AoPS Academy
denote the set of all real numbers. Find all functions 
such that
for all real numbers 
, 
.
, 
, 
of an acute-angled triangle 
intersect at point 
. The bisectors of angles 
and 
of triangle 
meet the segments 
and 
at points 
and 
respectively. Denote the value of the angle 
by 
. Define 
and 
similarly. Find the sum 
.
be an acute angled triangle with orthocenter and 
. Let 
be any other point on the arc 
of the circumcircle of and let line 
intersect line 
at 
and let line 
intersect line 
at 
. Let the circumcircles of 
and intersect again at (
). Let the lines 
intersect the circumcircle of 
again at 
(
). Prove that the lines 
concur.
be a triangle. The incircle of triangle touches the side 
at 
, and the line 
meets the incircle again at a point 
. Let the lines 
and 
meet the incircle of triangle again at 
and 
, respectively. Prove that the lines , 
and 
are concurrent.
with incenter 
. It is known that 
is center of the ex-circle tangent to . Likewise, 
and 
are the centers of the ex-circles tangent to and , respectively. Prove that is the orthocenter of the triangle 
.
with incenter . It is known that is center of the ex-circle tangent to . Likewise, and are the centers of the ex-circles tangent to and , respectively. Prove that is the orthocenter of the triangle .
is the orthic triangle of by angle chase so we're done.
perpendicular with 
via 
and 
,
.