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 
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 
.
with orthocenter 
.
to meet 
.
meets the midpoint of 
.
,
.
with side length of 
.
such that 
.
as 
.
Y by Adventure10, Mango247
Circles 
and 
meet at 
and 
. Let 
be a point on . Let 
and 
meet again at 
and 
respectively. The lines 
and 
meet at 
. Show that, as varies on , lies on a fixed circle.
and 
meet at 
and 
. Let 
be a point on . Let 
and 
meet again at 
and 
respectively. The lines 
and 
meet at 
. Show that, as varies on , lies on a fixed circle. [asy]
import graph; size(300); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen ttttff = rgb(0.2,0.2,1); pen xdxdff = rgb(0.49,0.49,1); pen ffttww = rgb(1,0.2,0.4); pen ttccqq = rgb(0.2,0.8,0);
draw(circle((2.16,2.22),3.5),ttttff+linewidth(1.2pt)); draw(circle((6.4,2.1),1.86),ttttff+linewidth(1.2pt)); draw((8.13,2.79)--(2.74,-1.23),ffttww+linewidth(2pt)); draw((8.13,2.79)--(0.34,5.21),ffttww+linewidth(2pt)); draw((0.34,5.21)--(5.27,0.62),ffttww+linewidth(2pt)); draw((5.36,3.64)--(2.74,-1.23),ffttww+linewidth(2pt)); draw(circle((5.71,2.12),1.56),ttccqq+linetype("5pt 5pt")); draw((5.36,3.64)--(5.27,0.62),ffttww+linewidth(2pt));
dot((2.16,2.22),ds); label("$S_1$", (1.72,2.12),NE*lsf); dot((6.4,2.1),ds); label("$S_2$", (6.48,2.22),NE*lsf); dot((5.36,3.64),ds); label("$L$", (5.34,3.92),NE*lsf); dot((5.27,0.62),ds); label("$M$", (5.24,0.18),NE*lsf); dot((8.13,2.79),ds); label("$P$", (8.34,2.82),NE*lsf); dot((2.74,-1.23),ds); label("$R$", (2.62,-1.66),NE*lsf); dot((0.34,5.21),ds); label("$Q$", (0.06,5.34),NE*lsf); dot((4.25,1.57),ds); label("$K$", (3.8,1.44),NE*lsf); clip((-4.3,-10.94)--(-4.3,6.3)--(16.18,6.3)--(16.18,-10.94)--cycle); [/asy]
on 
and their corresponding 
.
.
is cyclic. Now suppose we fix 
.
,
'
,
of 
being on the minor arc 
.
