,
is the circumcenter of 
and 
is its diameter. The line parallel to 
through 
at 
,
extended at 
and 
meet at 
,
and 
meet at 
.
.
of positive integers such that
such that![\[ x f(x + xy) = x f(x) + f \left( x^2 \right) f(y) \quad \forall x,y \in \mathbb{R}.\]](http://latex.artofproblemsolving.com/b/0/f/b0f1a1d3b9de5d23f1b5a2e9be3b88d8bd575883.png)
over 
is linear
Y by mathmax12
Suppose 
is any point on the side 
of the right trapezoid 
(
). Let the circumscribed circle 
of the triangle 
touch the sides 
at points 
respectively. Let the line 
intersects the segments 
and 
at the points 
and 
, respectively. Let 
and 
intersect at the points 
and 
, respectively. Prove that 
.
is any point on the side 
of the right trapezoid 
(
). Let the circumscribed circle 
of the triangle 
touch the sides 
at points 
respectively. Let the line 
intersects the segments 
and 
at the points 
and 
, respectively. Let 
and 
intersect at the points 
and 
, respectively. Prove that 
.
