[/quote]
,
is defined such that 
and 
is the perpendicular bisector of 
such that 
.
as 
,
and 
as 
.
.
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 
.
to 
.
)
.
and 
,
is not less than the area of 
,
inequalities, these given comparisons satisfy the following three constraints:
and 
,
and 
respectively, then the sum of elements of the union 
is greater than that of 
.
that satisfies all these conditions?
.
be an acute triangle. Denote by 
the foot of the perpendicular line drawn from the point 
to the side 
,
the midpoint of 
the orthocenter of 
be the point of intersection of the circumcircle 
of the triangle 
,
be the point of intersection (other than 
)
and the circle 
must hold.
the length of the line segment 
such that 
(where 
)
that satisfies 
for all 
is the function obtained by composing 
times, that is, 
)
are there such that 
and 
?
, where 
is a real number. Prove that for all positive 
Y by
In the triangle 
, from the incenter 
draw a perpendicular on the side 
, which intersects it at the point 
. From the foot of the bisector of the angle 
draw another perpendicular to the same side , which intersects it at the point 
. Prove that the segment 
is divisible by line 
in half, where 
is the point of intersection of the extension of the bisector of the angle with the circumcircle of the triangle .
(D. Basov)
, from the incenter 
draw a perpendicular on the side 
, which intersects it at the point 
. From the foot of the bisector of the angle 
draw another perpendicular to the same side , which intersects it at the point 
. Prove that the segment 
is divisible by line 
in half, where 
is the point of intersection of the extension of the bisector of the angle with the circumcircle of the triangle .(D. Basov)
