[/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
The inscribed circle 
of triangle 
with center 
touches the sides 
at points 
. The circle circumsrcibed around 
intersects the circumscribed circle of for second time at the point 
. Let 
be the midpoint 
, 
be the midpoint of 
. The circle circumsrcibed around 
cuts intersects for second time at the point 
. Prove that the circumscribed circles of triangles 
and 
are tangent.
of triangle 
with center 
touches the sides 
at points 
. The circle circumsrcibed around 
intersects the circumscribed circle of for second time at the point 
. Let 
be the midpoint 
, 
be the midpoint of 
. The circle circumsrcibed around 
cuts intersects for second time at the point 
. Prove that the circumscribed circles of triangles 
and 
are tangent.
