be an acute triangle with 
,
.
be a point on the extension of 
past 
.
be on the perpendicular bisector of 
such that ![\[\frac12\angle ALE=1.4\angle ABE+3.4\angle ACE-558^\circ\]](http://latex.artofproblemsolving.com/c/6/3/c63eeb053f1fa8a29bef93516814ae121382219e.png)
Let the reflection of 
and 
be 
and 
,
and 
such that 
.
and 
intersect the circumcircles of 
and 
at 
and 
,
and 
intersect 
and 
at 
and 
.
intersect 
.
.
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 
.
Y by Adventure10, Mango247
Given a regular dodecahedron of side 
. Take two pairs of opposite faces: 
and 
. For the pair 
take the line joining the centers of the faces and take points 
and 
on the line each a distance 
outside one of the faces. Similarly, take 
and 
on the line joining the centers of each a distance outside one of the faces. Show that 
is a rectangle and find the ratio of its side lengths.
. Take two pairs of opposite faces: 
and 
. For the pair 
take the line joining the centers of the faces and take points 
and 
on the line each a distance 
outside one of the faces. Similarly, take 
and 
on the line joining the centers of each a distance outside one of the faces. Show that 
is a rectangle and find the ratio of its side lengths.
