?
such that ![\[f(f(x)) + xf(xy) = x + f(y)\]](http://latex.artofproblemsolving.com/a/6/f/a6f84dd53eec985b90bad3df6580eeaf29d73195.png)
for all positive real numbers 
and 
.
its incenter. The lines 
and 
intersect the circumcircle of 
and 
,
and 
be the circumferences of diameters 
and 
,
at 
and 
,
and 
.
Y by
Let 
be the center of the circumscribed circle of triangle 
. Line 
and 
intersect sides 
and 
at points 
and 
, and intersect the circumcircle at points 
and 
respectively. Lines 
and 
meet at point 
. Prove that 
.
(B.Podkhalyuzin)
be the center of the circumscribed circle of triangle 
. Line 
and 
intersect sides 
and 
at points 
and 
, and intersect the circumcircle at points 
and 
respectively. Lines 
and 
meet at point 
. Prove that 
.(B.Podkhalyuzin)
This post has been edited 1 time. Last edited by parmenides51, Jun 26, 2021, 7:34 AM
and 
are isogonal wrt 
,
and 
are isogonal wrt 
,
and 
