[/quote]
,
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 
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 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, 
)
be an acute triangle with 
,
.
past 
.
be on the perpendicular bisector of 
such that ![\[\frac12\angle ALE=1.4\angle ABE+3.4\angle ACE-558^\circ\]](http://latex.artofproblemsolving.com/c/6/3/c63eeb053f1fa8a29bef93516814ae121382219e.png)
Let the reflection of 
and 
be 
and 
,
and 
such that 
.
and 
intersect the circumcircles of 
and 
at 
and 
,
and 
intersect 
and 
at 
and 
.
intersect 
.
.
such that the relation ![\[|z^{2n}+z^nw^n+w^{2n} | = 2^{2n}+2^n+1 \]](http://latex.artofproblemsolving.com/1/2/2/1229a609403904449c447225094b3d97e063efd4.png)
holds for all positive integers 
Y by Mango247
Let 
be a line and let 
be two points on . Circles 
and 
centred at 
and 
respectively are both tangent to at 
, with and being on opposite sides of . Circles 
and 
centred at 
and 
respectively are both tangent to at 
, with and being on opposite sides of . Moreover and are on the same side of . Prove that if there exists a circle tangent to all circles 
containing all of them in its interior, then the lines 
and are either concurrent or parallel.
be a line and let 
be two points on . Circles 
and 
centred at 
and 
respectively are both tangent to at 
, with and being on opposite sides of . Circles 
and 
centred at 
and 
respectively are both tangent to at 
, with and being on opposite sides of . Moreover and are on the same side of . Prove that if there exists a circle tangent to all circles 
containing all of them in its interior, then the lines 
and are either concurrent or parallel.
