[/quote]
,
are there such that 
and 
?
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.
, where 
is a real number. Prove that for all positive 
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 
.
then prove
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 
(not necessarily distinct from the previously chosen ones), and, if the red and blue numbers are 
and 
,
.
Y by
Inside a triangle with side lengths 
, and 
are drawn three line segments parallel to the sides so that all sides of the hexagon marked in the figure with bold lines, are equal. Find the length of the sides of this hexagon.
(source in case you are wondering which the bold ones are, I believe it's not that clear)
, and 
are drawn three line segments parallel to the sides so that all sides of the hexagon marked in the figure with bold lines, are equal. Find the length of the sides of this hexagon.(source in case you are wondering which the bold ones are, I believe it's not that clear)
This post has been edited 1 time. Last edited by parmenides51, May 4, 2021, 8:57 AM
is the length of the side of the hexagon parallel to the side of the triangle with length 
by similar triangles
,
