[/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 
,
.
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
An ant sits in a rectangular matchbox measuring 
cm, and a net of sugar crumbs. If we introduce a coordinate system with axes parallel to the edges of the box so that one vertex of the box is at the origin, and the second at point with coordinates (1
mm, 
mm, 
mm), then the ant will sit at point with coordinates (
mm, 
mm, mm), and the crumb will be coordinates at point with coordinates (
mm, 
mm, mm). What is the shortest distance that the ant will have to crawl around the sugar crumbs if it can move only over the surface of the box?
cm, and a net of sugar crumbs. If we introduce a coordinate system with axes parallel to the edges of the box so that one vertex of the box is at the origin, and the second at point with coordinates (1
mm, 
mm, 
mm), then the ant will sit at point with coordinates (
mm, 
mm, mm), and the crumb will be coordinates at point with coordinates (
mm, 
mm, mm). What is the shortest distance that the ant will have to crawl around the sugar crumbs if it can move only over the surface of the box?This post has been edited 1 time. Last edited by parmenides51, May 4, 2021, 8:21 AM
