** is tangent to , and due to harmonicity and projective duality, the polar of passes through ., , and must lie on a common line. is tangent at and is a chord, the angle between and chord is right. is perpendicular to .:** varies, the point moves. lie on a fixed line, which is perpendicular to 
, and is parallel to the isogonal (or isotropic) line determined by the centers and ., or of line , as varies?![\[y''+by'+cy=g(t)\]](http://latex.artofproblemsolving.com/7/8/8/7882dc954ab6dba2445b68d370d630ce1c66a6db.png)
with 
, where 
are constants has the form ![\[y(t)=\int^{t}_{t_0}K(t-s)g(s)ds\,\]](http://latex.artofproblemsolving.com/6/3/c/63c34df1c0013bf659bc76506a93873a5f879070.png)
that exist function 
such continuous on 
and discontinuous on 
is fixed and points 
are distributed in such a way that the angle 
(in radians). Cut the circle at points 
How many arcs with different lengths are obtained. ?
Then 
But I am stuck. I came across this functional equation by trying to find a limit of a sequence, I can show that if it is helpful. 
has solutio
and use variation of parameters in each case to find the solution. Is there any short way to calculate solution?
