[/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 
contains exactly 
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 
,
.
be a parallelogram. Let line 
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 
.
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 jhu08, x3yukari
Let 
be arbitrary points on sides 
and 
of triangle 
, respectively, and 
an arbitrary point of segment 
. Show that at least one of the triangles 
and 
has an area less than or equal to 
of the area of triangle .
be arbitrary points on sides 
and 
of triangle 
, respectively, and 
an arbitrary point of segment 
. Show that at least one of the triangles 
and 
has an area less than or equal to 
of the area of triangle .This post has been edited 2 times. Last edited by parmenides51, Dec 15, 2022, 7:06 PM
.
first and pick a line through point ![$[AMP]=[BNP]$](http://latex.artofproblemsolving.com/a/9/3/a93af07a10a296c9f04fad0d4b3a1b521308f2bd.png)
)
equal, obviously increases ![$[BMP]$](http://latex.artofproblemsolving.com/f/4/7/f471cc55e128f185cbd7c6745634c822ad56d0df.png)
,
to hold, we must pivot the line through 
.
and 
,
and 
are the new positions of 
by vertical angles, ![$[MPM']>[NPN']$](http://latex.artofproblemsolving.com/6/2/3/623ccd58d5579751eef784fd561aca47f8bd4fcd.png)
and thus ![$[CM'N']>[CMN]$](http://latex.artofproblemsolving.com/b/a/d/bad532ae8cbe57bb38dc9d88b313ae38d9cbd006.png)
,![$[ABC]$](http://latex.artofproblemsolving.com/d/3/3/d33cc80fa8f093e155c5be46d2e5d9da3d7e1ef5.png)
and ![$[ABM]$](http://latex.artofproblemsolving.com/9/5/0/950a26d0f7d54d46a857c798159310059aa21d72.png)
are staying constant, ![$[AMP]$](http://latex.artofproblemsolving.com/c/3/6/c36617a7598e5915069aa16e908303ae8397f3db.png)
and ![$[CMN]+[APB]$](http://latex.artofproblemsolving.com/5/e/6/5e6008d13ae70cff1b74b4afc93cd023e74702df.png)
.
and the side length of the equilateral be 
.
.
.![$[ABC]=\frac{\sqrt{3}}{4}$](http://latex.artofproblemsolving.com/9/b/2/9b2cb3a56b3259de6927efdb7fe44cb13fe66ba3.png)
,![$[AMP]+[BNP]$](http://latex.artofproblemsolving.com/3/5/2/3522f048c329202faf962d42c0c0653260903a1c.png)
is ![$\frac{\sqrt{3}}{16}=\frac{1}{4}\cdot [ABC]$](http://latex.artofproblemsolving.com/a/f/a/afabb55dae886213541e4f2107ec4bc9946a9a25.png)
,![$\operatorname{max}([AMP], [BNP]) \geq \frac{1}{8}[ABC]$](http://latex.artofproblemsolving.com/f/d/a/fda58c7d9ebcd70b27d26c3a5e278bd7c4b0a6c5.png)
.
