,
be the projection from 
onto 
.
be a point on the extension of 
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 
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
An equilateral triangle 
is given. Point 
lies on the shorter arc 
of the circle circumscribed around the triangle. Point 
is midpoints of segment 
. Point 
is symmetrical to point with respect to the point . Prove that 
.
is given. Point 
lies on the shorter arc 
of the circle circumscribed around the triangle. Point 
is midpoints of segment 
. Point 
is symmetrical to point with respect to the point . Prove that 
.This post has been edited 1 time. Last edited by parmenides51, Apr 13, 2021, 3:36 PM
be the center of 
be the midpoints of 
.
,
