[/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 a triangle, and let 
,
,
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 Adventure10, Mango247
Let 
be a triangle, and 
be a point where incircle touches side 
. 
is midpoint of , and 
is a point on such that 
. Let 
be a point on such that 
. Define 
to be circle with diameter 
. We define 
similarly. Prove that every two of these circles are tangent.
be a triangle, and 
be a point where incircle touches side 
. 
is midpoint of , and 
is a point on such that 
. Let 
be a point on such that 
. Define 
to be circle with diameter 
. We define 
similarly. Prove that every two of these circles are tangent.
the center of the nine-point circle and by 
the projection
form a harmonic division. Let 
be the intersection of the circles
- with diameters 
and 
be the projection
are in the involution
that is determined by the rotation of the angle 
around the vertex 
,
,
.
.
. Did anyone solve this in the exam time, Omid?
be a circle with center 
and radius 
and 
two point collinear with 
then any circle passing through 
is perpendicular to 
are two tangent circles and 
is circle perpendicular to both, then 
center of 
.
be the midpoint of 
then 
.
.
are also on the tangent line at 
,let a tanget at F to the incircle intersect BC at N, we have 
hence N is midpoint of DD' ,hence a circle with diameter DD' pass through Fuerbach point.
