and 
Prove that
and 
be two infinite sequences of integers. Prove that there exists an infinite sequence of integers 
such that for any positive integer 
,![\[
\sum_{k|n} k \cdot z_k^{\frac{n}{k}} = \left( \sum_{k|n} k \cdot x_k^{\frac{n}{k}} \right) \cdot \left( \sum_{k|n} k \cdot y_k^{\frac{n}{k}} \right).
\]](http://latex.artofproblemsolving.com/2/5/9/259ee32c89befe7a3ee2797e0d5b1df1c3066cf6.png)
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 
.
Y by tiendung2006
Let 
be an arbitrary interior point of 
, and 
, 
, 
intersect 
, 
, 
at 
, 
, 
, respectively. Suppose that 
be the midpoint of , 
and 
intersect at 
, 
intersects at 
, and 
intersects at 
. Show that 
is tangent to .
be an arbitrary interior point of 
, and 
, 
, 
intersect 
, 
, 
at 
, 
, 
, respectively. Suppose that 
be the midpoint of , 
and 
intersect at 
, 
intersects at 
, and 
intersects at 
. Show that 
is tangent to .
to 
also sends 
to 
where the last step follows by ceva. Since 
,
.
.
and the line 
lies on 
The rest of the problem can be finished by angle chasing. 
.
Thus 
and we are done.
