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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 cells. such that Florian has a strategy to always deduce the entire tiling.
, 
be the orthocenter of it and 
be any point on the side 
. The points 
are on the segments 
, respectively, such that the points 
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 
. Prove that the points 
and lie on the same line.
, 
, and 
. Points 
and 
of the sides 
and 
respectively, are such that 
and 
. Calculate the lenght of 
.
described above, let us examine the triangles 
and 
. If 
and 
, then we obtain 
(i) from . Next, an application of Law of Sines to 
and 
yields the following system of equations in 
and 
:
(ii);
(iii)
.
such that
