be an acute triangle with 
,
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 
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
A triangle 
has 
. A line 
is drawn so that 
where 
is inside the triangle. A point 
outside the triangle is constructed so that 
is parallel to 
, and 
is parallel to 
. 
is the point on 
(separated from by the line ) such that 
.
Prove that the circumcircle of touches the circumcircle of 
.
has 
. A line 
is drawn so that 
where 
is inside the triangle. A point 
outside the triangle is constructed so that 
is parallel to 
, and 
is parallel to 
. 
is the point on 
(separated from by the line ) such that 
.Prove that the circumcircle of
touches the circumcircle of 
.
be the circumcircles of triangles 
.
,
,
are collinear. Then 
is an cyclic trapezoid, so it is isosceles and 
lies on 
.
.
be a line tangent to 
,
at 
,
is an isosceles trapezoid, hence cyclic, and 
,
,
is cyclic as well; for the tangency, same as before.
