[/quote]
,
are there such that 
and 
?
sastifying 
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 
.
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 
,
.
, where 
is a real number. Prove that for all positive 
then prove
Y by Mango247
Isosceles triangles, 
, are built on the sides of the triangle 
in such a way that the circle circumscribed around is inscribed in a isosceles triangle 
and the angle 
is 
degrees. Find the ratio of the radii of the circles inscribed in the 
and 
.
(S. Usov)
, are built on the sides of the triangle 
in such a way that the circle circumscribed around is inscribed in a isosceles triangle 
and the angle 
is 
degrees. Find the ratio of the radii of the circles inscribed in the 
and 
.(S. Usov)
This post has been edited 1 time. Last edited by parmenides51, Apr 15, 2021, 12:40 PM
