[/quote]
,
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 
.
,
,
is a positive integer.
of positive integers 
such that 
.
denotes component-wise function multiplication)
and 
Prove that
distinct points 
connected if the line 
points from 
such that there exists a set 
is connected with exactly 
other points from 
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 
.
be a fixed positive integer, and 
be a sequence of nonnegative real numbers such that, for all 
,
.
is bounded above.
is bounded above.
are positiv real number, such that 
prove inequality 
be a positive integer. Consider the following equation: ![\[ \{x\}+\{2x\}+ \dots + \{nx\} = \lfloor x \rfloor + \lfloor 2x \rfloor + \dots + \lfloor 2nx \rfloor\]](http://latex.artofproblemsolving.com/8/2/9/8299b3ceb1650418c0a373536cb1d2931900efd9.png)
a) For 
,
.
externally bisect 
and let 
be the line passing through 
which is parallel to line 
and 
,
at point 
.
meet line 
at point 
and the internal bisector of 
at point 
.
,
are concyclic.
and 
.
be an acute-angled triangle, with circumcenter 
,
and orthocenter 
.
be a point on 
such that 
.
and 
similarly.
,
be a positive integer. There are two numbers written on a blackboard, one red and one blue. Initially, both are 0. We define the following procedure: at each step, we choose a nonnegative integer 
and 
,
Y by
The extensions of the sides 
and 
of the convex quadrilateral 
intersect at the point 
, and the diagonals 
and 
intersect at the point 
. The points 
and 
are the midpoints of the diagonals and respectively. The circumscribed circles of triangles 
and 
intersect at the point 
(
). Prove that if 
then the line 
bisects the segment 
.
and 
of the convex quadrilateral 
intersect at the point 
, and the diagonals 
and 
intersect at the point 
. The points 
and 
are the midpoints of the diagonals and respectively. The circumscribed circles of triangles 
and 
intersect at the point 
(
). Prove that if 
then the line 
bisects the segment 
.
