be a monic polynomial with integer coefficient. And suppose there exist 4 distinct integer 
such that 
.
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
The inscribed circle of an acute triangle 
touches the sides 
and 
at the points 
and 
, respectively. The altitude 
intersects the bisectors of the angles 
and 
at the points 
and 
, respectively. Cirumscribed circles of triangles 
and 
are denoted by 
and 
. Prove that if the midpoint of the altitude lies outside the circles and , then the tangents to the circles 
and , drawn from this midpoint, are equal.
(Hilko Danilo)
touches the sides 
and 
at the points 
and 
, respectively. The altitude 
intersects the bisectors of the angles 
and 
at the points 
and 
, respectively. Cirumscribed circles of triangles 
and 
are denoted by 
and 
. Prove that if the midpoint of the altitude lies outside the circles and , then the tangents to the circles 
and , drawn from this midpoint, are equal.(Hilko Danilo)
