such that for all 
,![\[f(x+yf(x))+y = xy + f(x+y).\]](http://latex.artofproblemsolving.com/c/f/3/cf3d20a041c27244e90876119b4568b7a3e13c03.png)
,
and 
are the midpoints of 
and 
and 
are points on the minor arcs 
and 
of the circumcircle of triangle 
.
and 
are tangent if and only if 
.
and 
on 
and 
.
with 
and 
are 
with 
and 
are 
,
and 
are concurrent.
,
,
is a parallellogram. Suppose that there exists a point 
on 
to 
.
be the midpoint of 
.
,
.
Y by Mango247
Let 
be an acute triangle, 
its circumcenter and 
its circumcircle. 
cuts again at 
. The tangents of at 
, intersect at 
. The perpendicular on 
from cuts 
at
and at 
and 
. 
cuts again at 
. Prove that is the midpoint of the arc 
of that does not contain 
.
be an acute triangle, 
its circumcenter and 
its circumcircle. 
cuts again at 
. The tangents of at 
, intersect at 
. The perpendicular on 
from cuts 
at
and at 
and 
. 
cuts again at 
. Prove that is the midpoint of the arc 
of that does not contain 
.
is a diameter of 
is given. For this, let 
.
)
,
,
.
.
is cyclic. 
and 
are collinear. Since 
we are done!
so we are basically done 
is an harmonic quadrilateral, thus:
But since 
we have that 
thus 
and we are done 
