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 
,
.
Y by PikaPika999
Let 
be a triangle with circumcenter 
. Let 
be a point on the perpendicular bisector of 
(see figure) and 
, 
be the intersections of the perpendicular bisectors of 
and 
, respectively, with 
and 
. Prove that the midpoints of 
and 
and the point are collinear.
be a triangle with circumcenter 
. Let 
be a point on the perpendicular bisector of 
(see figure) and 
, 
be the intersections of the perpendicular bisectors of 
and 
, respectively, with 
and 
. Prove that the midpoints of 
and 
and the point are collinear.Attachments:
and 
then by area chasing we get: ![$[AOP]=[BOP]$](http://latex.artofproblemsolving.com/b/b/c/bbc5abf9b1922a1caf86e6f4c0ff3fa165f4b55a.png)
![$[AOQ]-[POQ]=[BOR]-[POR]$](http://latex.artofproblemsolving.com/2/8/9/2895a61206d1c01d6f6b2e4a4fb6ed50e47794b3.png)
![$[AOQ]+[POR]=[BOR]+[POQ]$](http://latex.artofproblemsolving.com/8/b/9/8b93c7a441b32ae6454150c9ebef3e232e4233f9.png)
![$[COQ]+[POR]=[COR]+[POQ],$](http://latex.artofproblemsolving.com/f/2/f/f2fc7998ea0e8fe10d28a3f3c1acba81443046c9.png)
or ![$[COQ]+[POR]=\tfrac{1}{2}[CRPQ]$](http://latex.artofproblemsolving.com/3/e/6/3e6590dfa0dec31a3f6dd08fa4c7747a67cc0a18.png)
and the result follows.
