[/quote]
,
be a point in the plane, distinct from the vertices of 
.
the reflections of 
and 
,
are collinear if and only if 
coincide, prove 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 
.
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 
,
.
real solutions.
be a parallelogram. Let line 
externally bisect 
and let 
be the line passing through 
which is parallel to line 
.
at point 
and 
,
at point 
.
meet line 
at point 
and the internal bisector of 
at point 
.
,
are concyclic.
and 
.
,
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 
(not necessarily distinct from the previously chosen ones), and, if the red and blue numbers are 
and 
,
.
be the set of all positive integers. Determine all subsets 
of 
such that
Y by Adventure10, Mango247
Let A, B, C, A’, B’, C’ be points on a circle such that AA’ is perpendicular to BC, BB’ is perpendicular to CA, and CC' is
perpendicular to AB. Further, let D be a point on that circle and let DA' intersect BC in A”, DB' intersect CA in B", and DC' intersect AB in C", all line Segments being extended where required.
Prove that A", B", C and the orthocentre of triangle ABC are collinear.
perpendicular to AB. Further, let D be a point on that circle and let DA' intersect BC in A”, DB' intersect CA in B", and DC' intersect AB in C", all line Segments being extended where required.
Prove that A", B", C and the orthocentre of triangle ABC are collinear.
