[/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 Willie, Davi-8191, myh2910, jhu08, HamstPan38825, Adventure10, Mango247, Rounak_iitr
Let 
and 
be two orthogonal circles, and let the center of be 
. Diameter 
of is selected so that 
lies strictly inside . The two circles tangent to , passing through and 
, touch at 
and 
. Prove that 
is cyclic.
Proposed by Eric Chen
and 
be two orthogonal circles, and let the center of be 
. Diameter 
of is selected so that 
lies strictly inside . The two circles tangent to , passing through and 
, touch at 
and 
. Prove that 
is cyclic.Proposed by Eric Chen
Since 
Inversion WRT 
tangent to 
of 
These lines will meet at the double point 
is the polar of 
are collinear. Since 
lie on the inverse circle of 
,
,
,
,
are sent to the tangents from 
,
are tangents to 
,
be the center of 
,
.
by Power of a Point. In conjunction with 
,
![[asy]
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and 
be the circumcircles of 
respectively. Invert about 
![[asy]
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lie on 
and 
are both tangents to 
denotes the image of 
we need to show that 
is a straight line. Note that 
is the polar of 
where 
lies closer to 
Since 
so we're done. 
be the midpoint of 
;
.
are inverses under the inversion WRT 
and 
is harmonic and by perspective at 
.
as the intersection of the polar of 
.
as desired.
.
are tangent to 
If 
are the midpoints of 
then it suffices to prove 
by homothety, but this follows as 
and tangent to 
and 
respectively.
around 
as 
A line tangent to 
A line tangent to 
are the point of tangencies from 
and 
.
and 
.
Polar of 
are collinear, now inverting back we get that 
be a circle with center 
point outside of this circle. Let 
be a line passing through 
and 
lines tangent to 
,
lies on 
is concyclic. This is trivial, as 
will be diameter of this circle.
as the center of 
,
,
and 
,
were cyclic then the following must hold:
Thus we need to show that the points 
and 
,
because of the tangents (here we define this as 
)
,
)
are all collinear,then we must show that the point 
or in other words is on the circle of the cyclic quad 
,
Thus the points 
are all collinear.
and we get the following
be a triangle. Let 
be the intersection of tangents at 
to 
.
be a point on 
.
be reflection of 
be tangent from 
be the center of 
are collinear. But consider the second inversion around 
with 
is cyclic. Notice that both 
and 
are cyclic, which proves the desired result.
.
implying 
,
it is obvious since 
as desired. 
around 
,
are tangent 
are collinear
c
goes to 
and 
goes to 
.
.
are tangent to 
are collinear. Because 
by orthogonality, we know 
where the last equality follows from Thales', as desired. 
problems...
and notice that 
and 
are the tangents from 
and notice that 
is the midpoint of 
and also on 
and 
and 
wrt 
(because of orthogonality), which is the 
of 
so 
and 
are tangent to 
be the second intersection of 
with 
are collinear. Inverting about 
cyclic. In fact, 
is cyclic, since 
and 
;
and 
are the touch points of the tangents at 
.
,
,
to be concyclic, which holds as each point lies on 
.
