[/quote]
,
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 
.
be prime numbers such that their sum isn't divisible by 
.
positive integer quadruples satisfy:
![\[ \left\lfloor \frac{n}{1} \right\rfloor+ \left\lfloor \frac{n}{2} \right\rfloor + \cdots + \left\lfloor \frac{n}{n} \right\rfloor + \left\lfloor \sqrt{n} \right\rfloor \]](http://latex.artofproblemsolving.com/5/6/0/560a9568b2c09512cac1d317a011931a439e6a02.png)
and the sets 
If 
has exactly three elements, prove that 
cannot be an integer.
satisfy:
![$[(x-2)^2+(y-4)^2][ (x-1)^2 + (y-2)^2] = 36$](http://latex.artofproblemsolving.com/b/1/0/b108dc2cac5b7491fb83418eb36c7184f6d5d30c.png)
such that
holds for all 
.
such that:
with all 
,
,
be collinear points such that 
,
,
.
,
,
are the reflections of 
,
,
,
is a nondegenerate triangle, then its circumcenter lies on the circumcircle of 
be three positive real numbers satisfying 
Prove that
There is a very elegant proof :-D Could anyone think of it?
and has exactly 
then
Y by
Let 
be the circumcenter of the triangle 
be a point symmetric of 
wrt line 
is an arbitrary point on the ray 
(
). Angle bisector of angle 
intersects the circumcircle of triangle 
at point 
(
). Let 
be the midpoint of the segment 
. A line passing through point parallel to 
, intersects at point 
. Prove that angles 
and 
angles are equal.
be the circumcenter of the triangle 
be a point symmetric of 
wrt line 
is an arbitrary point on the ray 
(
). Angle bisector of angle 
intersects the circumcircle of triangle 
at point 
(
). Let 
be the midpoint of the segment 
. A line passing through point parallel to 
, intersects at point 
. Prove that angles 
and 
angles are equal.
.
be the reflection of 
is a parallelogram, also see that 
from here it follows that 
collinear and 
,
but also 
where 
(
then 
and the conclusion follows
