AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
be a triangle with circumcircle 
and circumcenter 
. Let 
and 
represent the 
-excircle and -excenter, respectively. Denote by 
the circle tangent to 
, 
, and on the arc not containing , and similarly for 
. Let the tangency points of 
with line be 
, respectively. Let 
be the intersection point of 
and . Define 
as the point on segment 
such that 
. Suppose that 
intersects again at 
. Let 
be the touch point of the -mixtilinear incircle and , and let 
be the antipode of with respect to . Let 
be the intersection of 
and 
.
is the radical axis of and .
such that
for all 
.
+1 w
, 
and 
be points on circle divided into three equal parts. Construct three equal circles 
, 
, and 
tangent to internally at points , , and respectively. Let 
be any point on arc 
, and draw tangents 
, 
, and 
to circles , , and respectively. Prove that 
.
, 
, 
, 
be points on circle divided into 
equal parts. Construct equal circles , , , 
tangent to internally at , , , . Let be any point on circle , and draw tangents 
, 
, , 
to circles , , , . If the sum of 
of , , , equals the sum of the remaining 
(where 
), find all such .
denote the set of positive integers
such that for all 
it holds that 
Note: consider 
and 
and ![$P_1(x),P_2(x), \ldots P_n(x) \in \mathbb{Z}[x]$](http://latex.artofproblemsolving.com/c/6/4/c64176e52d94032e571b4579e35f9e6adfde675a.png)
such that 
. Prove that exists many 
such that every equation: 
has no real roots
intersecting forming 
. By isogonal conjugate and pedal circle we get that 
are concurrent at 
isogonal conjugate to 
wrt. . Following the Miquel structure for 
so 
lie on a circle. Similarly so 
lie on a circle. By simple angle chasing so 
lie on a circle.
and 
Prove that
Where 
Where 
be projection point from 
.
cuts circumcircle 
at 
.
be antipode of 
are cyclic.
are equidistant from the circumcenter of 
.