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 the incentre and point 
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 
.
such that 
.
as 
.
Y by Mango247, Mango247, Mango247
Let 
and 
be the bisectors of the interior angles 
and 
, respectively (
, 
). Consider the circumcircle of 
with center 
and the excircle corresponding to the side 
with center 
. These two circles intersect at points 
and 
.
(a) Prove that
is parallel to 
.
(b) Prove that
is perpendicular to .
and 
be the bisectors of the interior angles 
and 
, respectively (
, 
). Consider the circumcircle of 
with center 
and the excircle corresponding to the side 
with center 
. These two circles intersect at points 
and 
.(a) Prove that
is parallel to 
.(b) Prove that
is perpendicular to .
