?
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 
.
be a triangle and 
its incenter. The lines 
and 
intersect the circumcircle of 
and 
,
and 
be the circumferences of diameters 
and 
,
at 
,
at 
and 
.
Y by Adventure10
Two noncongruent circles 
and 
are exterior to each other. Their common tangents intersect the line through their centers at points 
and 
. Let 
be any point of . Prove that there is a diameter of with one endpoint on line 
and the other on 
.
and 
are exterior to each other. Their common tangents intersect the line through their centers at points 
and 
. Let 
be any point of . Prove that there is a diameter of with one endpoint on line 
and the other on 
.
be the diameter of 
(where 
are the centers of 
respectively). 
are the centers of the two similarities which turn the circles into one another, one internal and one external. One of these similarities will take 
onto 
are endpoints of diameters which correspond to one another through the two similarities), while the other one will take 
onto 
