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 
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 
such that 
.
as 
.
Y by Adventure10
On sides 
of a triangle 
points 
respectively are chosen so that 
have a common point, say 
. Suppose that one can inscribe circles in the quadrilaterals 
so that each two of them have a common point. Prove that triangle is equilateral.
of a triangle 
points 
respectively are chosen so that 
have a common point, say 
. Suppose that one can inscribe circles in the quadrilaterals 
so that each two of them have a common point. Prove that triangle is equilateral.This post has been edited 1 time. Last edited by parmenides51, Jan 12, 2020, 8:05 AM
Analogously 
.
we have from angle bisector theorem 
It's equivalent to 
Analogously we can prove 
.
we have finished.
