,
onto 
past 
.
be on the perpendicular bisector of 
such that 
are on the same side of ![\[\frac12\angle ALE=1.4\angle ABE+3.4\angle ACE-558^\circ\]](http://latex.artofproblemsolving.com/c/6/3/c63eeb053f1fa8a29bef93516814ae121382219e.png)
Let the reflection of 
and 
,
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
Let 
be a triangle with area 
. Let 
be a point on segment 
. Let points 
and 
on 
and 
, respectively, satisfy 
and 
. Compute, with proof, the area of the quadrilateral with vertices at , , the midpoint of 
, and the midpoint of 
.
be a triangle with area 
. Let 
be a point on segment 
. Let points 
and 
on 
and 
, respectively, satisfy 
and 
. Compute, with proof, the area of the quadrilateral with vertices at , , the midpoint of 
, and the midpoint of 
.
.
,
into 
,
,
and split 
into 
,
,
.
