[/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 
,
.
, 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
Carlson cut a rectangular cake using 
straight line cuts parallel to the sides of the cake into 
pieces. The mass of the largest piece, together with the the smallest piece was equal to the masses of the other two pieces of cake. Prove that one of the cuts passed through the center of the cake.
straight line cuts parallel to the sides of the cake into 
pieces. The mass of the largest piece, together with the the smallest piece was equal to the masses of the other two pieces of cake. Prove that one of the cuts passed through the center of the cake.This post has been edited 2 times. Last edited by parmenides51, Nov 1, 2021, 7:28 PM
