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)
then prove \frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}+\frac{1}{2}\cdot\frac{abc}{ab+bc+ca}\ge \frac{5}{3}.$$
and 
Prove that
Let 
Prove that
Let 
Prove that
Let 
Prove that
is defined such that 
and 
is the perpendicular bisector of 
and 
be the circles passing through 
and 
respectively such that 
to be a point such that it lies on both the circles 
and 
are perpendicular.
weights, each weighing 
and another 
weights with 
each. Write a algorithm that determines the ways in which a scale can be balanced with a weight 
on the left pan, and display the number of possible solutions. (The weights can be placed on both pans and the program starts with the numbers 
.
.
to meet 
.
meets the midpoint of 
.
,
.
with side length of 
.
is selected on the diagonal 
such that 
.
as 
.
Y by
Let 
be a triangle, 
its incenter, 
its circumscribed circle, and 
the midpoint of side 
. Suppose that there are two different points 
on the line 
such that 
and 
. Let 
be the point of intersection of the lines 
and 
. Prove that the line 
and the perpendicular bisector of segment intersect at a point on .
be a triangle, 
its incenter, 
its circumscribed circle, and 
the midpoint of side 
. Suppose that there are two different points 
on the line 
such that 
and 
. Let 
be the point of intersection of the lines 
and 
. Prove that the line 
and the perpendicular bisector of segment intersect at a point on .This post has been edited 1 time. Last edited by parmenides51, Aug 22, 2021, 10:40 AM
Reason: latex
Reason: latex
then 
and 
or 
are collinear. Similarly, we have 
are collinear
