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 
and 
is the perpendicular bisector of 
and 
be the circles passing through 
and 
respectively such that 
is tangent to 
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 
.
as 
.
Y by
Points 
and 
are the midpoints of the sides 
and 
of triangle 
respectively. The equilateral triangles 
and 
are constructed on the sides 
and 
to the exterior of the triangle . Point 
is the midpoint of the segment 
. Find the value of the angle 
.
and 
are the midpoints of the sides 
and 
of triangle 
respectively. The equilateral triangles 
and 
are constructed on the sides 
and 
to the exterior of the triangle . Point 
is the midpoint of the segment 
. Find the value of the angle 
.
-midpoints of 
and 
is point of intersection of 
and 
and 
are triangles with angles 
so 
.
so 
and 
