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.
and 
of the same colour such that 
is the midpoint of 
.
a triangle with circumcircle 
and isogonal conjugates 
and 
. Take 
and 
points on 
and 
respectively such that 
. If 
cut again at 
. Prove that 
and 
cut at and are external just read it by directed angles
chess board. Every link of the chain connects the centers of adjacent squares either horizontally, vertically or diagonally, where the two squares are adjacent if they share an edge or a corner. For the interior polygon bounded by the chain, prove that the total area of black pieces equals the total area of white pieces. “ have incenter 
and intouch triangle 
. Let the circumcircle of intersect 
at 
and have center 
. Let 
be the midpoint of arc 
on the circumcircle. Suppose quadrilateral 
is cyclic such that 
lies on . Show that 
.
, 
, 
cut three common chords concurrent at . Consider two distinct circles 
, 
which are internally tangent to all 
. Determine, with proof, which of the following two statements is true. is the insimilicenter of and is the exsimilicenter of and .
.
be a point on 
, and let 
be a line passing through such that
. intersects 
at 
and the circumcircle of triangle 
at 
(
).
and 
are tangent at .
+1 w
, such that for 
, then 
and 
work, where 
.
be nonnegative real numbers such that 
.
satisfying 
.
satisfying:
)