such that
such that for all 
,![\[f(x+yf(x))+y = xy + f(x+y).\]](http://latex.artofproblemsolving.com/c/f/3/cf3d20a041c27244e90876119b4568b7a3e13c03.png)
,
be positive integers. Julia and Florian play a game on a 
board. Julia
,
be the orthocenter of it and 
be any point on the side 
.
are on the segments 
,
and 
are cyclic. The segments 
and 
intersect at 
is a point on 
such that 
is tangent to the circumcircle of triangle 
at 
and 
intersect at 
.
and 
Y by
Let 
and 
be two parallel lines. The circle 
is tangent to line at point 
and intersects the line at the distinct points 
and 
. The point 
is located on the line . 
and 
intersect again the circle at the points 
and 
, respectively . Show that the line 
bisects the segment ![$[AT]$](//latex.artofproblemsolving.com/6/e/3/6e397da3c2a38b457f7b3504c5cb2a488a64f3a6.png)
.
and 
be two parallel lines. The circle 
is tangent to line at point 
and intersects the line at the distinct points 
and 
. The point 
is located on the line . 
and 
intersect again the circle at the points 
and 
, respectively . Show that the line 
bisects the segment ![$[AT]$](http://latex.artofproblemsolving.com/6/e/3/6e397da3c2a38b457f7b3504c5cb2a488a64f3a6.png)
.
