be a monic polynomial with integer coefficient. And suppose there exist 4 distinct integer 
such that 
.
such that ![\[f(2x+y) + f(x+f(2y)) = f(x)f(y) - xy\]](http://latex.artofproblemsolving.com/0/6/2/0623b7c328af6d088cfc5dd288afe3e17a94ff29.png)
for all reals 
and 
.
sastifying 
Y by
Let 
be a circle with center 
. Let 
be a chord of this circle with center 
. The tangents from to 
and 
intersect at 
. The line 
goes through T and intersects at 
and 
, with 
and 
. Prove that the circumcenter of the triangle 
is the reflection of wrt the line 
.
be a circle with center 
. Let 
be a chord of this circle with center 
. The tangents from to 
and 
intersect at 
. The line 
goes through T and intersects at 
and 
, with 
and 
. Prove that the circumcenter of the triangle 
is the reflection of wrt the line 
.
and we shall denote 
the reflection of 
is the 
and the 
.
this implies that 
,
this implies that 
,
)
